Dimension

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Dimension
From Wikipedia, the free encyclopedia

Chapter 1

2 1/2D
2 1/2D is a conventional way of describing a process for obtaining an object where the third dimension is somehow
dependent from the transversal cross-section of the same.
2 1/2D is used for diﬀerentiating a 3D process, like for example injection moulding, casting or forging, as well as 2D
processes like plastics extrusion, extrusion, drawing or dipping from other processes, like for example microextrusion,
overmoulding, etc.
The product of the 2 1/2D process can, in fact, be described by mean of a transversal cross-section and a longitudinal
cross-section as any other, but in which the transversal cross-section varies, generating a variable longitudinal crosssection generally more simple than the transversal cross-section.
In other words, the third dimension, so the longitudinal cross-section can be represented as the projection of the
transversal cross-section in the inﬁnity of the horizon.
An example of 2 1/2D processing, can be 2 1/2D microextrusion, where, thanks to an optimal control of the process,
the obtained extrudate, which is generally, but not always a tube, can have:
• Tapered proﬁle, which means that diameters or characteristic dimensions, varies independently from each other.
• Bump proﬁle, which means that dimensions will vary in a dependent way, so the dimensions will be proportional.
• TIE (Totally Intermittent Extrusion) proﬁle, which means that at least one of the materials composing the
cross-section switch to another allowing a variation of hardness, radio-opacity or other speciﬁc characteristics.
• ILE (Intermittent Layer Extrusion) proﬁle, which means that the materials composing the cross-section can
vary the proportion between themselves, in case of a tube for example, the thickness of one layer can change
independently from others.

1.1 References
1.1.1

Bibliography

• Maccagnan, Simone; Perale, Giuseppe; Capelletti, Tiziano; Maccagnan, Davide (2011), The Next Generation
of Microextrusion Technology, UBM Canon

2

Chapter 2

Abscissa

y

(2,3)

3
2

(−3,1)

1

(0,0)
−3

−2

−1

1

x
2

3

−1
−2

(−1.5,−2.5)

−3

Illustration of a Cartesian coordinate plane. The ﬁrst value in each ordered pair is the abscissa of the corresponding point.

In mathematics, an abscissa (/æbˈsɪs.ə/; plural abscissae or abscissæ or abscissas) is the perpendicular distance of
a point from the vertical axis. Usually this is the horizontal coordinate of a point in a two-dimensional rectangular
Cartesian coordinate system. The term can also refer to the horizontal axis (typically x-axis) of a two-dimensional
3

4

CHAPTER 2. ABSCISSA

graph (because that axis is used to deﬁne and measure the horizontal coordinates of points in the space). An ordered
pair consists of two terms—the abscissa (horizontal, usually x) and the ordinate (vertical, usually y)—which deﬁne
the location of a point in two-dimensional rectangular space.

abscissa ordinate

z}|{ z}|{
( x , y )

2.1 Etymology
Though the word “abscissa” (Latin; “linea abscissa”, “a line cut oﬀ”) has been used at least since De Practica Geometrie
published in 1220 by Fibonacci (Leonardo of Pisa), its use in its modern sense may be due to Venetian mathematician
Stefano degli Angeli in his work Miscellaneum Hyperbolicum, et Parabolicum of 1659.[1]
In his 1892 work Vorlesungen über Geschichte der Mathematik, Volume 2, ("Lectures on history of mathematics")
German historian of mathematics Moritz Cantor writes
“Wir kennen keine ältere Benutzung des Wortes Abssisse in lateinischen Originalschriften [than degli
Angeli’s]. Vielleicht kommt das Wort in Übersetzungen der Apollonischen Kegelschnitte vor, wo Buch
I Satz 20 von ἀποτεμνομέναις die Rede ist, wofür es kaum ein entsprechenderes lateinisches Wort als
abscissa geben möchte.”[2]
“We know no earlier use of the word abscissa in Latin originals [than degli Angeli’s]. Maybe the
word descends from translations of the Apollonian conics, where in Book I, Chapter 20 there appears
ἀποτεμνομέναις, for which there would hardly be as an appropriate Latin word as abscissa."

2.2 In parametric equations
In a somewhat obsolete variant usage, the abscissa of a point may also refer to any number that describes the point’s
location along some path, e.g. the parameter of a parametric equation.[3] Used in this way, the abscissa can be
thought of as a coordinate-geometry analog to the independent variable in a mathematical model or experiment (with
any ordinates ﬁlling a role analogous to dependent variables).

2.3 Examples
• For the point (2, 3), 2 is called the abscissa and 3 the ordinate.
• For the point (−1.5, −2.5), −1.5 is called the abscissa and −2.5 the ordinate.

2.4 See also
• Ordinate
• Dependent and independent variables
• Function (mathematics)
• Relation (mathematics)
• Line chart

2.5. REFERENCES

5

2.5 References
[1] Dyer, Jason (March 8, 2009). “On the Word “Abscissa"". https://numberwarrior.wordpress.com. The number Warrior.
Retrieved September 10, 2015.
[2] Cantor, Moritz (1900). Vorlesungen über Geschichte der Mathematik, Volume 2. Leipzig: B.G. Teubner. p. 898. Retrieved
10 September 2015.
[3] Hedegaard, Rasmus; Weisstein, Eric W. “Abscissa”. MathWorld. Retrieved 14 July 2013.

This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008
and incorporated under the “relicensing” terms of the GFDL, version 1.3 or later.

2.6 External links
• The dictionary deﬁnition of abscissa at Wiktionary

Chapter 3

Cartesian coordinate system

y

(2,3)

3
2

(−3,1)

1

(0,0)
−3

−2

−1

1

x
2

3

−1
−2

(−1.5,−2.5)

−3

Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in
red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.

A Cartesian coordinate system is a coordinate system that speciﬁes each point uniquely in a plane by a pair of
numerical coordinates, which are the signed distances to the point from two ﬁxed perpendicular directed lines,
6

7
measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the
point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be deﬁned as the positions
of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian
coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real n-space)
specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign,
to distances from the point to n mutually perpendicular hyperplanes.

y
3
2

2

2

x +y = 4

1
-3

-2

-1

1
-1

2

3

x

-2
-3

Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (x − a)2 + (y
− b)2 = r2 where a and b are the coordinates of the center (a, b) and r is the radius.

The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the ﬁrst systematic link between Euclidean geometry and algebra. Using the
Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic
equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at
the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 +
y2 = 4.
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations
for many other branches of mathematics, such as linear algebra, complex analysis, diﬀerential geometry, multivariate

8

CHAPTER 3. CARTESIAN COORDINATE SYSTEM

calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates
are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering
and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric
design and other geometry-related data processing.

3.1 History
The adjective Cartesian refers to the French mathematician and philosopher René Descartes (who used the name
Cartesius in Latin).
The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat,
although Fermat also worked in three dimensions and did not publish the discovery.[1] Both authors used a single axis
in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes
was introduced later, after Descartes’ La Géométrie was translated into Latin in 1649 by Frans van Schooten and his
students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes’
work.[2]
The development of the Cartesian coordinate system would play a fundamental role in the development of the calculus
by Isaac Newton and Gottfried Wilhelm Leibniz.[3]
Nicole Oresme, a French cleric and friend of the Dauphin (later to become King Charles V) of the 14th Century,
used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat.
Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and
the spherical and cylindrical coordinates for three-dimensional space.

3.2 Description
3.2.1

One dimension

Main article: Number line
Choosing a Cartesian coordinate system for a one-dimensional space—that is, for a straight line—involves choosing
a point O of the line (the origin), a unit of length, and an orientation for the line. An orientation chooses which
of the two half-lines determined by O is the positive, and which is negative; we then say that the line “is oriented”
(or “points”) from the negative half towards the positive half. Then each point P of the line can be speciﬁed by its
distance from O, taken with a + or − sign depending on which half-line contains P.
A line with a chosen Cartesian system is called a number line. Every real number has a unique location on the line.
Conversely, every point on the line can be interpreted as a number in an ordered continuum such as the real numbers.

3.2.2

Two dimensions

Further information: Two-dimensional space
The Cartesian coordinate system in two dimensions (also called a rectangular coordinate system) is deﬁned by an
ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis. (Early
systems allowed “oblique” axes, that is, axes that did not meet at right angles.) The lines are commonly referred to
as the x- and y-axes where the x-axis is taken to be horizontal and the y-axis is taken to be vertical. The point where
the axes meet is taken as the origin for both, thus turning each axis into a number line. For a given point P, a line is
drawn through P perpendicular to the x-axis to meet it at X and second line is drawn through P perpendicular to the
y-axis to meet it at Y. The coordinates of P are then X and Y interpreted as numbers x and y on the corresponding
number lines. The coordinates are written as an ordered pair (x, y).
The point where the axes meet is the common origin of the two number lines and is simply called the origin. It is
often labeled O and if so then the axes are called Ox and Oy. A plane with x- and y-axes deﬁned is often referred to

3.2. DESCRIPTION

9

as the Cartesian plane or xy-plane. The value of x is called the x-coordinate or abscissa and the value of y is called
the y-coordinate or ordinate.
The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate
unknown values. The ﬁrst part of the alphabet was used to designate known values.
In the Cartesian plane, reference is sometimes made to a unit circle or a unit hyperbola.

3.2.3

Three dimensions

Further information: Three-dimensional space
Choosing a Cartesian coordinate system for a three-dimensional space means choosing an ordered triplet of lines

Z

(x,y,z)
O
x
y

z
Y

X

A three dimensional Cartesian coordinate system, with origin O and axis lines X, Y and Z, oriented as shown by the arrows. The
tick marks on the axes are one length unit apart. The black dot shows the point with coordinates x = 2, y = 3, and z = 4, or (2,3,4).

(axes) that are pair-wise perpendicular, have a single unit of length for all three axes and have an orientation for each
axis. As in the two-dimensional case, each axis becomes a number line. The coordinates of a point P are obtained
by drawing a line through P perpendicular to each coordinate axis, and reading the points where these lines meet the
axes as three numbers of these number lines.
Alternatively, the coordinates of a point P can also be taken as the (signed) distances from P to the three planes
deﬁned by the three axes. If the axes are named x, y, and z, then the x-coordinate is the distance from the plane
deﬁned by the y and z axes. The distance is to be taken with the + or − sign, depending on which of the two half-

10

CHAPTER 3. CARTESIAN COORDINATE SYSTEM

spaces separated by that plane contains P. The y and z coordinates can be obtained in the same way from the xz- and
xy-planes respectively.

The coordinate surfaces of the Cartesian coordinates (x, y, z). The z-axis is vertical and the x-axis is highlighted in green. Thus, the
red plane shows the points with x = 1, the blue plane shows the points with z = 1, and the yellow plane shows the points with y =
−1. The three surfaces intersect at the point P (shown as a black sphere) with the Cartesian coordinates (1, −1, 1).

3.2.4

Higher dimensions

A Euclidean plane with a chosen Cartesian system is called a Cartesian plane. Since Cartesian coordinates are
unique and non-ambiguous, the points of a Cartesian plane can be identiﬁed with pairs of real numbers; that is with
the Cartesian product R2 = R × R , where R is the set of all reals. In the same way, the points any Euclidean space
of dimension n be identiﬁed with the tuples (lists) of n real numbers, that is, with the Cartesian product Rn .

3.2.5

Generalizations

The concept of Cartesian coordinates generalizes to allow axes that are not perpendicular to each other, and/or different units along each axis. In that case, each coordinate is obtained by projecting the point onto one axis along a
direction that is parallel to the other axis (or, in general, to the hyperplane deﬁned by all the other axes). In such
an oblique coordinate system the computations of distances and angles must be modiﬁed from that in standard
Cartesian systems, and many standard formulas (such as the Pythagorean formula for the distance) do not hold.

3.3. NOTATIONS AND CONVENTIONS

11

3.3 Notations and conventions
The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10, 5) or
(3, 5, 7). The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates
are often denoted by the letters x and y on the plane, and x, y, and z in three-dimensional space. This custom comes
from a convention of algebra, which use letters near the end of the alphabet for unknown values (such as were the
coordinates of points in many geometric problems), and letters near the beginning for given quantities.
These conventional names are often used in other domains, such as physics and engineering, although other letters
may be used. For example, in a graph showing how a pressure varies with time, the graph coordinates may be denoted
t and p. Each axis is usually named after the coordinate which is measured along it; so one says the x-axis, the y-axis,
the t-axis, etc.
Another common convention for coordinate naming is to use subscripts, as in x1 , x2 , ... xn for the n coordinates in
an n-dimensional space; especially when n is greater than 3, or not speciﬁed. Some authors prefer the numbering x0 ,
x1 , ... xn₋₁. These notations are especially advantageous in computer programming: by storing the coordinates of a
point as an array, instead of a record, the subscript can serve to index the coordinates.
In mathematical illustrations of two-dimensional Cartesian systems, the ﬁrst coordinate (traditionally called the
abscissa) is measured along a horizontal axis, oriented from left to right. The second coordinate (the ordinate) is
then measured along a vertical axis, usually oriented from bottom to top.
However, computer graphics and image processing often use a coordinate system with the y axis oriented downwards
on the computer display. This convention developed in the 1960s (or earlier) from the way that images were originally
stored in display buﬀers.
For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z axis added to represent
height (positive up). Furthermore, there is a convention to orient the x-axis toward the viewer, biased either to the
right or left. If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically,
respectively, then the z axis should be shown pointing “out of the page” towards the viewer or camera. In such a 2D
diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down
and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation
of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always
comply with the right-hand rule, unless speciﬁcally stated otherwise. All laws of physics and math assume this righthandedness, which ensures consistency.
For 3D diagrams, the names “abscissa” and “ordinate” are rarely used for x and y, respectively. When they are, the
z-coordinate is sometimes called the applicate. The words abscissa, ordinate and applicate are sometimes used to
refer to coordinate axes rather than the coordinate values.[4]

3.3.1

Quadrants and octants

Main articles: Octant (solid geometry) and Quadrant (plane geometry)
The axes of a two-dimensional Cartesian system divide the plane into four inﬁnite regions, called quadrants, each
bounded by two half-axes. These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the
signs of the two coordinates are +,+), II (−,+), III (−,−), and IV (+,−). When the axes are drawn according to the
mathematical custom, the numbering goes counter-clockwise starting from the upper right (“north-east”) quadrant.
Similarly, a three-dimensional Cartesian system deﬁnes a division of space into eight regions or octants, according to
the signs of the coordinates of the points. The convention used for naming a speciﬁc octant is to list its signs, e.g. (+
+ +) or (− + −). The generalization of the quadrant and octant to an arbitrary number of dimensions is the orthant,
and a similar naming system applies.

3.4 Cartesian formulae for the plane
3.4.1

Distance between two points

The Euclidean distance between two points of the plane with Cartesian coordinates (x1 , y1 ) and (x2 , y2 ) is

12

CHAPTER 3. CARTESIAN COORDINATE SYSTEM

The four quadrants of a Cartesian coordinate system.

d=

√
(x2 − x1 )2 + (y2 − y1 )2 .

This is the Cartesian version of Pythagoras’s theorem. In three-dimensional space, the distance between points
(x1 , y1 , z1 ) and (x2 , y2 , z2 ) is

d=

√
(x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 ,

which can be obtained by two consecutive applications of Pythagoras’ theorem.

3.4.2

Euclidean transformations

The Euclidean transformations or Euclidean motions are the (bijective) mappings of points of the Euclidean plane to
themselves which preserve distances between points. There are four types of these mappings (also called isometries):
translations, rotations, reﬂections and glide reﬂections.[5]
Translation
Translating a set of points of the plane, preserving the distances and directions between them, is equivalent to adding
a ﬁxed pair of numbers (a, b) to the Cartesian coordinates of every point in the set. That is, if the original coordinates

3.4. CARTESIAN FORMULAE FOR THE PLANE

13

of a point are (x, y), after the translation they will be

(x′ , y ′ ) = (x + a, y + b).
Rotation
To rotate a ﬁgure counterclockwise around the origin by some angle θ is equivalent to replacing every point with
coordinates (x,y) by the point with coordinates (x',y'), where

x′ = x cos θ − y sin θ
y ′ = x sin θ + y cos θ.
Thus: (x′ , y ′ ) = ((x cos θ − y sin θ ), (x sin θ + y cos θ )).
Reﬂection
If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reﬂection across the second
coordinate axis (the Y-axis), as if that line were a mirror. Likewise, (x, −y) are the coordinates of its reﬂection across
the ﬁrst coordinate axis (the X-axis). In more generality, reﬂection across a line through the origin making an angle
θ with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′),
where

x′ = x cos 2θ + y sin 2θ
y ′ = x sin 2θ − y cos 2θ.
Thus: (x′ , y ′ ) = ((x cos 2θ + y sin 2θ ), (x sin 2θ − y cos 2θ )).
Glide reﬂection
A glide reﬂection is the composition of a reﬂection across a line followed by a translation in the direction of that
line. It can be seen that the order of these operations does not matter (the translation can come ﬁrst, followed by the
reﬂection).
General matrix form of the transformations
These Euclidean transformations of the plane can all be described in a uniform way by using matrices. The result
(x′ , y ′ ) of applying a Euclidean transformation to a point (x, y) is given by the formula

(x′ , y ′ ) = (x, y)A + b
where A is a 2×2 orthogonal matrix and b = (b1 , b2 ) is an arbitrary ordered pair of numbers;[6] that is,
x′ = xA11 + yA21 + b1
y ′ = xA12 + yA22 + b2 ,
where
(
)
A11 A12
A=
. [Note the use of row vectors for point coordinates and that the matrix
A21 A22
is written on the right.]

14

CHAPTER 3. CARTESIAN COORDINATE SYSTEM

To be orthogonal, the matrix A must have orthogonal rows with same Euclidean length of one, that is,

A11 A21 + A12 A22 = 0
and

A211 + A212 = A221 + A222 = 1.
This is equivalent to saying that A times its transpose must be the identity matrix. If these conditions do not hold, the
formula describes a more general aﬃne transformation of the plane provided that the determinant of A is not zero.
The formula deﬁnes a translation if and only if A is the identity matrix. The transformation is a rotation around some
point if and only if A is a rotation matrix, meaning that

A11 A22 − A21 A12 = 1.
A reﬂection or glide reﬂection is obtained when,

A11 A22 − A21 A12 = −1.
Assuming that translation is not used transformations can be combined by simply multiplying the associated transformation matrices.

Aﬃne transformation
Another way to represent coordinate transformations in Cartesian coordinates is through aﬃne transformations. In
aﬃne transformations an extra dimension is added and all points are given a value of 1 for this extra dimension. The
advantage of doing this is that point translations can be speciﬁed in the ﬁnal column of matrix A. In this way, all of
the euclidean transformations become transactable as matrix point multiplications. The aﬃne transformation is given
by:
 ′
 
x
A11 A21 b1
x
A12 A22 b2 y  = y ′ . [Note the matrix A from above was transposed. The
1
1
0
0
1
matrix is on the left and column vectors for point coordinates are used.]


Using aﬃne transformations multiple diﬀerent euclidean transformations including translation can be combined by
simply multiplying the corresponding matrices.

Scaling
An example of an aﬃne transformation which is not a Euclidean motion is given by scaling. To make a ﬁgure larger
or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If (x,
y) are the coordinates of a point on the original ﬁgure, the corresponding point on the scaled ﬁgure has coordinates

(x′ , y ′ ) = (mx, my).
If m is greater than 1, the ﬁgure becomes larger; if m is between 0 and 1, it becomes smaller.

3.5. ORIENTATION AND HANDEDNESS

15

Shearing
A shearing transformation will push the top of a square sideways to form a parallelogram. Horizontal shearing is
deﬁned by:

(x′ , y ′ ) = (x + ys, y)
Shearing can also be applied vertically:

(x′ , y ′ ) = (x, xs + y)

3.5 Orientation and handedness
Main article: Orientation (mathematics)
See also: right-hand rule and Axes conventions

3.5.1

In two dimensions

The right hand rule.

Fixing or choosing the x-axis determines the y-axis up to direction. Namely, the y-axis is necessarily the perpendicular
to the x-axis through the point marked 0 on the x-axis. But there is a choice of which of the two half lines on
the perpendicular to designate as positive and which as negative. Each of these two choices determines a diﬀerent
orientation (also called handedness) of the Cartesian plane.

16

CHAPTER 3. CARTESIAN COORDINATE SYSTEM

The usual way of orienting the axes, with the positive x-axis pointing right and the positive y-axis pointing up (and
the x-axis being the “ﬁrst” and the y-axis the “second” axis) is considered the positive or standard orientation, also
called the right-handed orientation.
A commonly used mnemonic for deﬁning the positive orientation is the right hand rule. Placing a somewhat closed
right hand on the plane with the thumb pointing up, the ﬁngers point from the x-axis to the y-axis, in a positively
oriented coordinate system.
The other way of orienting the axes is following the left hand rule, placing the left hand on the plane with the thumb
pointing up.
When pointing the thumb away from the origin along an axis towards positive, the curvature of the ﬁngers indicates
a positive rotation along that axis.
Regardless of the rule used to orient the axes, rotating the coordinate system will preserve the orientation. Switching
any two axes will reverse the orientation, but switching both will leave the orientation unchanged.

3.5.2

In three dimensions

Fig. 7 – The left-handed orientation is shown on the left, and the right-handed on the right.

Once the x- and y-axes are speciﬁed, they determine the line along which the z-axis should lie, but there are two
possible directions on this line. The two possible coordinate systems which result are called 'right-handed' and 'lefthanded'. The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis
form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane)
is called right-handed or positive.
The name derives from the right-hand rule. If the index ﬁnger of the right hand is pointed forward, the middle ﬁnger
bent inward at a right angle to it, and the thumb placed at a right angle to both, the three ﬁngers indicate the relative
directions of the x-, y-, and z-axes in a right-handed system. The thumb indicates the x-axis, the index ﬁnger the
y-axis and the middle ﬁnger the z-axis. Conversely, if the same is done with the left hand, a left-handed system
results.
Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on
the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also
meant to point towards the observer, whereas the “middle” axis is meant to point away from the observer. The red
circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). Hence
the red arrow passes in front of the z-axis.

3.5. ORIENTATION AND HANDEDNESS

17

z-axis

y=0
xz
plane

x = 0 plane
yz plane
origin
y-axis
z = 0 plane
xy plane

x-axis
Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes.

3D Cartesian Coordinate Handedness

Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused
by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as “ﬂipping in
and out” between a convex cube and a concave “corner”. This corresponds to the two possible orientations of the
coordinate system. Seeing the ﬁgure as convex gives a left-handed coordinate system. Thus the “correct” way to view

18

CHAPTER 3. CARTESIAN COORDINATE SYSTEM

Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

3.6 Representing a vector in the standard basis
A point in space in a Cartesian coordinate system may also be represented by a position vector, which can be thought
of as an arrow pointing from the origin of the coordinate system to the point.[7] If the coordinates represent spatial
positions (displacements), it is common to represent the vector from the origin to the point of interest as r . In two
dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as:

r = xi + yj
( )
( )
1
0
where i =
, and j =
are unit vectors in the direction of the x-axis and y-axis respectively, generally
0
1
referred to as the standard basis (in some application areas these may also be referred to as versors). Similarly, in
three dimensions, the vector from the origin to the point with Cartesian coordinates (x, y, z) can be written as:[8]

r = xi + yj + zk
 
0
where k = 0 is the unit vector in the direction of the z-axis.
1
There is no natural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however
there is a way to use complex numbers to provide such a multiplication. In a two dimensional cartesian plane, identify
the point with coordinates (x, y) with the complex number z = x + iy. Here, i is the imaginary unit and is identiﬁed
with the point with coordinates (0, 1), so it is not the unit vector in the direction of the x-axis. Since the complex
numbers can be multiplied giving another complex number, this identiﬁcation provides a means to “multiply” vectors.
In a three dimensional cartesian space a similar identiﬁcation can be made with a subset of the quaternions.

3.7 Applications
Cartesian coordinates are an abstraction that have a multitude of possible applications in the real world. However,
three constructive steps are involved in superimposing coordinates on a problem application. 1) Units of distance
must be decided deﬁning the spatial size represented by the numbers used as coordinates. 2) An origin must be
assigned to a speciﬁc spatial location or landmark, and 3) the orientation of the axes must be deﬁned using available
directional cues for (n-1) of the n axes.
Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D).
What units make sense? Kilometers are a good choice, since the original deﬁnition of the kilometer was geospatial...10,000 km equalling the surface distance from the Equator to the North Pole. Where to place the origin? Based
on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite orbits). Finally, how to orient X, Y and Z axis directions? The axis of Earth’s spin provides a natural direction strongly
associated with “up vs. down”, so positive Z can adopt the direction from geocenter to North Pole. A location on
the Equator is needed to deﬁne the X-axis, and the Prime Meridian stands out as a reference direction, so the X-axis
takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions,
and two perpendicular axes directions pinned down for X and Z, the Y-axis is determined by the ﬁrst two choices. In
order to obey the right hand rule, the Y-axis must point out from the geocenter to [ 90 degrees longitude, 0 degrees
latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City? Using [ longitude
= −73.985656, latitude = 40.748433 ], Earth radius = 40,000/2π, and transforming from spherical --> Cartesian
coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] = [ 1330.53 km,
–4635.75 km, 4155.46 km ]. GPS navigation relies on such geocentric coordinates.
In engineering projects, agreement on the deﬁnition of coordinates is a crucial foundation. One cannot assume that
coordinates come predeﬁned for a novel application, so knowledge of how to erect a coordinate system where there
is none is essential to applying René Descartes’ ingenious thinking.

3.8. SEE ALSO

19

While spatial apps employ identical units along all axes, in business and scientiﬁc apps, each axis may have diﬀerent
units of measurement associated with it (such as kilograms, seconds, pounds, etc.). Although four- and higherdimensional spaces are diﬃcult to visualize, the algebra of Cartesian coordinates can be extended relatively easily
to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic
extension is what is used to deﬁne the geometry of higher-dimensional spaces.) Conversely, it is often helpful to use
the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or
three of many non-spatial variables.
The graph of a function or relation is the set of all points satisfying that function or relation. For a function of one
variable, f, the set of all points (x, y), where y = f(x) is the graph of the function f. For a function g of two variables,
the set of all points (x, y, z), where z = g(x, y) is the graph of the function g. A sketch of the graph of such a function
or relation would consist of all the salient parts of the function or relation which would include its relative extrema,
its concavity and points of inﬂection, any points of discontinuity and its end behavior. All of these terms are more
fully deﬁned in calculus. Such graphs are useful in calculus to understand the nature and behavior of a function or
relation.

3.8 See also
• Horizontal and vertical
• Jones diagram, which plots four variables rather than two.
• Orthogonal coordinates
• Polar coordinate system
• Spherical coordinate system

3.9 Notes
3.10 References
[1] “Analytic geometry”. Encyclopædia Britannica (Encyclopædia Britannica Online ed.). 2008.
[2] Burton 2011, p. 374
[3] A Tour of the Calculus, David Berlinski
[4] Springer online reference Encyclopedia of Mathematics
[5] Smart 1998, Chap. 2
[6] Brannan, Esplen & Gray 1998, pg. 49
[7] Brannan, Esplen & Gray 1998, Appendix 2, pp. 377–382
[8] David J. Griﬃths (1999). Introduction to Electrodynamics. Prentice Hall. ISBN 0-13-805326-X.

3.11 Sources
• Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998), Geometry, Cambridge: Cambridge University
Press, ISBN 0-521-59787-0
• Burton, David M. (2011), The History of Mathematics/An Introduction (7th ed.), New York: McGraw-Hill,
ISBN 978-0-07-338315-6
• Smart, James R. (1998), Modern Geometries (5th ed.), Paciﬁc Grove: Brooks/Cole, ISBN 0-534-35188-3

20

CHAPTER 3. CARTESIAN COORDINATE SYSTEM

3.12 Further reading
• Descartes, René (2001). Discourse on Method, Optics, Geometry, and Meteorology. Trans. by Paul J. Oscamp
(Revised ed.). Indianapolis, IN: Hackett Publishing. ISBN 0-87220-567-3. OCLC 488633510.
• Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers (1st ed.). New York:
McGraw-Hill. pp. 55–79. LCCN 59-14456. OCLC 19959906.
• Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand.
LCCN 55-10911.
• Moon P, Spencer DE (1988). “Rectangular Coordinates (x, y, z)". Field Theory Handbook, Including Coordinate Systems, Diﬀerential Equations, and Their Solutions (corrected 2nd, 3rd print ed.). New York: SpringerVerlag. pp. 9–11 (Table 1.01). ISBN 978-0-387-18430-2.
• Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. ISBN
0-07-043316-X. LCCN 52-11515.
• Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. LCCN 6725285.

3.13 External links
• Cartesian Coordinate System
• Printable Cartesian Coordinates
• Cartesian coordinates at PlanetMath.org.
• MathWorld description of Cartesian coordinates
• Coordinate Converter – converts between polar, Cartesian and spherical coordinates
• Coordinates of a point Interactive tool to explore coordinates of a point
• open source JavaScript class for 2D/3D Cartesian coordinate system manipulation

Chapter 4

Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in
manifolds, and suitable subsets of algebraic varieties.
The dual concept is relative dimension.

4.1 Deﬁnition
Codimension is a relative concept: it is only deﬁned for one object inside another. There is no “codimension of a
vector space (in isolation)”, only the codimension of a vector subspace.
If W is a linear subspace of a ﬁnite-dimensional vector space V, then the codimension of W in V is the diﬀerence
between the dimensions:

codim(W ) = dim(V ) − dim(W ).
It is the complement of the dimension of W, in that, with the dimension of W, it adds up to the dimension of the
ambient space V:

dim(W ) + codim(W ) = dim(V ).
Similarly, if N is a submanifold or subvariety in M, then the codimension of N in M is

codim(N ) = dim(M ) − dim(N ).
Just as the dimension of a submanifold is the dimension of the tangent bundle (the number of dimensions that you
can move on the submanifold), the codimension is the dimension of the normal bundle (the number of dimensions
you can move oﬀ the submanifold).
More generally, if W is a linear subspace of a (possibly inﬁnite dimensional) vector space V then the codimension of
W in V is the dimension (possibly inﬁnite) of the quotient space V/W, which is more abstractly known as the cokernel
of the inclusion. For ﬁnite-dimensional vector spaces, this agrees with the previous deﬁnition

codim(W ) = dim(V /W ) = dim coker(W → V ) = dim(V ) − dim(W ),
and is dual to the relative dimension as the dimension of the kernel.
Finite-codimensional subspaces of inﬁnite-dimensional spaces are often useful in the study of topological vector
spaces.
21

22

CHAPTER 4. CODIMENSION

4.2 Additivity of codimension and dimension counting
The fundamental property of codimension lies in its relation to intersection: if W 1 has codimension k1 , and W 2 has
codimension k2 , then if U is their intersection with codimension j we have
max (k1 , k2 ) ≤ j ≤ k1 + k2 .
In fact j may take any integer value in this range. This statement is more perspicuous than the translation in terms of
dimensions, because the RHS is just the sum of the codimensions. In words
codimensions (at most) add.
If the subspaces or submanifolds intersect transversally (which occurs generically), codimensions add
exactly.
This statement is called dimension counting, particularly in intersection theory.

4.3 Dual interpretation
In terms of the dual space, it is quite evident why dimensions add. The subspaces can be deﬁned by the vanishing of
a certain number of linear functionals, which if we take to be linearly independent, their number is the codimension.
Therefore we see that U is deﬁned by taking the union of the sets of linear functionals deﬁning the Wᵢ. That union may
introduce some degree of linear dependence: the possible values of j express that dependence, with the RHS sum being
the case where there is no dependence. This deﬁnition of codimension in terms of the number of functions needed
to cut out a subspace extends to situations in which both the ambient space and subspace are inﬁnite dimensional.
In other language, which is basic for any kind of intersection theory, we are taking the union of a certain number of
constraints. We have two phenomena to look out for:
1. the two sets of constraints may not be independent;
2. the two sets of constraints may not be compatible.
The ﬁrst of these is often expressed as the principle of counting constraints: if we have a number N of parameters
to adjust (i.e. we have N degrees of freedom), and a constraint means we have to 'consume' a parameter to satisfy
it, then the codimension of the solution set is at most the number of constraints. We do not expect to be able to ﬁnd
a solution if the predicted codimension, i.e. the number of independent constraints, exceeds N (in the linear algebra
case, there is always a trivial, null vector solution, which is therefore discounted).
The second is a matter of geometry, on the model of parallel lines; it is something that can be discussed for linear
problems by methods of linear algebra, and for non-linear problems in projective space, over the complex number
ﬁeld.

4.4 In geometric topology
Codimension also has some clear meaning in geometric topology: on a manifold, codimension 1 is the dimension of
topological disconnection by a submanifold, while codimension 2 is the dimension of ramiﬁcation and knot theory.
In fact, the theory of high-dimensional manifolds, which starts in dimension 5 and above, can alternatively be said to
start in codimension 3, because higher codimensions avoid the phenomenon of knots. Since surgery theory requires
working up to the middle dimension, once one is in dimension 5, the middle dimension has codimension greater than
2, and hence one avoids knots.
This quip is not vacuous: the study of embeddings in codimension 2 is knot theory, and diﬃcult, while the study of
embeddings in codimension 3 or more is amenable to the tools of high-dimensional geometric topology, and hence
considerably easier.

4.5. SEE ALSO

23

4.5 See also
• glossary of diﬀerential geometry and topology

4.6 References
• Hazewinkel, Michiel, ed. (2001), “Codimension”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608010-4

Chapter 5

Complex dimension
In mathematics, complex dimension usually refers to the dimension of a complex manifold M, or a complex algebraic
variety V.[1] If the complex dimension is d, the real dimension will be 2d.[2] That is, the smooth manifold M has
dimension 2d; and away from any singular point V will also be a smooth manifold of dimension 2d.
However, for a real algebraic variety (that is a variety deﬁned by equations with real coeﬃcients), its dimension refers
commonly to its complex dimension, and its real dimension refers to the maximum of the dimensions of the manifolds
contained in the set of its real points. The real dimension is not greater than the dimension, and equals it if the variety
is irreducible and has real points that are nonsingular. For example, the equation x2 +y 2 +z 2 = 0 deﬁnes a variety of
(complex) dimension 2 (a surface), but of real dimension 0 — it has only one real point, (0, 0, 0), which is singular.[3]
The same points apply to codimension. For example a smooth complex hypersurface in complex projective space of
dimension n will be a manifold of dimension 2(n − 1). A complex hyperplane does not separate a complex projective
space into two components, because it has real codimension 2.

5.1 References
[1] Cavagnaro, Catherine; Haight, William T., II (2001), Dictionary of Classical and Theoretical Mathematics, CRC Press, p.
22, ISBN 9781584880509.
[2] Marsden, Jerrold E.; Ratiu, Tudor S. (1999), Introduction to Mechanics and Symmetry: A Basic Exposition of Classical
Mechanical Systems, Texts in Applied Mathematics 17, Springer, p. 152, ISBN 9780387986432.
[3] Bates, Daniel J.; Hauenstein, Jonathan D.; Sommese, Andrew J.; Wampler, Charles W. (2013), Numerically Solving Polynomial Systems with Bertini, Software, Environments, and Tools 25, SIAM, p. 225, ISBN 9781611972702.

24

Chapter 6

Complex network zeta function
Diﬀerent deﬁnitions have been given for the dimension of a complex network or graph. For example, metric dimension
is deﬁned in terms of the resolving set for a graph. Dimension has also been deﬁned based on the box covering
method applied to graphs.[1] Here we describe the deﬁnition based on the complex network zeta function.[2] This
generalises the deﬁnition based on the scaling property of the volume with distance.[3] The best deﬁnition depends
on the application.

6.1 Deﬁnition
One usually thinks of dimension for a set which is dense, like the points on a line, for example. Dimension makes
sense in a discrete setting, like for graphs, only in the large system limit, as the size tends to inﬁnity. For example, in
Statistical Mechanics, one considers discrete points which are located on regular lattices of diﬀerent dimensions. Such
studies have been extended to arbitrary networks, and it is interesting to consider how the deﬁnition of dimension can
be extended to cover these cases. A very simple and obvious way to extend the deﬁnition of dimension to arbitrary
large networks is to consider how the volume (number of nodes within a given distance from a speciﬁed node) scales
as the distance (shortest path connecting two nodes in the graph) is increased. For many systems arising in physics,
this is indeed a useful approach. This deﬁnition of dimension could be put on a strong mathematical foundation,
similar to the deﬁnition of Hausdorﬀ dimension for continuous systems. The mathematically robust deﬁnition uses
the concept of a zeta function for a graph. The complex network zeta function and the graph surface function were
introduced to characterize large graphs. They have also been applied to study patterns in Language Analysis. In this
section we will brieﬂy review the deﬁnition of the functions and discuss further some of their properties which follow
from the deﬁnition.
We denote by rij the distance from node i to node j , i.e., the length of the shortest path connecting the ﬁrst node to
the second node. rij is ∞ if there is no path from node i to node j . With this deﬁnition, the nodes of the complex
network become points in a metric space.[2] Simple generalisations of this deﬁnition can be studied, e.g., we could
consider weighted edges. The graph surface function, S(r) , is deﬁned as the number of nodes which are exactly at a
distance r from a given node, averaged over all nodes of the network. The complex network zeta function ζG (α) is
deﬁned as

ζG (α) :=

1 ∑ ∑ −α
rij ,
N i
j̸=i

where N is the graph size, measured by the number of nodes. When α is zero all nodes contribute equally to the sum
in the previous equation. This means that ζG (0) is N − 1 , and it diverges when N → ∞ . When the exponent α
tends to inﬁnity, the sum gets contributions only from the nearest neighbours of a node. The other terms tend to zero.
Thus, ζG (α) tends to the average degree < k > for the graph as α → ∞ .

⟨k⟩ = lim ζG (α).
α→∞

25

26

CHAPTER 6. COMPLEX NETWORK ZETA FUNCTION

The need for taking an average over all nodes can be avoided by using the concept of supremum over nodes, which
makes the concept much easier to apply for formally inﬁnite graphs.[4] The deﬁnition can be expressed as a weighted
sum over the node distances. This gives the Dirichlet series relation

ζG (α) =

∑

S(r)/rα .

r

This deﬁnition has been used in the shortcut model to study several processes and their dependence on dimension.

6.2 Properties
ζG (α) is a decreasing function of α , ζG (α1 ) > ζG (α2 ) , if α1 < α2 . If the average degree of the nodes (the mean
coordination number for the graph) is ﬁnite, then there is exactly one value of α , αtransition , at which the complex
network zeta function transitions from being inﬁnite to being ﬁnite. This has been deﬁned as the dimension of the
complex network. If we add more edges to an existing graph, the distances between nodes will decrease. This results
in an increase in the value of the complex network zeta function, since S(r) will get pulled inward. If the new links
connect remote parts of the system, i.e., if the distances change by amounts which do not remain ﬁnite as the graph
size N → ∞ , then the dimension tends to increase. For regular discrete d-dimensional lattices Zd with distance
deﬁned using the L1 norm

∥⃗n∥1 = ∥n1 ∥ + · · · + ∥nd ∥,
the transition occurs at α = d . The deﬁnition of dimension using the complex network zeta function satisﬁes
properties like monotonicity (a subset has a lower or the same dimension as its containing set), stability (a union of
sets has the maximum dimension of the component sets forming the union) and Lipschitz invariance,[5] provided
the operations involved change the distances between nodes only by ﬁnite amounts as the graph size N goes to ∞ .
Algorithms to calculate the complex network zeta function have been presented.[6]

6.3 Values for discrete regular lattices
For a one-dimensional regular lattice the graph surface function S1 (r) is exactly two for all values of r (there are two
nearest neighbours, two next-nearest neighbours, and so on). Thus, the complex network zeta function ζG (α) is equal
to 2ζ(α) , where ζ(α) is the usual Riemann zeta function. By choosing a given axis of the lattice and summing over
cross-sections for the allowed range of distances along the chosen axis the recursion relation below can be derived

Sd+1 (r) = 2 + Sd (r) + 2

r−1
∑

Sd (i).

i=1

From combinatorics the surface function for a regular lattice can be written[7] as

Sd (r) =

( )(
)
d−1
∑
d d+r−i−1
(−1)i 2d−i
.
i
d−i−1
i=0

The following expression for the sum of positive integers raised to a given power k will be useful to calculate the
surface function for higher values of d :

r
∑
i=1

ik =

rk+1
rk
+
+
(k + 1)
2

⌊(k+1)/2⌋

∑
j=1

(−1)j+1 2ζ(2j)k!rk+1−2j
.
(2π)2j (k + 1 − 2j)!

Another formula for the sum of positive integers raised to a given power k is

6.4. RANDOM GRAPH ZETA FUNCTION

n
∑
(
k=1

n+1 k

r
)∑

27

ik = (r + 1)((r + 1)n − 1).

i=1

Sd (r) → O(2d rd−1 /Γ(d)) as r → ∞ .
The Complex network zeta function for some lattices is given below.
d = 1 : ζG (α) = 2ζ(α)
d = 2 : ζG (α) = 4ζ(α − 1)
d = 3 : ζG (α) = 4ζ(α − 2) + 2ζ(α )
d = 4 : ζG (α) = 83 ζ(α − 3) +

16
3 ζ(α

− 1)

r → ∞ : ζG (α) = 2d ζ(α − d + 1)/Γ(d) (for α near the transition point.)

6.4 Random graph zeta function
Random graphs are networks having some number N of vertices, in which each pair is connected with probability p
, or else the pair is disconnected. Random graphs have a diameter of two with probability approaching one, in the
inﬁnite limit ( N → ∞ ). To see this, consider two nodes A and B . For any node C diﬀerent from A or B , the
probability that C is not simultaneously connected to both A and B is (1 − p2 ) . Thus, the probability that none of
the N − 2 nodes provides a path of length 2 between nodes A and B is (1 − p2 )N −2 . This goes to zero as the system
size goes to inﬁnity, and hence most random graphs have their nodes connected by paths of length at most 2 . Also,
the mean vertex degree will be p(N − 1) . For large random graphs almost all nodes are at a distance of one or two
from any given node, S(1) is p(N − 1) , S(2) is (N − 1)(1 − p) , and the graph zeta function is
ζG (α) = p(N − 1) + (N − 1)(1 − p)2−α .

6.5 References
[1] K.-I. Goh, G. Salvi, B. Kahng and D. Kim, Phys. Rev. Lett. 96, 018701 (2006).
[2] O. Shanker (2007). “Graph Zeta Function and Dimension of Complex Network”. Modern Physics Letters B 21 (11):
639–644. Bibcode:2007MPLB...21..639S. doi:10.1142/S0217984907013146.
[3] O. Shanker (2007). “Deﬁning Dimension of a Complex Network”. Modern Physics Letters B 21 (6): 321–326. Bibcode:2007MPLB...21..321S.
doi:10.1142/S0217984907012773.
[4] O. Shanker (2010). “Complex Network Dimension and Path Counts”. Theoretical Computer Science 411 (26–28): 2454–
2458. doi:10.1016/j.tcs.2010.02.013.
[5] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, second edition, 2003

[6] O. Shanker, (2008). “Algorithms for Fractal Dimension Calculation”. Modern Physics Letters B 22 (7): 459–466. Bibcode:2008MPLB...22..459S
doi:10.1142/S0217984908015048.
[7] O. Shanker (2008). “Sharp dimension transition in a shortcut model”. J. Phys. A: Math. Theor. 41 (28): 285001.
Bibcode:2008JPhA...41B5001S. doi:10.1088/1751-8113/41/28/285001.

Chapter 7

Concentration dimension
In mathematics — speciﬁcally, in probability theory — the concentration dimension of a Banach space-valued
random variable is a numerical measure of how “spread out” the random variable is compared to the norm on the
space.

7.1 Deﬁnition
Let (B, || ||) be a Banach space and let X be a Gaussian random variable taking values in B. That is, for every linear
functional ℓ in the dual space B∗ , the real-valued random variable ⟨ℓ, X⟩ has a normal distribution. Deﬁne

σ(X) = sup

{√

}

E[⟨ℓ, X⟩2 ] ℓ ∈ B ∗ , ∥ℓ∥ ≤ 1 .

Then the concentration dimension d(X) of X is deﬁned by

d(X) =

E[∥X∥2 ]
.
σ(X)2

7.2 Examples
• If B is n-dimensional Euclidean space Rn with its usual Euclidean norm, and X is a standard Gaussian random
variable, then σ(X) = 1 and E[||X||2 ] = n, so d(X) = n.
• If B is Rn with the supremum norm, then σ(X) = 1 but E[||X||2 ] (and hence d(X)) is of the order of log(n).

7.3 References
• Ledoux, Michel; Talagrand, Michel (1991), Probability in Banach spaces: Isoperimetry and processes, Ergebnisse der Mathematik und ihrer Grenzgebiete 23, Berlin: Springer-Verlag, p. 237, doi:10.1007/978-3-64220212-4, ISBN 3-540-52013-9, MR 1102015.
• Pisier, Gilles (1989), The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics 94, Cambridge University Press, Cambridge, pp. 42–43, doi:10.1017/CBO9780511662454, ISBN
0-521-36465-5, MR 1036275.

28

Chapter 8

Degrees of freedom
In many scientiﬁc ﬁelds, the degrees of freedom of a system is the number of parameters of the system that may
vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates;
a non-inﬁnitesimal object on the plane might have additional degrees of freedoms related to its orientation.
In mathematics, this notion is formalized as the dimension of a manifold or an algebraic variety. When degrees of
freedom is used instead of dimension, this usually means that the manifold or variety that models the system is only
implicitly deﬁned. See:
• Degrees of freedom (mechanics), number of independent motions that are allowed to the body or, in case of
a mechanism made of several bodies, number of possible independent relative motions between the pieces of
the mechanism
• Degrees of freedom (physics and chemistry), a term used in explaining dependence on parameters, or the
dimensions of a phase space
• Degrees of freedom (statistics), the number of values in the ﬁnal calculation of a statistic that are free to vary
• Degrees of freedom problem, the problem of controlling motor movement given abundant degrees of freedom

8.1 See also
• Six degrees of freedom

29

Chapter 9

Degrees of freedom (physics and
chemistry)
This article is about physics and chemistry. For other ﬁelds, see Degrees of freedom.
In physics, a degree of freedom is an independent physical parameter in the formal description of the state of a
physical system. The set of all dimensions of a system is known as a phase space, and degrees of freedom are
sometimes referred to as its dimensions.

9.1 Deﬁnition
A degree of freedom of a physical system is an independent parameter that is necessary to characterize the state of a
physical system. In general, a degree of freedom may be any useful property that is not dependent on other variables.
The location of a particle in three-dimensional space requires three position coordinates. Similarly, the direction and
speed at which a particle moves can be described in terms of three velocity components, each in reference to the
three dimensions of space. If the time evolution of the system is deterministic, where the state at one instant uniquely
determines its past and future position and velocity as a function of time, such a system has six degrees of freedom. If
the motion of the particle is constrained to a lower number of dimensions, for example, the particle must move along
a wire or on a ﬁxed surface, then the system has fewer than six degrees of freedom. On the other hand, a system with
an extended object that can rotate or vibrate can have more than six degrees of freedom.
In classical mechanics, the state of a point particle at any given time is often described with position and velocity
coordinates in the Lagrangian formalism, or with position and momentum coordinates in the Hamiltonian formalism.
In statistical mechanics, a degree of freedom is a single scalar number describing the microstate of a system.[1] The
speciﬁcation of all microstates of a system is a point in the system’s phase space.
In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer.
It is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic
function to the energy of the system.

9.2 Degrees of freedom of gas molecules
In three-dimensional space, three degrees of freedom are associated with the movement of a particle. A diatomic
gas molecule thus has 6 degrees of freedom. This set may be decomposed in terms of translations, rotations, and
vibrations of the molecule. The center of mass motion of the entire molecule accounts for 3 degrees of freedom. In
addition, the molecule has two rotational degrees of motion and one vibrational mode. The rotations occur around the
two axes perpendicular to the line between the two atoms. The rotation around the atom–atom bond is not a physical
rotation. This yields, for a diatomic molecule, a decomposition of:
30

9.2. DEGREES OF FREEDOM OF GAS MOLECULES

31

Diﬀerent ways of visualizing the 6 degrees of freedom of a diatomic molecule. (CM: center of mass of the system, T: translational
motion, R: rotational motion, V: vibrational motion.)

3N = 6 = 3 + 2 + 1.
For a general (non-linear) molecule with N > 2 atoms, all 3 rotational degrees of freedom are considered, resulting
in the decomposition:

3N = 3 + 3 + (3N − 6)
which means that an N-atom molecule has 3N − 6 vibrational degrees of freedom for N > 2. In special cases, such
as adsorbed large molecules, the rotational degrees of freedom can be limited to only one.[2]
As deﬁned above one can also count degrees of freedom using the minimum number of coordinates required to
specify a position. This is done as follows:
1. For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space.
Thus its degree of freedom in a 3-D space is 3.
2. For a body consisting of 2 particles (ex. a diatomic molecule) in a 3-D space with constant distance between
them (let’s say d) we can show (below) its degrees of freedom to be 5.

32

CHAPTER 9. DEGREES OF FREEDOM (PHYSICS AND CHEMISTRY)

Let’s say one particle in this body has coordinate (x1 , y1 , z1 ) and the other has coordinate (x2 , y2 , z2 ) with z2 unknown.
Application of the formula for distance between two coordinates

d=

√
(x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2

results in one equation with one unknown, in which we can solve for z2 . One of x1 , x2 , y1 , y2 , z1 , or z2 can be
unknown.
Contrary to the classical equipartition theorem, at room temperature, the vibrational motion of molecules typically
makes negligible contributions to the heat capacity. This is because these degrees of freedom are frozen because
the spacing between the energy eigenvalues exceeds the energy corresponding to ambient temperatures (kBT). In the
following table such degrees of freedom are disregarded because of their low eﬀect on total energy. However, at very
high temperatures they cannot be neglected.

9.3 Independent degrees of freedom
The set of degrees of freedom X1 , ... , XN of a system is independent if the energy associated with the set can be
written in the following form:

E=

N
∑

Ei (Xi ),

i=1

where Eᵢ is a function of the sole variable Xᵢ.
example: if X1 and X2 are two degrees of freedom, and E is the associated energy:
• If E = X14 + X24 , then the two degrees of freedom are independent.
• If E = X14 + X1 X2 + X24 , then the two degrees of freedom are not independent. The term
involving the product of X1 and X2 is a coupling term, that describes an interaction between the
two degrees of freedom.
For i from 1 to N, the value of the ith degree of freedom Xᵢ is distributed according to the Boltzmann distribution.
Its probability density function is the following:

pi (Xi ) = ∫

e

Ei
BT

−k

dXi e

Ei
BT

−k

In this section, and throughout the article the brackets ⟨⟩ denote the mean of the quantity they enclose.
The internal energy of the system is the sum of the average energies associated with each of the degrees of freedom:

⟨E⟩ =

N
∑
⟨Ei ⟩.
i=1

9.4 Quadratic degrees of freedom
A degree of freedom Xᵢ is quadratic if the energy terms associated with this degree of freedom can be written as

E = αi Xi2 + βi Xi Y
where Y is a linear combination of other quadratic degrees of freedom.
example: if X1 and X2 are two degrees of freedom, and E is the associated energy:

9.5. GENERALIZATIONS
•
•
•
•

33

If E = X14 +X13 X2 +X24 , then the two degrees of freedom are not independent and non-quadratic.
If E = X14 + X24 , then the two degrees of freedom are independent and non-quadratic.
If E = X12 +X1 X2 +2X22 , then the two degrees of freedom are not independent but are quadratic.
If E = X12 + 2X22 , then the two degrees of freedom are independent and quadratic.

For example, in Newtonian mechanics, the dynamics of a system of quadratic degrees of freedom are controlled by
a set of homogeneous linear diﬀerential equations with constant coeﬃcients.

9.4.1

Quadratic and independent degree of freedom

X1 , ... , XN are quadratic and independent degrees of freedom if the energy associated with a microstate of the
system they represent can be written as:

E=

N
∑

αi Xi2

i=1

9.4.2

Equipartition theorem

In the classical limit of statistical mechanics, at thermodynamic equilibrium, the internal energy of a system of N
quadratic and independent degrees of freedom is:
kB T
2
Here, the mean energy associated with a degree of freedom is:
U = ⟨E⟩ = N

∫

∫
⟨Ei ⟩ =

dXi αi Xi2

pi (Xi ) =

∫

−

αi Xi2

dXi αi Xi2 e kB T
α X2
∫
− i i
dXi e kB T

x2

dx x2 e− 2
kB T
=
∫
x2
−
2
dx e 2
Since the degrees of freedom are independent, the internal energy of the system is equal to the sum of the mean
energy associated with each degree of freedom, which demonstrates the result.
kB T
⟨Ei ⟩ =
2

9.5 Generalizations
The description of a system’s state as a point in its phase space, although mathematically convenient, is thought to be
fundamentally inaccurate. In quantum mechanics, the motion degrees of freedom are superseded with the concept
of wave function, and operators which correspond to other degrees of freedom have discrete spectra. For example,
intrinsic angular momentum operator (which corresponds to the rotational freedom) for an electron or photon have
only two eigenvalues. This discreteness becomes apparent when action has an order of magnitude of the Planck
constant, and individual degrees of freedom can be distinguished.

9.6 References
[1] Reif, F. (2009). Fundamentals of Statistical and Thermal Physics. Long Grove, IL: Waveland Press, Inc. p. 51. ISBN
1-57766-612-7.
[2] Thomas Waldmann, Jens Klein, Harry E. Hoster, R. Jürgen Behm (2012), “Stabilization of Large Adsorbates by Rotational
Entropy: A Time-Resolved Variable-Temperature STM Study” (in German), ChemPhysChem: pp. n/a–n/a, doi:10.1002/cphc.201200531

Chapter 10

Dimension
This article is about dimensions of space. For the dimension of a quantity, see Dimensional analysis. For other uses,
see Dimension (disambiguation).
In physics and mathematics, the dimension of a mathematical space (or object) is informally deﬁned as the minimum

From left to right: the square, the cube and the tesseract. The two-dimensional (2d) square is bounded by one-dimensional (1d) lines;
the three-dimensional (3d) cube by two-dimensional areas; and the four-dimensional (4d) tesseract by three-dimensional volumes.
For display on a two-dimensional surface such as a screen, the 3d cube and 4d tesseract require projection.

The ﬁrst four spatial dimensions.

number of coordinates needed to specify any point within it.[1][2] Thus a line has a dimension of one because only
one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a
plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a
point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The
inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point
within these spaces.
34

10.1. IN MATHEMATICS

35

In classical mechanics, space and time are diﬀerent categories and refer to absolute space and time. That conception
of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The
four dimensions of spacetime consist of events that are not absolutely deﬁned spatially and temporally, but rather are
known relative to the motion of an observer. Minkowski space ﬁrst approximates the universe without gravity; the
pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are
used to describe string theory, and the state-space of quantum mechanics is an inﬁnite-dimensional function space.
The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or conﬁguration spaces such as in Lagrangian or Hamiltonian
mechanics; these are abstract spaces, independent of the physical space we live in.

10.1 In mathematics
In mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is
embedded. For example, a point on the unit circle in the plane can be speciﬁed by two Cartesian coordinates, but
a single polar coordinate (the angle) would be suﬃcient, so the circle is 1-dimensional even though it exists in the
2-dimensional plane. This intrinsic notion of dimension is one of the chief ways the mathematical notion of dimension
diﬀers from its common usages.
The dimension of Euclidean n-space En is n. When trying to generalize to other types of spaces, one is faced with
the question “what makes En n-dimensional?" One answer is that to cover a ﬁxed ball in En by small balls of radius ε,
one needs on the order of ε−n such small balls. This observation leads to the deﬁnition of the Minkowski dimension
and its more sophisticated variant, the Hausdorﬀ dimension, but there are also other answers to that question. For
example, the boundary of a ball in En looks locally like En−1 and this leads to the notion of the inductive dimension.
While these notions agree on En , they turn out to be diﬀerent when one looks at more general spaces.
A tesseract is an example of a four-dimensional object. Whereas outside mathematics the use of the term “dimension”
is as in: “A tesseract has four dimensions", mathematicians usually express this as: “The tesseract has dimension 4",
or: “The dimension of the tesseract is 4”.
Although the notion of higher dimensions goes back to René Descartes, substantial development of a higher-dimensional
geometry only began in the 19th century, via the work of Arthur Cayley, William Rowan Hamilton, Ludwig Schläﬂi
and Bernhard Riemann. Riemann’s 1854 Habilitationsschrift, Schläﬂi’s 1852 Theorie der vielfachen Kontinuität,
Hamilton’s 1843 discovery of the quaternions and the construction of the Cayley algebra marked the beginning of
higher-dimensional geometry.
The rest of this section examines some of the more important mathematical deﬁnitions of the dimensions.

10.1.1

Dimension of a vector space

Main article: Dimension (vector space)
The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates
necessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the Hamel
dimension or algebraic dimension to distinguish it from other notions of dimension.

10.1.2

Manifolds

A connected topological manifold is locally homeomorphic to Euclidean n-space, and the number n is called the
manifold’s dimension. One can show that this yields a uniquely deﬁned dimension for every connected topological
manifold.
For connected diﬀerentiable manifolds, the dimension is also the dimension of the tangent vector space at any point.
In geometric topology, the theory of manifolds is characterized by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases n > 4 are simpliﬁed by having extra space in which to “work"; and the cases
n = 3 and 4 are in some senses the most diﬃcult. This state of aﬀairs was highly marked in the various cases of the
Poincaré conjecture, where four diﬀerent proof methods are applied.

36

10.1.3

CHAPTER 10. DIMENSION

Varieties

Main article: Dimension of an algebraic variety
The dimension of an algebraic variety may be deﬁned in various equivalent ways. The most intuitive way is probably
the dimension of the tangent space at any regular point. Another intuitive way is to deﬁne the dimension as the number
of hyperplanes that are needed in order to have an intersection with the variety that is reduced to a ﬁnite number of
points (dimension zero). This deﬁnition is based on the fact that the intersection of a variety with a hyperplane reduces
the dimension by one unless if the hyperplane contains the variety.
An algebraic set being a ﬁnite union of algebraic varieties, its dimension is the maximum of the dimensions of its
components. It is equal to the maximal length of the chains V0 ⊊ V1 ⊊ . . . ⊊ Vd of sub-varieties of the given
algebraic set (the length of such a chain is the number of " ⊊ ").
Each variety can be considered as an algebraic stack, and its dimension as variety agrees with its dimension as stack.
There are however many stacks which do not correspond to varieties, and some of these have negative dimension.
Speciﬁcally, if V is a variety of dimension m and G is an algebraic group of dimension n acting on V, then the quotient
stack [V/G] has dimension m−n.[3]

10.1.4

Krull dimension

Main article: Krull dimension
The Krull dimension of a commutative ring is the maximal length of chains of prime ideals in it, a chain of length n
being a sequence P0 ⊊ P1 ⊊ . . . ⊊ Pn of prime ideals related by inclusion. It is strongly related to the dimension
of an algebraic variety, because of the natural correspondence between sub-varieties and prime ideals of the ring of
the polynomials on the variety.
For an algebra over a ﬁeld, the dimension as vector space is ﬁnite if and only if its Krull dimension is 0.

10.1.5

Lebesgue covering dimension

Main article: Lebesgue covering dimension
For any normal topological space X, the Lebesgue covering dimension of X is deﬁned to be n if n is the smallest integer
for which the following holds: any open cover has an open reﬁnement (a second open cover where each element is a
subset of an element in the ﬁrst cover) such that no point is included in more than n + 1 elements. In this case dim
X = n. For X a manifold, this coincides with the dimension mentioned above. If no such integer n exists, then the
dimension of X is said to be inﬁnite, and one writes dim X = ∞. Moreover, X has dimension −1, i.e. dim X = −1 if
and only if X is empty. This deﬁnition of covering dimension can be extended from the class of normal spaces to all
Tychonoﬀ spaces merely by replacing the term “open” in the deﬁnition by the term "functionally open".

10.1.6

Inductive dimension

Main article: Inductive dimension
An inductive deﬁnition of dimension can be created as follows. Consider a discrete set of points (such as a ﬁnite
collection of points) to be 0-dimensional. By dragging a 0-dimensional object in some direction, one obtains a 1dimensional object. By dragging a 1-dimensional object in a new direction, one obtains a 2-dimensional object. In
general one obtains an (n + 1)-dimensional object by dragging an n-dimensional object in a new direction.
The inductive dimension of a topological space may refer to the small inductive dimension or the large inductive
dimension, and is based on the analogy that (n + 1)-dimensional balls have n-dimensional boundaries, permitting an
inductive deﬁnition based on the dimension of the boundaries of open sets.

10.2. IN PHYSICS

10.1.7

37

Hausdorﬀ dimension

Main article: Hausdorﬀ dimension
For structurally complicated sets, especially fractals, the Hausdorﬀ dimension is useful. The Hausdorﬀ dimension is
deﬁned for all metric spaces and, unlike the dimensions considered above, can also attain non-integer real values.[4]
The box dimension or Minkowski dimension is a variant of the same idea. In general, there exist more deﬁnitions of
fractal dimensions that work for highly irregular sets and attain non-integer positive real values. Fractals have been
found useful to describe many natural objects and phenomena.[5][6]

10.1.8

Hilbert spaces

Every Hilbert space admits an orthonormal basis, and any two such bases for a particular space have the same
cardinality. This cardinality is called the dimension of the Hilbert space. This dimension is ﬁnite if and only if
the space’s Hamel dimension is ﬁnite, and in this case the above dimensions coincide.

10.2 In physics
10.2.1

Spatial dimensions

Classical physics theories describe three physical dimensions: from a particular point in space, the basic directions in
which we can move are up/down, left/right, and forward/backward. Movement in any other direction can be expressed
in terms of just these three. Moving down is the same as moving up a negative distance. Moving diagonally upward
and forward is just as the name of the direction implies; i.e., moving in a linear combination of up and forward.
In its simplest form: a line describes one dimension, a plane describes two dimensions, and a cube describes three
dimensions. (See Space and Cartesian coordinate system.)

10.2.2

Time

A temporal dimension is a dimension of time. Time is often referred to as the "fourth dimension" for this reason,
but that is not to imply that it is a spatial dimension. A temporal dimension is one way to measure physical change.
It is perceived diﬀerently from the three spatial dimensions in that there is only one of it, and that we cannot move
freely in time but subjectively move in one direction.
The equations used in physics to model reality do not treat time in the same way that humans commonly perceive it.
The equations of classical mechanics are symmetric with respect to time, and equations of quantum mechanics are
typically symmetric if both time and other quantities (such as charge and parity) are reversed. In these models, the
perception of time ﬂowing in one direction is an artifact of the laws of thermodynamics (we perceive time as ﬂowing
in the direction of increasing entropy).
The best-known treatment of time as a dimension is Poincaré and Einstein's special relativity (and extended to general
relativity), which treats perceived space and time as components of a four-dimensional manifold, known as spacetime,
and in the special, ﬂat case as Minkowski space.

10.2.3

Additional dimensions

In physics, three dimensions of space and one of time is the accepted norm. However, there are theories that attempt
to unify the four fundamental forces by introducing more dimensions. Most notably, superstring theory requires 10
spacetime dimensions, and originates from a more fundamental 11-dimensional theory tentatively called M-theory
which subsumes ﬁve previously distinct superstring theories. To date, no experimental or observational evidence is
available to conﬁrm the existence of these extra dimensions. If extra dimensions exist, they must be hidden from us by
some physical mechanism. One well-studied possibility is that the extra dimensions may be “curled up” at such tiny
scales as to be eﬀectively invisible to current experiments. Limits on the size and other properties of extra dimensions
are set by particle experiments such as those at the Large Hadron Collider.[7]

38

CHAPTER 10. DIMENSION

At the level of quantum ﬁeld theory, Kaluza–Klein theory uniﬁes gravity with gauge interactions, based on the realization that gravity propagating in small, compact extra dimensions is equivalent to gauge interactions at long distances.
In particular when the geometry of the extra dimensions is trivial, it reproduces electromagnetism. However at suﬃciently high energies or short distances, this setup still suﬀers from the same pathologies that famously obstruct direct
attempts to describe quantum gravity. Therefore, these models still require a UV completion, of the kind that string
theory is intended to provide. Thus Kaluza-Klein theory may be considered either as an incomplete description on
its own, or as a subset of string theory model building.
In addition to small and curled up extra dimensions, there may be extra dimensions that instead aren't apparent
because the matter associated with our visible universe is localized on a (3 + 1)-dimensional subspace. Thus the extra
dimensions need not be small and compact but may be large extra dimensions. D-branes are dynamical extended
objects of various dimensionalities predicted by string theory that could play this role. They have the property that
open string excitations, which are associated with gauge interactions, are conﬁned to the brane by their endpoints,
whereas the closed strings that mediate the gravitational interaction are free to propagate into the whole spacetime, or
“the bulk”. This could be related to why gravity is exponentially weaker than the other forces, as it eﬀectively dilutes
itself as it propagates into a higher-dimensional volume.
Some aspects of brane physics have been applied to cosmology. For example, brane gas cosmology[8][9] attempts to
explain why there are three dimensions of space using topological and thermodynamic considerations. According to
this idea it would be because three is the largest number of spatial dimensions where strings can generically intersect.
If initially there are lots of windings of strings around compact dimensions, space could only expand to macroscopic
sizes once these windings are eliminated, which requires oppositely wound strings to ﬁnd each other and annihilate.
But strings can only ﬁnd each other to annihilate at a meaningful rate in three dimensions, so it follows that only three
dimensions of space are allowed to grow large given this kind of initial conﬁguration.
Extra dimensions are said to be universal if all ﬁelds are equally free to propagate within them.

10.3 Networks and dimension
Some complex networks are characterized by fractal dimensions.[10] The concept of dimension can be generalized to
include networks embedded in space.[11] The dimension characterize their spatial constraints.

10.4 In literature
Main article: Fourth dimension in literature
Science ﬁction texts often mention the concept of “dimension” when referring to parallel or alternate universes or other
imagined planes of existence. This usage is derived from the idea that to travel to parallel/alternate universes/planes of
existence one must travel in a direction/dimension besides the standard ones. In eﬀect, the other universes/planes are
just a small distance away from our own, but the distance is in a fourth (or higher) spatial (or non-spatial) dimension,
not the standard ones.
One of the most heralded science ﬁction stories regarding true geometric dimensionality, and often recommended as
a starting point for those just starting to investigate such matters, is the 1884 novella Flatland by Edwin A. Abbott.
Isaac Asimov, in his foreword to the Signet Classics 1984 edition, described Flatland as “The best introduction one
can ﬁnd into the manner of perceiving dimensions.”
The idea of other dimensions was incorporated into many early science ﬁction stories, appearing prominently, for
example, in Miles J. Breuer's The Appendix and the Spectacles (1928) and Murray Leinster's The Fifth-Dimension
Catapult (1931); and appeared irregularly in science ﬁction by the 1940s. Classic stories involving other dimensions
include Robert A. Heinlein's —And He Built a Crooked House (1941), in which a California architect designs a house
based on a three-dimensional projection of a tesseract; and Alan E. Nourse's Tiger by the Tail and The Universe
Between (both 1951). Another reference is Madeleine L'Engle's novel A Wrinkle In Time (1962), which uses the ﬁfth
dimension as a way for “tesseracting the universe” or “folding” space in order to move across it quickly. The fourth
and ﬁfth dimensions were also a key component of the book The Boy Who Reversed Himself by William Sleator.

10.5. IN PHILOSOPHY

39

10.5 In philosophy
Immanuel Kant, in 1783, wrote: “That everywhere space (which is not itself the boundary of another space) has three
dimensions and that space in general cannot have more dimensions is based on the proposition that not more than
three lines can intersect at right angles in one point. This proposition cannot at all be shown from concepts, but rests
immediately on intuition and indeed on pure intuition a priori because it is apodictically (demonstrably) certain.”[12]
“Space has Four Dimensions” is a short story published in 1846 by German philosopher and experimental psychologist
Gustav Fechner under the pseudonym “Dr. Mises”. The protagonist in the tale is a shadow who is aware of and able
to communicate with other shadows, but who is trapped on a two-dimensional surface. According to Fechner, this
“shadow-man” would conceive of the third dimension as being one of time.[13] The story bears a strong similarity to
the "Allegory of the Cave" presented in Plato's The Republic (c. 380 BC).
Simon Newcomb wrote an article for the Bulletin of the American Mathematical Society in 1898 entitled “The Philosophy of Hyperspace”.[14] Linda Dalrymple Henderson coined the term “hyperspace philosophy”, used to describe
writing that uses higher dimensions to explore metaphysical themes, in her 1983 thesis about the fourth dimension
in early-twentieth-century art.[15] Examples of “hyperspace philosophers” include Charles Howard Hinton, the ﬁrst
writer, in 1888, to use the word “tesseract";[16] and the Russian esotericist P. D. Ouspensky.

10.6 More dimensions
• Degrees of freedom in mechanics / physics and chemistry / statistics

10.7 See also
10.7.1

Topics by dimension

Zero
• Point
• Zero-dimensional space
• Integer
One
• Line
• Graph (combinatorics)
• Real number
Two
• Complex number
• Cartesian coordinate system
• List of uniform tilings
• Surface
Three
• Platonic solid
• Stereoscopy (3-D imaging)

40

CHAPTER 10. DIMENSION
• Euler angles
• 3-manifold
• Knots

Four
• Spacetime
• Fourth spatial dimension
• Convex regular 4-polytope
• Quaternion
• 4-manifold
• Fourth dimension in art
• Fourth dimension in literature
Higher dimensions
in mathematics
• Octonion
• Vector space
• Manifold
• Calabi–Yau spaces
• Curse of dimensionality
in physics
• Kaluza–Klein theory
• String theory
• M-theory
Inﬁnite
• Hilbert space
• Function space

10.8 References
[1] “Curious About Astronomy”. Curious.astro.cornell.edu. Retrieved 2014-03-03.
[2] “MathWorld: Dimension”. Mathworld.wolfram.com. 2014-02-27. Retrieved 2014-03-03.
[3] Fantechi, Barbara (2001), “Stacks for everybody” (PDF), European Congress of Mathematics Volume I, Progr. Math. 201,
Birkhäuser, pp. 349–359
[4] Fractal Dimension, Boston University Department of Mathematics and Statistics
[5] Bunde, Armin; Havlin, Shlomo, eds. (1991). Fractals and Disordered Systems. Springer.
[6] Bunde, Armin; Havlin, Shlomo, eds. (1994). Fractals in Science. Springer.
[7] CMS Collaboration, “Search for Microscopic Black Hole Signatures at the Large Hadron Collider” (arxiv.org)
[8] Brandenberger, R., Vafa, C., Superstrings in the early universe

10.9. FURTHER READING

41

[9] Scott Watson, Brane Gas Cosmology (pdf).
[10] Song, Chaoming; Havlin, Shlomo; Makse, Hernán A. (2005). “Self-similarity of complex networks”. Nature 433 (7024).
arXiv:cond-mat/0503078v1. Bibcode:2005Natur.433..392S. doi:10.1038/nature03248.
[11] Daqing, Li; Kosmidis, Kosmas; Bunde, Armin; Havlin, Shlomo (2011). “Dimension of spatially embedded networks”.
Nature Physics 7 (6). Bibcode:2011NatPh...7..481D. doi:10.1038/nphys1932.
[12] Prolegomena, § 12
[13] Banchoﬀ, Thomas F. (1990). “From Flatland to Hypergraphics: Interacting with Higher Dimensions”. Interdisciplinary
Science Reviews 15 (4): 364. doi:10.1179/030801890789797239.
[14] Newcomb, Simon (1898). “The Philosophy of Hyperspace”. Bulletin of the American Mathematical Society 4 (5): 187.
doi:10.1090/S0002-9904-1898-00478-0.
[15] Kruger, Runette (2007). “Art in the Fourth Dimension: Giving Form to Form – The Abstract Paintings of Piet Mondrian”
(PDF). Spaces of Utopia: an Electronic Journal (5): 11.
[16] Pickover, Cliﬀord A. (2009), “Tesseract”, The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the
History of Mathematics, Sterling Publishing Company, Inc., p. 282, ISBN 9781402757969.

10.9 Further reading
• Katta G Murty, “Systems of Simultaneous Linear Equations” (Chapter 1 of Computational and Algorithmic
Linear Algebra and n-Dimensional Geometry, World Scientiﬁc Publishing: 2014 (ISBN 978-981-4366-62-5).
• Edwin A. Abbott, Flatland: A Romance of Many Dimensions (1884) (Public domain: Online version with
ASCII approximation of illustrations at Project Gutenberg).
• Thomas Banchoﬀ, Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Second Edition, W. H. Freeman and Company: 1996.
• Cliﬀord A. Pickover, Surﬁng through Hyperspace: Understanding Higher Universes in Six Easy Lessons, Oxford
University Press: 1999.
• Rudy Rucker, The Fourth Dimension, Houghton-Miﬄin: 1984.
• Kaku, Michio (1994). Hyperspace, a Scientiﬁc Odyssey Through the 10th Dimension. Oxford University Press.
ISBN 0-19-286189-1.
• Krauss, Lawrence M. (2005). Hiding in the Mirror. Viking Press. ISBN 0670033952.

10.10 External links
• Copeland, Ed (2009). “Extra Dimensions”. Sixty Symbols. Brady Haran for the University of Nottingham.

Chapter 11

Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over
its base ﬁeld.[1][lower-alpha 1]
For every vector space there exists a basis,[lower-alpha 2] and all bases of a vector space have equal cardinality;[lower-alpha 3]
as a result, the dimension of a vector space is uniquely deﬁned. We say V is ﬁnite-dimensional if the dimension of
V is ﬁnite, and inﬁnite-dimensional if its dimension is inﬁnite.
The dimension of the vector space V over the ﬁeld F can be written as dimF(V) or as [V : F], read “dimension of V
over F". When F can be inferred from context, dim(V) is typically written.

11.1 Examples
The vector space R3 has
      
0
0 
 1
0, 1, 0


0
0
1
as a basis, and therefore we have dimR(R3 ) = 3. More generally, dimR(Rn ) = n, and even more generally, dimF(F n )
= n for any ﬁeld F.
The complex numbers C are both a real and complex vector space; we have dimR(C) = 2 and dimC(C) = 1. So the
dimension depends on the base ﬁeld.
The only vector space with dimension 0 is {0}, the vector space consisting only of its zero element.

11.2 Facts
If W is a linear subspace of V, then dim(W) ≤ dim(V).
To show that two ﬁnite-dimensional vector spaces are equal, one often uses the following criterion: if V is a ﬁnitedimensional vector space and W is a linear subspace of V with dim(W) = dim(V), then W = V.
Rn has the standard basis {e1 , ..., en}, where ei is the i-th column of the corresponding identity matrix. Therefore
Rn has dimension n.
Any two vector spaces over F having the same dimension are isomorphic. Any bijective map between their bases
can be uniquely extended to a bijective linear map between the vector spaces. If B is some set, a vector space with
dimension |B| over F can be constructed as follows: take the set F (B) of all functions f : B → F such that f(b) = 0
for all but ﬁnitely many b in B. These functions can be added and multiplied with elements of F, and we obtain the
desired F-vector space.
An important result about dimensions is given by the rank–nullity theorem for linear maps.
42

11.3. GENERALIZATIONS

43

If F/K is a ﬁeld extension, then F is in particular a vector space over K. Furthermore, every F-vector space V is also
a K-vector space. The dimensions are related by the formula
dimK(V) = dimK(F) dimF(V).
In particular, every complex vector space of dimension n is a real vector space of dimension 2n.
Some simple formulae relate the dimension of a vector space with the cardinality of the base ﬁeld and the cardinality
of the space itself. If V is a vector space over a ﬁeld F then, denoting the dimension of V by dim V, we have:
If dim V is ﬁnite, then |V| = |F|dim V .
If dim V is inﬁnite, then |V| = max(|F|, dim V).

11.3 Generalizations
One can see a vector space as a particular case of a matroid, and in the latter there is a well-deﬁned notion of dimension.
The length of a module and the rank of an abelian group both have several properties similar to the dimension of
vector spaces.
The Krull dimension of a commutative ring, named after Wolfgang Krull (1899–1971), is deﬁned to be the maximal
number of strict inclusions in an increasing chain of prime ideals in the ring.

11.3.1

Trace

See also: Trace (linear algebra)
The dimension of a vector space may alternatively be characterized as the trace of the identity operator. For instance,
tr idR2 = tr ( 10 01 ) = 1 + 1 = 2. This appears to be a circular deﬁnition, but it allows useful generalizations.
Firstly, it allows one to deﬁne a notion of dimension when one has a trace but no natural sense of basis. For example,
one may have an algebra A with maps η : K → A (the inclusion of scalars, called the unit) and a map ϵ : A → K
(corresponding to trace, called the counit). The composition ϵ ◦ η : K → K is a scalar (being a linear operator on a
1-dimensional space) corresponds to “trace of identity”, and gives a notion of dimension for an abstract algebra. In
practice, in bialgebras one requires that this map be the identity, which can be obtained by normalizing the counit by
dividing by dimension ( ϵ := n1 tr ), so in these cases the normalizing constant corresponds to dimension.
Alternatively, one may be able to take the trace of operators on an inﬁnite-dimensional space; in this case a (ﬁnite)
trace is deﬁned, even though no (ﬁnite) dimension exists, and gives a notion of “dimension of the operator”. These
fall under the rubric of "trace class operators” on a Hilbert space, or more generally nuclear operators on a Banach
space.
A subtler generalization is to consider the trace of a family of operators as a kind of “twisted” dimension. This
occurs signiﬁcantly in representation theory, where the character of a representation is the trace of the representation,
hence a scalar-valued function on a group χ : G → K, whose value on the identity 1 ∈ G is the dimension of the
representation, as a representation sends the identity in the group to the identity matrix: χ(1G ) = tr IV = dim V.
One can view the other values χ(g) of the character as “twisted” dimensions, and ﬁnd analogs or generalizations of
statements about dimensions to statements about characters or representations. A sophisticated example of this occurs
in the theory of monstrous moonshine: the j-invariant is the graded dimension of an inﬁnite-dimensional graded
representation of the Monster group, and replacing the dimension with the character gives the McKay–Thompson
series for each element of the Monster group.[2]

11.4 See also
• Basis (linear algebra)
• Topological dimension, also called Lebesgue covering dimension

44

CHAPTER 11. DIMENSION (VECTOR SPACE)
• Fractal dimension
• Krull dimension
• Matroid rank
• Rank (linear algebra)

11.5 Notes
[1] It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension.
[2] if one assumes the axiom of choice
[3] see dimension theorem for vector spaces

11.6 References
[1] Itzkov, Mikhail (2009). Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics.
Springer. p. 4. ISBN 978-3-540-93906-1.
[2] Gannon, Terry (2006), Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, ISBN
0-521-83531-3

11.7 External links
• MIT Linear Algebra Lecture on Independence, Basis, and Dimension by Gilbert Strang at MIT OpenCourseWare

Chapter 12

Dimension of an algebraic variety
In mathematics and speciﬁcally in algebraic geometry, the dimension of an algebraic variety may be deﬁned in various
equivalent ways.
Some of these deﬁnitions are of geometric nature, while some other are purely algebraic and rely on commutative
algebra. Some are restricted to algebraic varieties while others apply also to any algebraic set. Some are intrinsic,
as independent of any embedding of the variety into an aﬃne or projective space, while other are related to such an
embedding.

12.1 Dimension of an aﬃne algebraic set
Let K be a ﬁeld, and L ⊇ K be an algebraically closed extension. An aﬃne algebraic set V is the set of the common
zeros in Ln of the elements of an ideal I in a polynomial ring R = K[x1 , . . . , xn ]. Let A=R/I be the algebra of the
polynomials over V. The dimension of V is any of the following integers. It does not change if K is enlarged, if L is
replaced by another algebraically closed extension of K and if I is replaced by another ideal having the same zeros
(that is having the same radical). The dimension is also independent of the choice of coordinates; in other words is
does not change if the xi are replaced by linearly independent linear combinations of them. The dimension of V is
• The maximal length d of the chains V0 ⊂ V1 ⊂ . . . ⊂ Vd of distinct nonempty subvarieties.
This deﬁnition generalizes a property of the dimension of a Euclidean space or a vector space. It is thus probably the
deﬁnition that gives the easiest intuitive description of the notion.
• The Krull dimension of A.
This is the transcription of the preceding deﬁnition in the language of commutative algebra, the Krull dimension
being the maximal length of the chains p0 ⊂ p1 ⊂ . . . ⊂ pd of prime ideals of A.
• The maximal Krull dimension of the local rings at the points of V.
This deﬁnition shows that the dimension is a local property.
• If V is a variety, the Krull dimension of the local ring at any regular point of V
This shows that the dimension is constant on a variety
• The maximal dimension of the tangent vector spaces at the non singular points of V.
This relies the dimension of a variety to that of a diﬀerentiable manifold. More precisely, if V if deﬁned over the
reals, then the set of its real regular points is a diﬀerentiable manifold that has the same dimension as variety and as
a manifold.
45

46

CHAPTER 12. DIMENSION OF AN ALGEBRAIC VARIETY
• If V is a variety, the dimension of the tangent vector space at any non singular point of V.

This is the algebraic analogue to the fact that a connected manifold has a constant dimension.
• The number of hyperplanes or hypersurfaces in general position which are needed to have an intersection with
V which is reduced to a nonzero ﬁnite number of points.
This deﬁnition is not intrinsic as it apply only to algebraic sets that are explicitly embedded in an aﬃne or projective
space.
• The maximal length of a regular sequence in A.
This the algebraic translation of the preceding deﬁnition.
• The diﬀerence between n and the maximal length of the regular sequences contained in I.
This is the algebraic translation of the fact that the intersection of n-d hypersurfaces is, in general, an algebraic set of
dimension d.
• The degree of the Hilbert polynomial of A.
• The degree of the denominator of the Hilbert series of A.
This allows, through a Gröbner basis computation to compute the dimension of the algebraic set deﬁned by a given
system of polynomial equations
• If I is a prime ideal (i.e. V is an algebraic variety), the transcendence degree over K of the ﬁeld of fractions of
A.
This allows to prove easily that the dimension is invariant under birational equivalence.

12.2 Dimension of a projective algebraic set
Let V be a projective algebraic set deﬁned as the set of the common zeros of a homogeneous ideal I in a polynomial
ring R = K[x0 , x1 , . . . , xn ] over a ﬁeld K, and let A=R/I be the graded algebra of the polynomials over V.
All the deﬁnitions of the previous section apply, with the change that, when A or I appear explicitly in the deﬁnition,
the value of the dimension must be reduced by one. For example, the dimension of V is one less than the Krull
dimension of A.

12.3 Computation of the dimension
Given a system of polynomial equations, it may be diﬃcult to compute the dimension of the algebraic set that it
deﬁnes.
Without further information on the system, there is only one practical method that consists to compute a Gröbner
basis and to deduce the degree of the denominator of the Hilbert series of the ideal generated by the equations.
The second step, which is usually the fastest, may be accelerated in the following way: Firstly, the Gröbner basis is
replaced by the list of its leading monomials (this is already done for the computation of the Hilbert series). Then
min(e ,1)
min(e ,1)
each monomial like x1 e1 · · · xn en is replaced by the product of the variables in it: x1 1 · · · xn n . Then the
dimension is the maximal size of a subset S of the variables, such that none of these products of variables depends
only on the variables in S.
This algorithm is implemented in several computer algebra systems. For example in Maple, this is the function
Groebner[HilbertDimension].

12.4. REAL DIMENSION

47

12.4 Real dimension
See also: Complex dimension
The dimension of a set of real points, typically a semialgebraic set, is the dimension of its Zariski closure. For
an algebraic set deﬁned over the reals (that is deﬁned by polynomials with real coeﬃcients), it may occur that the
dimension of the set of its real points diﬀers from its dimension. For example, the algebraic surface of equation
x2 + y 2 + z 2 = 0 is an algebraic variety of dimension two, which has only one real point (0, 0, 0), and thus the real
dimension zero.
The real dimension is more diﬃcult to compute than the algebraic dimension, and, to date, there is no available
software to compute it.

12.5 See also
• Dimension theory (algebra)

12.6 External links
• Hazewinkel, Michiel, ed. (2001), “Algebraic function”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4

Chapter 13

Dimension theory (algebra)
In mathematics, dimension theory is a branch of commutative algebra studying the notion of the dimension of a
commutative ring, and by extension that of a scheme.
The theory is much simpler for an aﬃne ring; i.e., an integral domain that is a ﬁnitely generated algebra over a
ﬁeld. By Noether’s normalization lemma, the Krull dimension of such a ring is the transcendence degree over the
base ﬁeld and the theory runs in parallel with the counterpart in algebraic geometry; cf. Dimension of an algebraic
variety. The general theory tends to be less geometrical; in particular, very little works/is known for non-noetherian
rings. (Kaplansky’s commutative rings gives a good account of the non-noetherian case.) Today, a standard approach
is essentially that of Bourbaki and EGA, which makes essential use of graded modules and, among other things,
emphasizes the role of multiplicities, the generalization of the degree of a projective variety. In this approach, Krull’s
principal ideal theorem appears as a corollary.
Throughout the article, dim denotes Krull dimension of a ring and ht the height of a prime ideal (i.e., the Krull
dimension of the localization at that prime ideal.) Rings are assumed to be commutative except in the last section on
dimensions of non-commutative rings.

13.1 Basic results
Let R be a noetherian ring or valuation ring. Then

dim R[x] = dim R + 1.
If R is noetherian, this follows from the fundamental theorem below (in particular, Krull’s principal ideal theorem.)
But it is also a consequence of the more precise result. For any prime ideal p in R,
ht(pR[x]) = ht(p) .
ht(q) = ht(p) + 1 for any prime ideal q ⊋ pR[x] in R[x] that contracts to p .
This can be shown within basic ring theory (cf. Kaplansky, commutative rings). By the way, it says in particular that
in each ﬁber of Spec R[x] → Spec R , one cannot have a chain of primes ideals of length ≥ 2 .
Since an artinian ring (e.g., a ﬁeld) has dimension zero, by induction, one gets the formula: for an artinian ring R,

dim R[x1 , . . . , xn ] = n.

13.2 Local rings
48

13.2. LOCAL RINGS

13.2.1

49

Fundamental theorem

Let (R, m) be a noetherian local ring and I a m -primary ideal (i.e., it sits between some power of m and m ). Let
n
n+1
. That is,
F (t) be the Poincaré series of the associated graded ring grI R = ⊕∞
0 I /I

F (t) =

∞
∑

ℓ(I n /I n+1 )tn

0

where ℓ refers to the length of a module (over an artinian ring (grI R)0 = R/I ). If x1 , . . . , xs generate I, then
their image in I/I 2 have degree 1 and generate grI R as R/I -algebra. By the Hilbert–Serre theorem, F is a rational
function with exactly one pole at t = 1 of order d ≤ s . Since

(1 − t)−d =

)
∞ (
∑
d−1+j j
t
d−1
0

we ﬁnd that the coeﬃcient of tn in F (t) = (1 − t)d F (t)(1 − t)−d is of the form
N
∑
0

)
(
d−1+n−k
nd−1
+ O(nd−2 ).
ak
= (1 − t)d F (t)|t=1
d − 1!
d−1

That is to say, ℓ(I n /I n+1 ) is a polynomial P in n of degree d − 1 . P is called the Hilbert polynomial of grI R .
We set d(R) = d . We also set δ(R) to be the minimum number of elements of R that can generate a m -primary
ideal of R. Our ambition is to prove the fundamental theorem:

δ(R) = d(R) = dim R
Since we can take s to be δ(R) , we already have δ(R) ≥ d(R) from the above. Next we prove d(R) ≥ dim R by
induction on d(R) . Let p0 ⊊ · · · ⊊ pm be a chain of prime ideals in R. Let D = R/p0 and x a nonzero nonunit
element in D. Since x is not a zero-divisor, we have the exact sequence

x

0 → D→D → D/xD → 0
The degree bound of the Hilbert-Samuel polynomial now implies that d(D) > d(D/xD) ≥ d(R/p1 ) . (This
essentially follows from the Artin-Rees lemma; see Hilbert-Samuel function for the statement and the proof.) In
R/p1 , the chain pi becomes a chain of length m − 1 and so, by inductive hypothesis and again by the degree
estimate,

m − 1 ≤ dim(R/p1 ) ≤ d(R/p1 ) ≤ d(D) − 1 ≤ d(R) − 1
The claim follows. It now remains to show dim R ≥ δ(R). More precisely, we shall show:
Lemma: R contains elements x1 , . . . , xs such that, for any i, any prime ideal containing (x1 , . . . , xi )
has height ≥ i .
(Notice: (x1 , . . . , xs ) is then m -primary.) The proof is omitted. It appears, for example, in Atiyah–MacDonald.
But it can also be supplied privately; the idea is to use prime avoidance.

13.2.2

Consequences of the fundamental theorem

Let (R, m) be a noetherian local ring and put k = R/m . Then

50

CHAPTER 13. DIMENSION THEORY (ALGEBRA)
• dim R ≤ dimk m/m2 , since a basis of m/m2 lifts to a generating set of m by Nakayama. If the equality holds,
then R is called a regular local ring.
b = dim R , since gr R = gr R
b.
• dim R
• (Krull’s principal ideal theorem) The height of the ideal generated by elements x1 , . . . , xs in a noetherian ring
is at most s. Conversely, a prime ideal of height s can be generated by s elements. (Proof: Let p be a prime
ideal minimal over such an ideal. Then s ≥ dim Rp = ht p . The converse was shown in the course of the
proof of the fundamental theorem.)

Theorem — If A → B is a morphism of noetherian local rings, then
dim B/mA B ≥ dim B − dim A. [1]
The equality holds if A → B is ﬂat or more generally if it has the going-down property.
Proof: Let x1 , . . . , xn generate a mA -primary ideal and y1 , . . . , ym be such that their images generate a mB /mA B
-primary ideal. Then mB s ⊂ (y1 , . . . , ym ) + mA B for some s. Raising both sides to higher powers, we see some
power of mB is contained in (y1 , . . . , ym , x1 , . . . , xn ) ; i.e., the latter ideal is mB -primary; thus, m + n ≥ dim B .
The equality is a straightforward application of the going-down property. □
Proposition — If R is a noetherian ring, then
dim R + 1 = dim R[x] = dim R[[x]]
Proof: If p0 ⊊ p1 ⊊ · · · ⊊ pn are a chain of prime ideals in R, then pi R[x] are a chain of prime ideals in R[x] while
pn R[x] is not a maximal ideal. Thus, dim R + 1 ≤ dim R[x] . For the reverse inequality, let m be a maximal ideal of
R[x] and p = R ∩ m . Clearly, R[x]m = Rp [x]m . Since R[x]m /pRp R[x]m = (Rp /pRp )[x]m is then a localization
of a principal ideal domain and has dimension at most one, we get 1 + dim R ≥ 1 + dim Rp ≥ dim R[x]m by the
previous inequality. Since m is arbitrary, it follows 1 + dim R ≥ dim R[x] . □

13.2.3

Nagata’s altitude formula

Theorem — Let R ⊂ R′ be integral domains, p′ ⊂ R′ be a prime ideal and p = R ∩ p′ . If R is a Noetherian ring,
then
dim Rp′ ′ + tr. degR/p R′ /p′ ≤ dim Rp + tr. degR R′
where the equality holds if either (a) R is universally catenary and R ' is ﬁnitely generated R-algebra or (b) R ' is a
polynomial ring over R.
Proof:[2] First suppose R′ is a polynomial ring. By induction on the number of variables, it is enough to consider the
case R′ = R[x] . Since R ' is ﬂat over R,
dim Rp′ ′ = dim Rp + dim κ(p) ⊗R R′ p′
By Noether’s normalization lemma, the second term on the right side is:
dim κ(p) ⊗R R′ − dim κ(p) ⊗R R′ /p′ = 1 − tr. degκ(p) κ(p′ ) = tr. degR R′ − tr. deg κ(p′ ).
Next, suppose R′ is generated by a single element; thus, R′ = R[x]/I . If I = 0, then we are already done. Suppose
not. Then R′ is algebraic over R and so tr. degR R′ = 0 . Since R is a subring of R ', I ∩ R = 0 and so ht I =
dim R[x]I = dim Q(R)[x]I = 1 − tr. degQ(R) κ(I) = 1 since κ(I) = Q(R′ ) is algebraic over Q(R) . Let p′c
denote the pre-image in R[x] of p′ . Then, as κ(p′c ) = κ(p) , by the polynomial case,
ht p′ = ht p′c /I ≤ ht p′c − ht I = dim Rp − tr. degκ(p) κ(p′ ).
Here, note that the inequality is the equality if R ' is catenary. Finally, working with a chain of prime ideals, it is
straightforward to reduce the general case to the above case. □

13.3. HOMOLOGICAL METHODS

51

13.3 Homological methods
13.3.1

Regular rings

Let R be a noetherian ring. The projective dimension of a ﬁnite R-module M is the shortest length of any projective
resolution of R (possibly inﬁnite) and is denoted by pdR M . We set gl. dim R = sup{pdR M |module ﬁnite a is M}
; it is called the global dimension of R.
Assume R is local with residue ﬁeld k.
Lemma — pdR k = gl. dim R (possibly inﬁnite).
Proof: We claim: for any ﬁnite R-module M,

pdR M ≤ n ⇔ TorR
n+1 (M, k) = 0
By dimension shifting (cf. the proof of Theorem of Serre below), it is enough to prove this for n = 0 . But then, by
the local criterion for ﬂatness, TorR
1 (M, k) = 0 ⇒ M ﬂat ⇒ M free ⇒ pdR (M ) ≤ 0. Now,
gl. dim R ≤ n ⇒ pdR k ≤ n ⇒ TorR
n+1 (−, k) = 0 ⇒ pdR − ≤ n ⇒ gl. dim R ≤ n,
completing the proof. □
Remark: The proof also shows that pdR K = pdR M − 1 if M is not free and K is the kernel of some surjection
from a free module to M.
Lemma — Let R1 = R/f R , f a non-zerodivisor of R. If f is a non-zerodivisor on M, then

pdR M ≥ pdR1 (M ⊗ R1 )
Proof: If pdR M = 0 , then M is R-free and thus M ⊗ R1 is R1 -free. Next suppose pdR M > 0 . Then we have:
pdR K = pdR M − 1 as in the remark above. Thus, by induction, it is enough to consider the case pdR M = 1 .
Then there is a projective resolution: 0 → P1 → P0 → M → 0 , which gives:

TorR
1 (M, R1 ) → P1 ⊗ R1 → P0 ⊗ R1 → M ⊗ R1 → 0
But TorR
1 (M, R1 ) = f M = {m ∈ M |f m = 0} = 0. Hence, pdR (M ⊗ R1 ) is at most 1. □
Theorem of Serre — R regular ⇔ gl. dim R < ∞ ⇔ gl. dim R = dim R.
Proof:[3] If R is regular, we can write k = R/(f1 , . . . , fn ) , fi a regular system of parameters. An exact sequence
f

0 → M →M → M1 → 0 , some f in the maximal ideal, of ﬁnite modules, pdR M < ∞ , gives us:
f

R
R
R
0 = TorR
i+1 (M, k) → Tori+1 (M1 , k) → Tori (M, k)→ Tori (M, k),

i ≥ pdR M.

R
But f here is zero since it kills k. Thus, TorR
i+1 (M1 , k) ≃ Tori (M, k) and consequently pdR M1 = 1 + pdR M .
Using this, we get:

pdR k = 1 + pdR (R/(f1 , . . . , fn−1 )) = · · · = n.
The proof of the converse is by induction on dim R . We begin with the inductive step. Set R1 = R/f1 R , f1 among
a system of parameters. To show R is regular, it is enough to show R1 is regular. But, since dim R1 < dim R , by
inductive hypothesis and the preceding lemma with M = m ,

gl. dim R < ∞ ⇒ gl. dim R1 = pdR1 k ≤ pdR1 m/f1 m < ∞ ⇒ R1 regular .

52

CHAPTER 13. DIMENSION THEORY (ALGEBRA)

The basic step remains. Suppose dim R = 0 . We claim gl. dim R = 0 if it is ﬁnite. (This would imply that R is a
semisimple local ring; i.e., a ﬁeld.) If that is not the case, then there is some ﬁnite module M with 0 < pdR M < ∞
and thus in fact we can ﬁnd M with pdR M = 1 . By Nakayama’s lemma, there is a surjection F → M from a free
module F to M whose kernel K is contained in mF . Since dim R = 0 , the maximal ideal m is an associated prime
of R; i.e., m = ann(s) for some nonzero s in R. Since K ⊂ mF , sK = 0 . Since K is not zero and is free, this
implies s = 0 , which is absurd. □
Corollary — A regular local ring is a unique factorization domain.
Proof: Let R be a regular local ring. Then gr R ≃ k[x1 , . . . , xd ] , which is an integrally closed domain. It is a standard
algebra exercise to show this implies that R is an integrally closed domain. Now, we need to show every divisorial ideal
is principal; i.e., the divisor class group of R vanishes. But, according to Bourbaki, Algèbre commutative, chapitre 7,
§. 4. Corollary 2 to Proposition 16, a divisorial ideal is principal if it admits a ﬁnite free resolution, which is indeed
the case by the theorem. □
Theorem — Let R be a ring. Then gl. dim R[x1 , . . . , xn ] = gl. dim R + n .

13.3.2

Depths

Let R be a ring and M a module over it. A sequence of elements x1 , . . . , xn in R is called an M-regular sequence if
x1 is not a zero-divisor on M and xi is not a zero divisor on M /(x1 , . . . , xi−1 )M for each i = 2, . . . , n . A priori, it
is not obvious whether any permutation of a regular sequence is still regular (see the section below for some positive
answer.)
Let R be a local Noetherian ring with maximal ideal m and put k = R/m . Then, by deﬁnition, the depth of a ﬁnite Rmodule M is the supremum of the lengths of all M-regular sequences in m . For example, we have depth M = 0 ⇔ m
consists of zerodivisors on M ⇔ m is associated with M. By induction, we ﬁnd
depth M ≤ dim R/p
for any associated primes p of M. In particular, depth M ≤ dim M . If the equality holds for M = R, R is called a
Cohen–Macaulay ring.
Example: A regular Noetherian local ring is Cohen–Macaulay (since a regular system of parameters is an R-regular
sequence.)
In general, a Noetherian ring is called a Cohen–Macaulay ring if the localizations at all maximal ideals are Cohen–
Macaulay. We note that a Cohen–Macaulay ring is universally catenary. This implies for example that a polynomial
ring k[x1 , . . . , xd ] is universally catenary since it is regular and thus Cohen–Macaulay.
Proposition (Rees) — Let M be a ﬁnite R-module. Then depth M = sup{n| ExtiR (k, M ) = 0, i < n} .
More generally, for any ﬁnite R-module N whose support is exactly {m} ,

depth M = sup{n| ExtiR (N, M ) = 0, i < n}
Proof: We ﬁrst prove by induction on n the following statement: for every R-module M and every M-regular sequence
x1 , . . . , xn in m ,

ExtnR (N, M ) ≃ HomR (N, M /(x1 , . . . , xn )M ).
The basic step n = 0 is trivial. Next, by inductive hypothesis, Extn−1
R (N, M ) ≃ HomR (N, M /(x1 , . . . , xn−1 )M )
. But the latter is zero since the annihilator of N contains some power of xn . Thus, from the exact sequence
x1
0 → M →M
→ M1 → 0 and the fact that x1 kills N, using the inductive hypothesis again, we get
ExtnR (N, M ) ≃ Extn−1
R (N, M /x1 M ) ≃ HomR (N, M /(x1 , . . . , xn )M )
proving (*). Now, if n < depth M , then we can ﬁnd an M-regular sequence of length more than n and so by (*) we
see ExtnR (N, M ) = 0 . It remains to show ExtnR (N, M ) ̸= 0 if n = depth M . By (*) we can assume n = 0. Then

13.3. HOMOLOGICAL METHODS

53

m is associated with M; thus is in the support of M. On the other hand, m ∈ Supp(N ). It follows by linear algebra
that there is a nonzero homomorphism from N to M modulo m ; hence, one from N to M by Nakayama’s lemma. □
The Auslander–Buchsbaum formula relates depth and projective dimension.
Theorem — Let M be a ﬁnite module over a noetherian local ring R. If pdR M < ∞ , then

pdR M + depth M = depth R.
Proof: We argue by induction on pdR M , the basic case (i.e., M free) being trivial. By Nakayama’s lemma, we
f

have the exact sequence 0 → K →F → M → 0 where F is free and the image of f is contained in mF . Since
pdR K = pdR M − 1, what we need to show is depth K = depth M + 1 . Since f kills k, the exact sequence yields:
for any i,

ExtiR (k, F ) → ExtiR (k, M ) → Exti+1
R (k, K) → 0.
Note the left-most term is zero if i < depth R . If i < depth K − 1 , then since depth K ≤ depth R by inductive
i
hypothesis, we see ExtiR (k, M ) = 0. If i = depth K − 1 , then Exti+1
R (k, K) ̸= 0 and it must be ExtR (k, M ) ̸= 0.
□
As a matter of notation, for any R-module M, we let

Γm (M ) = {s ∈ M | supp(s) ⊂ {m}} = {s ∈ M |mj s = 0 some for j}.
j
= Rj Γm be its j-th right derived functor,
One sees without diﬃculty that Γm is a left-exact functor and then let Hm
j
called the local cohomology of R. Since Γm (M ) = lim HomR (R/m , M ) , via abstract nonsense,
−→

i
(M ) = lim ExtiR (R/mj , M )
Hm
−→

This observation proves the ﬁrst part of the theorem below.
Theorem (Grothendieck) — Let M be a ﬁnite R-module. Then
i
1. depth M = sup{n|Hm
(M ) = 0, i < n} .
i
2. Hm
(M ) = 0, i > dim M and ̸= 0 if i = dim M.

3. If R is complete and d its Krull dimension and if E is the injective hull of k, then
d
(−), E)
HomR (Hm

is representable (the representing object is sometimes called the canonical module especially
if R is Cohen–Macaulay.)
Proof: 1. is already noted (except to show the nonvanishing at the degree equal to the depth of M; use induction
to see this) and 3. is a general fact by abstract nonsense. 2. is a consequence of an explicit computation of a local
cohomology by means of Koszul complexes (see below). □

13.3.3

Koszul complex

Main article: Koszul complex
Let R be a ring and x an element in it. We form the chain complex K(x) given by K(x)i = R for i = 0, 1 and
K(x)i = 0 for any other i with the diﬀerential

54

CHAPTER 13. DIMENSION THEORY (ALGEBRA)

d : K1 (R) → K0 (R), r 7→ xr.
For any R-module M, we then get the complex K(x, M ) = K(x)⊗R M with the diﬀerential d⊗1 and let H∗ (x, M ) =
H∗ (K(x, M )) be its homology. Note:

H0 (x, M ) = M /xM
H1 (x, M ) = x M = {m ∈ M |xm = 0}
More generally, given a ﬁnite sequence x1 , . . . , xn of elements in a ring R, we form the tensor product of complexes:

K(x1 , . . . , xn ) = K(x1 ) ⊗ · · · ⊗ K(xn )
and let H∗ (x1 , . . . , xn , M ) = H∗ (K(x1 , . . . , xn , M )) its homology. As before,

H0 (x, M ) = M /(x1 , . . . , xn )M
Hn (x, M ) = AnnM ((x1 , . . . , xn ))
We now have the homological characterization of a regular sequence.
Theorem — Suppose R is Noetherian, M is a ﬁnite module over R and xi are in the Jacobson radical of R. Then the
following are equivalent
(i) x is an M-regular sequence.
(ii) Hi (x, M ) = 0, i ≥ 1 .
(iii) H1 (x, M ) = 0 .
Corollary — The sequence xi is M-regular if and only if any of its permutations is so.
Corollary — If x1 , . . . , xn is an M-regular sequence, then xj1 , . . . , xjn is also an M-regular sequence for each positive
integer j.
A Koszul complex is a powerful computational tool. For instance, it follows from the theorem and the corollary

Him (M ) ≃ lim Hi (K(xj1 , . . . , xjn ; M ))
−→
(Here, one uses the self-duality of a Koszul complex; see Proposition 17.15. of Eisenbud, Commutative Algebra with
a View Toward Algebraic Geometry.)
Another instance would be
Theorem — Assume R is local. Then let

s = dimk m/m2
the dimension of the Zariski tangent space (often called the embedding dimension of R). Then
( )
s
≤ dimk TorR
i (k, k)
i
Remark: The theorem can be used to give a second quick proof of Serre’s theorem, that R is regular if and only
if it has ﬁnite global dimension. Indeed, by the above theorem, TorR
s (k, k) ̸= 0 and thus gl. dim R ≥ s . On

13.4. MULTIPLICITY THEORY

55

the other hand, as gl. dim R = pdR k , the Auslander–Buchsbaum formula gives gl. dim R = dim R . Hence,
dim R ≤ s ≤ gl. dim R = dim R .
We next use a Koszul homology to deﬁne and study complete intersection rings. Let R be a Noetherian local ring.
By deﬁnition, the ﬁrst deviation of R is the vector space dimension

ϵ1 (R) = dimk H1 (x)
where x = (x1 , . . . , xd ) is a system of parameters. By deﬁnition, R is a complete intersection ring if dim R + ϵ1 (R)
is the dimension of the tangent space. (See Hartshorne for a geometric meaning.)
Theorem — R is a complete intersection ring if and only if its Koszul algebra is an exterior algebra.

13.3.4

Injective dimension and Tor dimensions

Let R be a ring. The injective dimension of an R-module M denoted by idR M is deﬁned just like a projective
dimension: it is the minimal length of an injective resolution of M. Let ModR be the category of R-modules.
Theorem — For any ring R,
gl. dim R = sup{idR M |M ∈ ModR }
= inf{n| ExtiR (M, N ) = 0, i > n, M, N ∈ ModR }
Proof: Suppose gl. dim R ≤ n . Let M be an R-module and consider a resolution
ϕ0

ϕn−1

0 → M → I0 →I1 → · · · → In−1 → N → 0
where Ii are injective modules. For any ideal I,

Ext1R (R/I, N ) ≃ Ext2R (R/I, ker(ϕn−1 )) ≃ · · · ≃ Extn+1
R (R/I, M ),
which is zero since Extn+1
R (R/I, −) is computed via a projective resolution of R/I . Thus, by Baer’s criterion, N
is injective. We conclude that sup{idR M |M } ≤ n . Essentially by reversing the arrows, one can also prove the
implication in the other way. □
The theorem suggests that we consider a sort of a dual of a global dimension:

w. gl. dim = inf{n| TorR
i (M, N ) = 0, i > n, M, N ∈ ModR }
It was originally called the weak global dimension of R but today it is more commonly called the Tor dimension of R.
Remark: for any ring R, w. gl. dim R ≤ gl. dim R .
Proposition — A ring has weak global dimension zero if and only if it is von Neumann regular.

13.4 Multiplicity theory
See also: intersection theory

13.5 Dimensions of non-commutative rings
Let A be a graded algebra over a ﬁeld k. If V is a ﬁnite-dimensional generating subspace of A, then we let f (n) =
dimk V n and then put

56

CHAPTER 13. DIMENSION THEORY (ALGEBRA)

gk(A) = lim sup
n→∞

log f (n)
log n

It is called the Gelfand–Kirillov dimension of A. It is easy to show gk(A) is independent of a choice of V.
Example: If A is ﬁnite-dimensional, then gk(A) = 0. If A is an aﬃne ring, then gk(A) = Krull dimension of A.
Bernstein’s inequality — See
See also: Goldie dimension, Krull–Gabriel dimension.

13.6 See also
• Bass number
• Perfect complex
• amplitude

13.7 Notes
[1] Eisenbud, Theorem 10.10
[2] Matsumura, Theorem 15.5.
[3] Weibel 1994, Theorem 4.4.16

13.8 References
• Bruns, Winfried; Herzog, Jürgen (1993), Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics
39, Cambridge University Press, ISBN 978-0-521-41068-7, MR 1251956
• Part II of Eisenbud, David (1995), Commutative algebra. With a view toward algebraic geometry, Graduate
Texts in Mathematics 150, New York: Springer-Verlag, ISBN 0-387-94268-8, MR 1322960.
• Chapter 10 of Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8.
• Kaplansky, Irving, Commutative rings, Allyn and Bacon, 1970.
• H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge
Studies in Advanced Mathematics, 8.
• Serre, Jean-Pierre (1975), Algèbre locale. Multiplicités, Cours au Collège de France, 1957-−1958, rédigé par
Pierre Gabriel. Troisième édition, 1975. Lecture Notes in Mathematics (in French) 11, Berlin, New York:
Springer-Verlag
• Weibel, Charles A. (1995). An Introduction to Homological Algebra. Cambridge University Press.

Chapter 14

Dimensional metrology
Dimensional Metrology is the science of calibrating and using physical measurement equipment to quantify the
physical size of or distance from any given object. Inspection is a critical step in product development and quality
control. Dimensional Metrology requires the use of a variety of physical scales to determine dimension, with the
most accurate of these being holographic etalons or laser interferometers. The realization of dimension using these
accurate scale technologies is the end goal of dimensional metrologists.
Early Mesopotamian and Egyptian metrologists created a set of measurement standards based on body measures
such as ﬁngers, palms, hands, feet, Cubits, and paces and agricultural measures such as feet, yards, paces, fathoms,
rods, cords, perch, stadia, miles and degrees of the Earth’s circumference. Early Egyptian rulers based on units of
ﬁngers, palms and feet based on inscription grids that incorporated standards of measure as canons of proportion were
made commensurate with Mesopotamian standards based on ﬁngers, hands and feet so that four palms or three hands
equaled one foot and ten hands equaled one meter. These standards which were used to measure and deﬁne property
such as buildings and ﬁelds were adopted by the Greeks, Romans and Persians as legal standards and became the
basis of European standards of measure. They were also used to relate length to area with units such as the khet, setat
and aroura, area to volume with units such as the artaba and space to time with units such as the Egyptian minute of
march, the itrw which recorded an hours travel on a river, and the days sail. Specialized units for carpenters, masons
and other craftsmen such as the remen were worked into a system of unit fractions that allowed calculations utilizing
analytic geometry. Carpenters and surveyors were some of the ﬁrst dimensional inspectors.
Modern measurement equipment include hand tools, CMMs (Coordinate-Measurement Machine), machine vision
systems, laser trackers, and optical comparators. For hand tools, see Caliper and micrometer. A CMM is based on
CNC technology to automate measurement of Cartesian coordinates using a touch probe, contact scanning probe, or
non-contact sensor. Optical comparators are used when physically touching the part is undesirable. Optical comparators can now build 3D models of a scanned part and internal passages using x-ray technology. Furthermore, optical
3D (laser) scanners are becoming more and common. By using a light sensitive detector (e.g. digital camera) and a
light source (laser, line projector) the triangulation principle is employed to generate 3D data, which is evaluated in
order to compare the measures against nominal geometries.
Data is collected in or compared to a print. A print is a blueprint illustrating crucial features. Prints can be hand
drawn or automatically generated by a CAD model.

14.1 See also
• Mechanical Engineering
Industrial Metrology for manufacturing quality and for Standards room/calibration activities are called as ﬁrst principle
method of deriving the actual value of measurement. the ﬁeld of in process gauging, incycle gauging and post process
gauging are excluded from this topic
Equipment generally used for manufacturing quality depends on industry, but broadly around mechanical engineering
and in particular automotive aerospace, machine tool, any other precision parts suitable for instrumentation. This is
broadly listed below:
1. Basic hand held instruments like Vernier Caliper, digital caliper, micrometer etc.
57

58

CHAPTER 14. DIMENSIONAL METROLOGY

14.2 References
14.3 Notes
14.4 External links
• An example of Industrial Metrology equipment.

14.5 Further reading
• Doiron, T. (2007). “20 °C—A Short History of the Standard Reference Temperature for Industrial Dimensional Measurements” (PDF). Journal of Research of the National Institute of Standards and Technology (National Institute of Science and Technology) 112 (1): 1. doi:10.6028/jres.112.001.

Chapter 15

Eight-dimensional space
In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 8,
the set of all such locations is called 8-dimensional space. Often such spaces are studied as vector spaces, without
any notion of distance. Eight-dimensional Euclidean space is eight-dimensional space equipped with a Euclidean
metric, which is deﬁned by the dot product.
More generally the term may refer to an eight-dimensional vector space over any ﬁeld, such as an eight-dimensional
complex vector space, which has 16 real dimensions. It may also refer to an eight-dimensional manifold such as an
8-sphere, or a variety of other geometric constructions.

15.1 Geometry
15.1.1

8-polytope

Main article: 8-polytope
A polytope in eight dimensions is called an 8-polytope. The most studied are the regular polytopes, of which there
are only three in eight dimensions: the 8-simplex, 8-cube, and 8-orthoplex. A broader family are the uniform 8polytopes, constructed from fundamental symmetry domains of reﬂection, each domain deﬁned by a Coxeter group.
Each uniform polytope is deﬁned by a ringed Coxeter-Dynkin diagram. The 8-demicube is a unique polytope from
the D8 family, and 421 , 241 , and 142 polytopes from the E8 family.

15.1.2

7-sphere

The 7-sphere or hypersphere in eight dimensions is the seven-dimensional surface equidistant from a point, e.g. the
origin. It has symbol S7 , with formal deﬁnition for the 7-sphere with radius r of
{
}
S 7 = x ∈ R8 : ∥x∥ = r .
The volume of the space bounded by this 7-sphere is

V8 =

π4 8
R
24

which is 4.05871 × r8 , or 0.01585 of the 8-cube that contains the 7-sphere.

15.1.3

Kissing number problem

Main article: Kissing number problem

59

60

CHAPTER 15. EIGHT-DIMENSIONAL SPACE

The kissing number problem has been solved in eight dimensions, thanks to the existence of the 421 polytope and its
associated lattice. The kissing number in eight dimensions is 240.

15.2 Octonions
Main article: Octonion
The octonions are a normed division algebra over the real numbers, the largest such algebra. Mathematically they
can be speciﬁed by 8-tuplets of real numbers, so form an 8-dimensional vector space over the reals, with addition of
vectors being the addition in the algebra. A normed algebra is one with a product that satisﬁes

∥xy∥ ≤ ∥x∥∥y∥
for all x and y in the algebra. A normed division algebra additionally must be ﬁnite-dimensional, and have the
property that every non-zero vector has a unique multiplicative inverse. Hurwitz’s theorem prohibits such a structure
from existing in dimensions other than 1, 2, 4, or 8.

15.3 Biquaternions
The complexiﬁed quaternions C ⊗ H , or "biquaternions,” are an eight-dimensional algebra dating to William Rowan
Hamilton's work in the 1850s. This algebra is equivalent (that is, isomorphic) to the Cliﬀord algebra Cℓ2 (C) and the
Pauli algebra. It has also been proposed as a practical or pedagogical tool for doing calculations in special relativity,
and in that context goes by the name Algebra of physical space (not to be confused with the Spacetime algebra, which
is 16-dimensional.)

15.4 References
• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony
C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 Wiley::Kaleidoscopes:
Selected Writings of H.S.M. Coxeter
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR
2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Table of the Highest Kissing Numbers Presently Known maintained by Gabriele Nebe and Neil Sloane (lower
bounds)
• Conway, John Horton; Smith, Derek A. (2003), On Quaternions and Octonions: Their Geometry, Arithmetic,
and Symmetry, A. K. Peters, Ltd., ISBN 1-56881-134-9. (Review).
• Duplij, Steven; Siegel, Warren; Bagger, Jonathan, eds. (2005), Concise Encyclopedia of Supersymmetry And
Noncommutative Structures in Mathematics and Physics, Berlin, New York: Springer, ISBN 978-1-4020-13386 (Second printing)

Chapter 16

Exterior dimension
In geometry, exterior dimension is a type of dimension that can be used to characterize fat fractals.
A fat fractal is a Cantor set with Lebesgue measure (an extension of the classical notions of length and area to more
complicated sets) greater than 0.
The Cantor set, sometimes also called the Cantor comb or no-middle-third set (Cullen 1968, pp. 78–81), is given
by taking the interval (set), removing the open middle third, removing the middle third of each of the two remaining
pieces, and continuing this procedure ad inﬁnitum. It is therefore the set of points in the interval whose ternary
expansions do not contain 1.
Iterating the process 1 -> 101, 0 -> 000 starting with 1 gives the sequence 1, 101, 101000101, 101000101000000000101000101,
.... The sequence of binary bits thus produced is therefore 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0,
0, 1, 0, 1, 0, ... (Sloane’s A088917) whose nth term is D(n, n) = Pn (mod 3), where D(n, n) is a (central) Delannoy
number and Pn(x) is a Legendre polynomial (E. W. Weisstein, Apr. 9, 2006).

16.1 References
• Cullen, H. F. (1968). Introduction to General Topology. Heath.

61

Chapter 17

Five-dimensional space
This article is about the hypothetical extra dimension. For the musical group, see The 5th Dimension.
A ﬁve-dimensional space is a space with ﬁve dimensions. If interpreted physically, that is one more than the usual

A perspective projection 3D to 2D of stereographic projection 4D to 3D of Schlegel diagram 5D to 4D of the 5-cube (or penteract)

three spatial dimensions and the fourth dimension of time used in relativitistic physics.[1] It is an abstraction which
62

17.1. PHYSICS

63

occurs frequently in mathematics, where it is a legitimate construct. In physics and mathematics, a sequence of N
numbers can be understood to represent a location in an N-dimensional space. Whether or not the real universe in
which we live is somehow ﬁve-dimensional is a topic of debate.

17.1 Physics
Much of the early work on ﬁve dimensional space was in an attempt to develop a theory that uniﬁes the four
fundamental forces in nature: strong and weak nuclear forces, gravity and electromagnetism. German mathematician
Theodor Kaluza and Swedish physicist Oskar Klein independently developed the Kaluza–Klein theory in 1921, which
used the ﬁfth dimension to unify gravity with electromagnetic force. Although their approaches were later found to
be at least partially inaccurate, their concept provided the basis for further research over the past century.[1]
To explain why this dimension would not be directly observable, Klein suggested that the ﬁfth dimension would be
rolled up into a tiny, compact loop on the order of 10−33 centimeters.[1] Under his reasoning, he envisioned light
as a disturbance caused by rippling in the higher dimension just beyond human perception, similar to how ﬁsh in
a pond can only see shadows of ripples across the surface of the water caused by raindrops.[2] As such, while not
detectable, it would provide a means of uniﬁcation. Kaluza-Klein theory experienced a revival in the 1970s due to
the emergence of superstring theory and supergravity: the concept that reality is composed of vibrating strands of
energy, a postulate only mathematically viable in ten dimensions or more. Superstring theory then evolved into a more
generalized approach known as M-theory. M-theory suggested a potentially observable extra dimension in addition
to the ten essential dimensions which would allow for the existence of superstrings. The other 10 dimensions are
compacted, or “rolled up”, to a size below the subatomic level.[1][2] Kaluza–Klein theory today is seen as essentially
a gauge theory, with the gauge being the circle group.
The ﬁfth dimension is diﬃcult to directly observe, though the Large Hadron Collider provides an opportunity for
indirect evidence of its existence.[1] Physicists theorize that collisions of subatomic particles in turn produce new
particles as a result of the collision, including a graviton that escapes from the fourth dimension, or brane, leaking
oﬀ into a ﬁve-dimensional bulk.[3] M-theory would explain the weakness of gravity relative to the other fundamental
forces of nature, as can be seen, for example, when using a magnet to lift a pin oﬀ a table — the magnet is able to
overcome the gravitational pull of the entire earth with ease.[1]
Mathematical approaches were developed in the early 20th century that viewed the ﬁfth dimension as a theoretical
construct. These theories make reference to Hilbert space, a mathematical concept that postulates an inﬁnite number
of mathematical dimensions to allow for a limitless number of quantum states. Einstein, Bergmann and Bargmann
later tried to extend the four-dimensional spacetime of general relativity into an extra physical dimension to incorporate electromagnetism, though they were unsuccessful.[1] In their 1938 paper, Einstein and Bergmann were among
the ﬁrst to introduce the modern viewpoint that a four-dimensional theory, which coincides with Einstein-Maxwell
theory at long distances, is derived from a ﬁve-dimensional theory with complete symmetry in all ﬁve dimensions.
They suggested that electromagnetism resulted from a gravitational ﬁeld that is “polarized” in the ﬁfth dimension.[4]
The main novelty of Einstein and Bergmann was to seriously consider the ﬁfth dimension as a physical entity, rather
than an excuse to combine the metric tensor and the electromagnetic potential. But they then reneged, modifying the
theory to break the ﬁve-dimensional symmetry. Their reasoning, as suggested by Edward Witten, was that the more
symmetric version of the theory predicted the existence of a new long range ﬁeld, one that was both massless and
scalar, which would have required a fundamental modiﬁcation to Einstein’s theory of general relativity.[5] Minkowski
space and Maxwell’s equations in vacuum can be embedded in a ﬁve-dimensional Riemann curvature tensor.
In 1993, the physicist Gerard 't Hooft put forward the holographic principle, which explains that the information
about an extra dimension is visible as a curvature in a spacetime with one fewer dimension. For example, holograms
are three-dimensional pictures placed on a two-dimensional surface, which gives the image a curvature when the
observer moves. Similarly, in general relativity, the fourth dimension is manifested in observable three dimensions
as the curvature path of a moving inﬁnitesimal (test) particle. Hooft has speculated that the ﬁfth dimension is really
the spacetime fabric.

17.2 Five-dimensional geometry
According to Klein’s deﬁnition, “a geometry is the study of the invariant properties of a spacetime, under transformations within itself.” Therefore, the geometry of the 5th dimension studies the invariant properties of such space-time,

64

CHAPTER 17. FIVE-DIMENSIONAL SPACE

as we move within it, expressed in formal equations.[6]

17.2.1

Polytopes

Main article: 5-polytope
In ﬁve or more dimensions, only three regular polytopes exist. In ﬁve dimensions, they are:
1. The 5-simplex of the simplex family, with 6 vertices, 15 edges, 20 faces (each an equilateral triangle), 15 cells
(each a regular tetrahedron), and 6 hypercells (each a 5-cell).
2. The 5-cube of the hypercube family, with 32 vertices, 80 edges, 80 faces (each a square), 40 cells (each a
cube), and 10 hypercells (each a tesseract).
3. The 5-orthoplex of the cross polytope family, with 10 vertices, 40 edges, 80 faces (each a triangle), 80 cells
(each a tetrahedron), and 32 hypercells (each a 5-cell).
A fourth polytope, a demihypercube, can be constructed as an alternation of the 5-cube, and is called a 5-demicube,
with half the vertices (16), bounded by alternating 5-cell and 16-cell hypercells.

17.2.2

Hypersphere

A hypersphere in 5-space (also called a 4-sphere due to its surface being 4-dimensional) consists of the set of all
points in 5-space at a ﬁxed distance r from a central point P. The hypervolume enclosed by this hypersurface is:

V =

8π 2 r5
15

17.3 References
[1] Paul Halpern (April 3, 2014). “How Many Dimensions Does the Universe Really Have”. Public Broadcasting Service.
Retrieved September 12, 2015.
[2] Oulette, Jennifer (March 6, 2011). “Black Holes on a String in the Fifth Dimension”. Discovery News. Retrieved September 12, 2015.
[3] Boyle, Alan (June 6, 2006). “Physicists probe ﬁfth dimension”. NBC news. Retrieved September 12, 2015.
[4] Einstein, Albert; Bergmann, Peter (1938). “On A Generalization Of Kaluza’s Theory Of Electricity”. Annals of Mathematics 39: 683.
[5] Witten, Edward (January 31, 2014). “A Note On Einstein, Bergmann, and the Fifth Dimension”. Institute for Advanced
Study. Retrieved September 12, 2015.
[6] Sancho, Luis (October 4, 2011). Absolute Relativity: The 5th dimension (abridged). p. 442.

17.4 See also
• 5-manifold
• Hypersphere
• List of regular 5-polytopes

17.5. FURTHER READING

65

17.5 Further reading
• Wesson, Paul S. (1999). Space-Time-Matter, Modern Kaluza-Klein Theory. Singapore: World Scientiﬁc. ISBN
981-02-3588-7.
• Wesson, Paul S. (2006). Five-Dimensional Physics: Classical and Quantum Consequences of Kaluza-Klein
Cosmology. Singapore: World Scientiﬁc. ISBN 981-256-661-9.
• Weyl, Hermann, Raum, Zeit, Materie, 1918. 5 edns. to 1922 ed. with notes by Jūrgen Ehlers, 1980. trans. 4th
edn. Henry Brose, 1922 Space Time Matter, Methuen, rept. 1952 Dover. ISBN 0-486-60267-2.

17.6 External links
• Anaglyph of a ﬁve dimensional hypercube in hyper perspective

Chapter 18

Flatland
This article is about the novella. For other uses, see Flatland (disambiguation).
Flatland: A Romance of Many Dimensions is an 1884 satirical novella by the English schoolmaster Edwin Abbott
Abbott.
Writing pseudonymously as “A Square”,[1] the book used the ﬁctional two-dimensional world of Flatland to comment
on the hierarchy of Victorian culture, but the novella’s more enduring contribution is its examination of dimensions.[2]
Several ﬁlms have been made from the story, including the feature ﬁlm Flatland (2007). Other eﬀorts have been
short or experimental ﬁlms, including one narrated by Dudley Moore and the short ﬁlms Flatland: The Movie (2007)
and Flatland 2: Sphereland (2012) starring Martin Sheen and Kristen Bell.[3]

18.1 Plot
The story describes a two-dimensional world occupied by geometric ﬁgures, whereof women are simple line-segments,
while men are polygons with various numbers of sides. The narrator is a square named A Square, a member of the
caste of gentlemen and professionals, who guides the readers through some of the implications of life in two dimensions. The ﬁrst half of the story goes through the practicalities of existing in a two-dimensional universe as well as a
history leading up to the year 1999 on the eve of the 3rd Millennium.
On New Year’s Eve, the Square dreams about a visit to a one-dimensional world (Lineland) inhabited by “lustrous
points”, in which he attempts to convince the realm’s monarch of a second dimension; but is unable to do so. In the
end, the monarch of Lineland tries to kill A Square rather than tolerate his nonsense any further.
Following this vision, he is himself visited by a three-dimensional sphere named A Sphere, which he cannot comprehend until he sees Spaceland (a tridimensional world) for himself. This Sphere visits Flatland at the turn of each
millennium to introduce a new apostle to the idea of a third dimension in the hopes of eventually educating the population of Flatland. From the safety of Spaceland, they are able to observe the leaders of Flatland secretly acknowledging
the existence of the sphere and prescribing the silencing of anyone found preaching the truth of Spaceland and the
third dimension. After this proclamation is made, many witnesses are massacred or imprisoned (according to caste),
including A Square’s brother, B.
After the Square’s mind is opened to new dimensions, he tries to convince the Sphere of the theoretical possibility of
the existence of a fourth (and ﬁfth, and sixth ...) spatial dimension; but the Sphere returns his student to Flatland in
disgrace.
The Square then has a dream in which the Sphere visits him again, this time to introduce him to Pointland, whereof
the point (sole inhabitant, monarch, and universe in one) perceives any communication as a thought originating in his
own mind (cf. Solipsism):
“You see,” said my Teacher, “how little your words have done. So far as the Monarch understands
them at all, he accepts them as his own – for he cannot conceive of any other except himself – and
plumes himself upon the variety of Its Thought as an instance of creative Power. Let us leave this God
66

18.2. SOCIAL ELEMENTS

67

Illustration of a simple house in Flatland.

of Pointland to the ignorant fruition of his omnipresence and omniscience: nothing that you or I can do
can rescue him from his self-satisfaction.”[4]
— the Sphere

The Square recognizes the identity of the ignorance of the monarchs of Pointland and Lineland with his own (and
the Sphere’s) previous ignorance of the existence of higher dimensions. Once returned to Flatland, the Square cannot
convince anyone of Spaceland’s existence, especially after oﬃcial decrees are announced that anyone preaching the
existence of three dimensions will be imprisoned (or executed, depending on caste). Eventually the Square himself is
imprisoned for just this reason, with only occasional contact with his brother who is imprisoned in the same facility.
He does not manage to convince his brother, even after all they have both seen. Seven years after being imprisoned,
A Square writes out the book Flatland in the form of a memoir, hoping to keep it as posterity for a future generation
that can see beyond their two-dimensional existence.

18.2 Social elements
Men are portrayed as polygons whose social status is determined by their regularity and the number of their sides,
with a Circle considered the “perfect” shape. On the other hand, females consist only of lines and are required by

68

CHAPTER 18. FLATLAND

law to sound a “peace-cry” as they walk, lest she be mistaken face-to-face for a point. The Square evinces accounts
of cases where women have accidentally or deliberately stabbed men to death, as evidence of the need for separate
doors for women and men in buildings.
In the world of Flatland, classes are distinguished by the “Art of Hearing”, the “Art of Feeling”, and the “Art of Sight
Recognition”. Classes can be distinguished by the sound of one’s voice, but the lower classes have more developed
vocal organs, enabling them to feign the voice of a Polygon or even a Circle. Feeling, practised by the lower classes
and women, determines the conﬁguration of a person by feeling one of its angles. The “Art of Sight Recognition”,
practised by the upper classes, is aided by “Fog”, which allows an observer to determine the depth of an object.
With this, polygons with sharp angles relative to the observer will fade more rapidly than polygons with more gradual
angles. Colour of any kind is banned in Flatland after Isosceles workers painted themselves to impersonate noble
Polygons. The Square describes these events, and the ensuing class war at length.
The population of Flatland can “evolve” through the “Law of Nature”, which states: “a male child shall have one more
side than his father, so that each generation shall rise (as a rule) one step in the scale of development and nobility.
Thus the son of a Square is a Pentagon, the son of a Pentagon, a Hexagon; and so on”.
This rule is not the case when dealing with Isosceles Triangles (Soldiers and Workmen) with only two congruent sides.
The smallest angle of an Isosceles Triangle gains thirty arc minutes (half a degree) each generation. Additionally,
the rule does not seem to apply to many-sided Polygons. For example, the sons of several hundred-sided Polygons
will often develop ﬁfty or more sides more than their parents. Furthermore, the angle of an Isosceles Triangle or the
number of sides of a (regular) Polygon may be altered during life by deeds or surgical adjustments.
An Equilateral Triangle is a member of the craftsman class. Squares and Pentagons are the “gentlemen” class, as
doctors, lawyers, and other professions. Hexagons are the lowest rank of nobility, all the way up to (near) Circles,
who make up the priest class. The higher-order Polygons have much less of a chance of producing sons, preventing
Flatland from being overcrowded with noblemen.
Regular Polygons were considered in isolation until chapter seven of the book when the issue of irregularity, or
physical deformity, became considered. In a two dimensional world a regular polygon can be identiﬁed by a single
angle and/or vertex. In order to maintain social cohesion, irregularity is to be abhorred, with moral irregularity and
criminality cited, “by some” (in the book), as inevitable additional deformities, a sentiment with which the Square
concurs. If the error of deviation is above a stated amount, the irregular Polygon faces euthanasia; if below, he
becomes the lowest rank of civil servant. An irregular Polygon is not destroyed at birth, but allowed to develop to
see if the irregularity can be “cured” or reduced. If the deformity remains, the irregular is “painlessly and mercifully
consumed.”[5]

18.3 As a social satire
In Flatland Abbott describes a society rigidly divided into classes. Social ascent is the main aspiration of its inhabitants, apparently granted to everyone but strictly controlled by the top of the hierarchy. Freedom is despised and
the laws are cruel. Innovators are imprisoned or suppressed. Members of lower classes who are intellectually valuable, and potential leaders of riots, are either killed, or promoted to the higher classes. Every attempt for change is
considered dangerous and harmful. This world, as ours, is not prepared to receive “Revelations from another world”.
The satirical part is mainly concentrated in the ﬁrst part of the book, “This World”, which describes Flatland. The
main points of interest are the Victorian concept on women’s roles in the society and in the class-based hierarchy of
men.[6] Abbott has been accused of misogyny due to his portrait of women in Flatland. In his Preface to the Second
and Revised Edition, 1884, he answers such critics by stating that the Square:

was writing as a Historian, he has identiﬁed himself (perhaps too closely) with the views generally
adopted by Flatland and (as he has been informed) even by Spaceland, Historians; in whose pages (until
very recent times) the destinies of Women and of the masses of mankind have seldom been deemed
worthy of mention and never of careful consideration.
— the Editor

18.4. CRITICAL RECEPTION

69

18.4 Critical reception
Although Flatland was not ignored when it was published,[7] it did not obtain a great success. In the entry on Edwin
Abbott in the Dictionary of National Biography, Flatland is not even mentioned.[2]
The book was discovered again after Albert Einstein's general theory of relativity was published, which introduced the
concept of a fourth dimension. Flatland was mentioned in a letter entitled “Euclid, Newton and Einstein” published
in Nature on February 12, 1920. In this letter Abbott is depicted, in a sense, as a prophet due to his intuition of the
importance of time to explain certain phenomena:[8][9]
Some thirty or more years ago a little jeu d'esprit was written by Dr. Edwin Abbott entitled Flatland.
At the time of its publication it did not attract as much attention as it deserved... If there is motion of
our three-dimensional space relative to the fourth dimension, all the changes we experience and assign
to the ﬂow of time will be due simply to this movement, the whole of the future as well as the past always
existing in the fourth dimension.
— from a “Letter to the Editor” by William Garnett. in Nature on February 12, 1920.

The Oxford Dictionary of National Biography now contains a reference to Flatland.

18.5 Editions in print
• Flatland (5th edition, 1963), 1983 reprint with foreword by Isaac Asimov, HarperCollins, ISBN 0-06-4635732
• bound together back-to-back with Dionys Burger's Sphereland (1994), HarperCollins, ISBN 0-06-2732765
• The Annotated Flatland (2002), coauthor Ian Stewart, Perseus Publishing, ISBN 0-7382-0541-9
• Signet Classics edition (2005), ISBN 0-451-52976-6
• Oxford University Press (2006), ISBN 0-19-280598-3
• Dover Publications thrift edition (2007), ISBN 0-486-27263-X
• CreateSpace edition (2008), ISBN 1-4404-1778-4
• FLATLAND - A Romance of Many Dimensions (The Distinguished Chiron Edition 2015, Chiron Academic
Press), ISBN 978-9187751165 (reproduction of the 1884 edition)

18.6 Adaptations and parodies
Numerous imitations or sequels to Flatland have been written, and multiple other works have alluded to it. Examples
include:

18.6.1

In ﬁlm

Flatland (1965), an animated short ﬁlm based on the novella, was directed by Eric Martin and based on an idea by
John Hubley.[10][11][12]
Flatland (2007), a 98-minute animated independent feature ﬁlm version directed by Ladd Ehlinger Jr,[13] updates the
satire from Victorian England to the modern-day United States.[13]
Flatland: The Movie (2007), by Dano Johnson and Jeﬀrey Travis,[14] is a 34-minute animated educational ﬁlm voice
acted by Martin Sheen, Kristen Bell, Michael York, and Tony Hale.[15] Its sequel was Flatland 2: Sphereland (2012),
inspired by the novel Sphereland by Dionys Burger and starring Kristen Bell, Danny Pudi, Michael York, Tony Hale,
Danica McKellar, and Kate Mulgrew.[16][17][18]

70

CHAPTER 18. FLATLAND

18.6.2

In literature

VAS: An Opera in Flatland is a novel of biotechnology by Steve Tomasula with art and design by Stephen Farrell, and
serves as a kind of adaptation of Flatland. It uses Abbott’s characters Square and Circle and the ﬂat, two-dimensional
world in which they live to critique contemporary society during the rise of genetic engineering and other body
manipulations[19] The text demonstrates a strong correlation between biology and art: “Utilizing a wide and historical
sweep of representations of the body, from pedigree charts to genetic sequences, this hybrid novel recounts how
diﬀering ways of imagining the body generate diﬀering stories of knowledge, power, history, gender, politics, art,
and, of course, the literature of who we are. It is the intersection of one tidy family’s life with the broader times in
which they live. "[20]
An Episode on Flatland: Or How a Plain Folk Discovered the Third Dimension by Charles Howard Hinton (1907),
Sphereland by Dionys Burger (1965), The Planiverse by A. K. Dewdney (1984), Flatterland by Ian Stewart (2001),
and Spaceland by Rudy Rucker (2002). Short stories inspired by Flatland include "The Dot and the Line: A Romance
in Lower Mathematics" by Norton Juster (1963), “The Incredible Umbrella” by Marvin Kaye (1980), and “Message
Found in a Copy of Flatland" by Rudy Rucker (1983)
Physicists and science popularizers Carl Sagan and Stephen Hawking have both commented on and postulated about
the eﬀects of Flatland. Sagan recreates the thought experiment as a set-up to discussing the possibilities of higher
dimensions of the physical universe in both the book and television series Cosmos,[21] whereas Dr. Hawking notes
the impossibility of life in two-dimensional space, as any inhabitants would necessarily be unable to digest their own
food.[22]

18.6.3

In television

Flatland features in The Big Bang Theory episode “The Psychic Vortex”,[23] when Sheldon Cooper declares it one of
his favorite imaginary places to visit.[24]
It also features in the Futurama episode “2-D Blacktop”, when Professor Farnsworth’s adventures in drag racing lead
to a foray of drifting in and out of inter-dimensional spaces.[25]

18.7 See also
• The Planiverse (1984), book by A.K. Dewdney
• Animal Farm (1945), novella by George Orwell
• Blind men and an elephant, Indian parable
• Fourth dimension in literature
• Dimension
• Sphere-world
• Triangle and Robert (1999-2007 webcomic)
• The Dot and the Line (1963 book)
• "—And He Built a Crooked House—" (1941 short story)
• Dimension-bending video games:
•
•
•
•
•
•
•
•

Super Paper Mario (2007)
Crush (2007)
Echochrome (2008)
Lost in Shadow (2010)
Fez (2012)
The Legend of Zelda: A Link Between Worlds (2013)
The Bridge (video game) (2013)
Miegakure (in development)

18.8. REFERENCES

71

18.8 References
• Tuck, Donald H. (1974). The Encyclopedia of Science Fiction and Fantasy. Chicago: Advent. p. 1. ISBN
0-911682-20-1.
[1] Abbott, Edwin A. (1884). Flatland: A Romance in Many Dimensions. New York: Dover Thrift Edition (1992 unabridged).
p. ii.
[2] Stewart, Ian (2008). The Annotated Flatland: A Romance of Many Dimensions. New York: Basic Books. pp. xiii. ISBN
0-465-01123-3.
[3] “Review of Flatland: The Movie and Flatland 2: Sphereland". Science News.
[4] Abbott, Edwin A. (1884) Flatland, Part II, § 20.—How the Sphere encouraged me in a Vision, p 92
[5] Abbott, Edwin A. (1952) [1884], Flatland: A Romance of Many Dimensions (6th ed.), New York: Dover, p. 31, ISBN
0-486-20001-9
[6] Stewart, Ian (2008). The Annotated Flatland: A Romance of Many Dimensions. New York: Basic Books. pp. xvii. ISBN
0-465-01123-3.
[7] “Flatland Reviews”. Retrieved 2011-04-02.
[8] Stewart, Ian (2008). The Annotated Flatland: A Romance of Many Dimensions. New York: Basic Books. p. 11. ISBN
0-465-01123-3.
[9] “Flatland Reviews - Nature, February 1920”. Retrieved 2011-04-02.
[10] Flatland at the Internet Movie Database
[11] “DER Documentary: Flatland”. Retrieved 11 October 2012.
[12] “Flatland Animation: The project”. Retrieved 11 October 2012.
[13] "Flatland the Film". Retrieved 2007-01-14.
[14] "Flatland: The Movie". Retrieved 2007-01-14.
[15] “IMDB Flatland: The Movie”.
[16] "Flatland 2: Sphereland".
[17] Flatland 2: Sphereland at the Internet Movie Database
[18] GeekDad.com Review of Flatland: The Movie and Flatland 2: Sphereland
[19] Vanderborg, Susan (Fall 2008). “Of 'Men and Mutations’: The Art of Reproduction in Fatland”. Journal of Artistic Books
(24): 4-11.
[20] Tomasula, Steve. “VAS”.
[21] Tremlin, Todd (2006). Minds and Gods: The Cognitive Foundations of Religion. USA: Oxford University Press. p. 91.
ISBN 978-0199739011.
[22] Gott, J. Richard (2001-05-21). Time Travel in Einstein’s Universe: The Physical Possibilities of Travel through Time. USA:
Houghton Miﬄin Company. p. 61. ISBN 978-0395955635.
[23] VanDerWerﬀ, Todd. “The Big Bang Theory: “The Psychic Vortex"". A.V. Club. Retrieved 2014-03-14.
[24] “Flatland Featured on The Big Bang Theory on CBS Television”. Giant Screen Cinema Association. Retrieved 2014-03-14.
[25] DeNitto, Nick. “Futurama Invades Flatland”. Stage Buddy. Retrieved 2014-03-14.

18.9 External links
• “Sci-Fri Bookclub”—recording of National Public Radio discussion of Flatland, featuring mathematician Ian
Stewart (Sept. 21, 2012)

72

CHAPTER 18. FLATLAND

18.9.1

Online and downloadable versions of the text

eBooks
• Flatland, a Romance of Many Dimensions (ﬁrst edition) on Wikisource
• Flatland, a Romance of Many Dimensions (second edition) on Wikisource
•
• Flatland at Project Gutenberg, text, no illustrations
•
• Flatland at Project Gutenberg, with ASCII illustrations
• Flatland, digitized copy of the ﬁrst edition from the Internet Archive
• Flatland (Second Edition), Revised with original illustrations (HTML format, one page)
• Flatland (Fifth Edition), Revised, with original illustrations (HTML format, one chapter per page)
• Flatland (Fifth Edition), Revised, with original illustrations (PDF format, all pages, with LaTeX source on
github)
• Flatland (illustrated version) on Manybooks
• Flatland on Open Library at the Internet Archive
Recording
• Flatland audio book (mp3 format from Librivox)

Chapter 19

Four-dimensional space

3D projection of a tesseract undergoing a simple rotation in four dimensional space.

In mathematics, four-dimensional space (“4D”) is a geometric space with four dimensions. It typically is more
speciﬁcally four-dimensional Euclidean space, generalizing the rules of three-dimensional Euclidean space. It has
been studied by mathematicians and philosophers for over two centuries, both for its own interest and for the insights
73

74

CHAPTER 19. FOUR-DIMENSIONAL SPACE

it oﬀered into mathematics and related ﬁelds.
Algebraically, it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions.
In particular a vector with four elements (a 4-tuple) can be used to represent a position in four-dimensional space.
The space is a Euclidean space, so has a metric and norm, and so all directions are treated as the same: the additional
dimension is indistinguishable from the other three.
In modern physics, space and time are uniﬁed in a four-dimensional Minkowski continuum called spacetime, whose
metric treats the time dimension diﬀerently from the three spatial dimensions (see below for the deﬁnition of the
Minkowski metric/pairing). Spacetime is not a Euclidean space.

19.1 History
See also: n-dimensional space § History
Lagrange wrote in his Mécanique analytique (published 1788, based on work done around 1755) that mechanics can be
viewed as operating in a four-dimensional space — three of dimensions of space, and one of time.[1] In 1827 Möbius
realized that a fourth dimension would allow a three-dimensional form to be rotated onto its mirror-image,[2] and by
1853 Ludwig Schläﬂi had discovered many polytopes in higher dimensions, although his work was not published until
after his death.[3] Higher dimensions were soon put on ﬁrm footing by Bernhard Riemann's 1854 Habilitationsschrift,
Über die Hypothesen welche der Geometrie zu Grunde liegen, in which he considered a “point” to be any sequence of
coordinates (x1 , ..., xn). The possibility of geometry in higher dimensions, including four dimensions in particular,
was thus established.
An arithmetic of four dimensions called quaternions was deﬁned by William Rowan Hamilton in 1843. This associative
algebra was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis. Soon after tessarines and coquaternions were introduced as other four-dimensional algebras over R.
One of the ﬁrst major expositors of the fourth dimension was Charles Howard Hinton, starting in 1880 with his essay
What is the Fourth Dimension?; published in the Dublin University magazine.[4] He coined the terms tesseract, ana
and kata in his book A New Era of Thought, and introduced a method for visualising the fourth dimension using cubes
in the book Fourth Dimension.[5][6] In 1886 Victor Schlegel described[7] his method of visualizing four-dimensional
objects with Schlegel diagrams.
In 1908, Hermann Minkowski presented a paper[8] consolidating the role of time as the fourth dimension of spacetime,
the basis for Einstein’s theories of special and general relativity.[9] But the geometry of spacetime, being nonEuclidean, is profoundly diﬀerent from that popularised by Hinton. The study of Minkowski space required new
mathematics quite diﬀerent from that of four-dimensional Euclidean space, and so developed along quite diﬀerent
lines. This separation was less clear in the popular imagination, with works of ﬁction and philosophy blurring the
distinction, so in 1973 H. S. M. Coxeter felt compelled to write:

Little, if anything, is gained by representing the fourth Euclidean dimension as time. In fact, this idea,
so attractively developed by H. G. Wells in The Time Machine, has led such authors as John William
Dunne (An Experiment with Time) into a serious misconception of the theory of Relativity. Minkowski’s
geometry of space-time is not Euclidean, and consequently has no connection with the present investigation.
— H. S. M. Coxeter, Regular Polytopes[10]

19.2 Vectors
Mathematically four-dimensional space is simply a space with four spatial dimensions, that is a space that needs four
parameters to specify a point in it. For example, a general point might have position vector a, equal to

19.2. VECTORS

75

 
a1
a 2 

a=
a3 .
a4
This can be written in terms of the four standard basis vectors (e1 , e2 , e3 , e4 ), given by
 
 
 
 
1
0
0
0
0
1
0
0







e1 =  ; e2 =  ; e3 =  ; e4 =  
,
0
0
1
0
0
0
0
1
so the general vector a is

a = a1 e1 + a2 e2 + a3 e3 + a4 e4 .
Vectors add, subtract and scale as in three dimensions.
The dot product of Euclidean three-dimensional space generalizes to four dimensions as

a · b = a1 b1 + a2 b2 + a3 b3 + a4 b4 .
It can be used to calculate the norm or length of a vector,

|a| =

√
√
a · a = a1 2 + a2 2 + a3 2 + a4 2 ,

and calculate or deﬁne the angle between two vectors as

θ = arccos

a·b
.
|a| |b|

Minkowski spacetime is four-dimensional space with geometry deﬁned by a nondegenerate pairing diﬀerent from the
dot product:

a · b = a1 b1 + a2 b2 + a3 b3 − a4 b4 .
As an example, the distance squared between the points (0,0,0,0) and (1,1,1,0) is 3 in both the Euclidean and
Minkowskian 4-spaces, while the distance squared between (0,0,0,0) and (1,1,1,1) is 4 in Euclidean space and 2
in Minkowski space; increasing b4 actually decreases the metric distance. This leads to many of the well known
apparent “paradoxes” of relativity.
The cross product is not deﬁned in four dimensions. Instead the exterior product is used for some applications, and
is deﬁned as follows:

a ∧ b = (a1 b2 − a2 b1 )e12 + (a1 b3 − a3 b1 )e13 + (a1 b4 − a4 b1 )e14 + (a2 b3 − a3 b2 )e23
+(a2 b4 − a4 b2 )e24 + (a3 b4 − a4 b3 )e34 .
This is bivector valued, with bivectors in four dimensions forming a six-dimensional linear space with basis (e12 , e13 ,
e14 , e23 , e24 , e34 ). They can be used to generate rotations in four dimensions.

76

CHAPTER 19. FOUR-DIMENSIONAL SPACE

19.3 Orthogonality and vocabulary
In the familiar 3-dimensional space in which we live there are three coordinate axes — usually labeled x, y, and z
— with each axis orthogonal (i.e. perpendicular) to the other two. The six cardinal directions in this space can be
called up, down, east, west, north, and south. Positions along these axes can be called altitude, longitude, and latitude.
Lengths measured along these axes can be called height, width, and depth.
Comparatively, 4-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually
labeled w. To describe the two additional cardinal directions, Charles Howard Hinton coined the terms ana and kata,
from the Greek words meaning “up toward” and “down from”, respectively. A position along the w axis can be called
spissitude, as coined by Henry More.

19.4 Geometry
See also: Rotations in 4-dimensional Euclidean space
The geometry of 4-dimensional space is much more complex than that of 3-dimensional space, due to the extra degree
of freedom.
Just as in 3 dimensions there are polyhedra made of two dimensional polygons, in 4 dimensions there are 4-polytopes
made of polyhedra. In 3 dimensions there are 5 regular polyhedra known as the Platonic solids. In 4 dimensions
there are 6 convex regular 4-polytopes, the analogues of the Platonic solids. Relaxing the conditions for regularity
generates a further 58 convex uniform 4-polytopes, analogous to the 13 semi-regular Archimedean solids in three
dimensions. Relaxing the conditions for convexity generates a further 10 nonconvex regular 4-polytopes.
In 3 dimensions, a circle may be extruded to form a cylinder. In 4 dimensions, there are several diﬀerent cylinderlike objects. A sphere may be extruded to obtain a spherical cylinder (a cylinder with spherical “caps”, known as a
spherinder), and a cylinder may be extruded to obtain a cylindrical prism (a cubinder). The Cartesian product of two
circles may be taken to obtain a duocylinder. All three can “roll” in 4-dimensional space, each with its own properties.
In 3 dimensions, curves can form knots but surfaces cannot (unless they are self-intersecting). In 4 dimensions,
however, knots made using curves can be trivially untied by displacing them in the fourth direction, but 2-dimensional
surfaces can form non-trivial, non-self-intersecting knots in 4-dimensional space.[11] Because these surfaces are 2dimensional, they can form much more complex knots than strings in 3-dimensional space can. The Klein bottle is
an example of such a knotted surface . Another such surface is the real projective plane.

19.4.1

Hypersphere

Main article: Hypersphere
The set of points in Euclidean 4-space having the same distance R from a ﬁxed point P0 forms a hypersurface known
as a 3-sphere. The hyper-volume of the enclosed space is:

V = 12 π 2 R4
This is part of the Friedmann–Lemaître–Robertson–Walker metric in General relativity where R is substituted by
function R(t) with t meaning the cosmological age of the universe. Growing or shrinking R with time means expanding
or collapsing universe, depending on the mass density inside.[12]

19.5 Cognition
Research using virtual reality ﬁnds that humans in spite of living in a three-dimensional world can without special practice make spatial judgments based on the length of, and angle between, line segments embedded in four-dimensional
space.[13] The researchers noted that “the participants in our study had minimal practice in these tasks, and it remains
an open question whether it is possible to obtain more sustainable, deﬁnitive, and richer 4D representations with

19.6. DIMENSIONAL ANALOGY

77

Stereographic projection of a Cliﬀord torus: the set of points (cos(a), sin(a), cos(b), sin(b)), which is a subset of the 3-sphere.

increased perceptual experience in 4D virtual environments.”[13] In another study,[14] the ability of humans to orient
themselves in 2D, 3D and 4D mazes has been tested. Each maze consisted of four path segments of random length
and connected with orthogonal random bends, but without branches or loops (i.e. actually labyrinths). The graphical
interface was based on John McIntosh’s free 4D Maze game.[15] The participating persons had to navigate through
the path and ﬁnally estimate the linear direction back to the starting point. The researchers found that some of the
participants were able to mentally integrate their path after some practice in 4D (the lower-dimensional cases were
for comparison and for the participants to learn the method).

19.6 Dimensional analogy
To understand the nature of four-dimensional space, a device called dimensional analogy is commonly employed. Dimensional analogy is the study of how (n − 1) dimensions relate to n dimensions, and then inferring how n dimensions
would relate to (n + 1) dimensions.[16]
Dimensional analogy was used by Edwin Abbott Abbott in the book Flatland, which narrates a story about a square
that lives in a two-dimensional world, like the surface of a piece of paper. From the perspective of this square, a threedimensional being has seemingly god-like powers, such as ability to remove objects from a safe without breaking it
open (by moving them across the third dimension), to see everything that from the two-dimensional perspective is

78

CHAPTER 19. FOUR-DIMENSIONAL SPACE

A net of a tesseract

enclosed behind walls, and to remain completely invisible by standing a few inches away in the third dimension.
By applying dimensional analogy, one can infer that a four-dimensional being would be capable of similar feats from
our three-dimensional perspective. Rudy Rucker illustrates this in his novel Spaceland, in which the protagonist
encounters four-dimensional beings who demonstrate such powers.

19.6.1

Cross-sections

As a three-dimensional object passes through a two-dimensional plane, a two-dimensional being would only see a
cross-section of the three-dimensional object. For example, if a spherical balloon passed through a sheet of paper, a
being on the paper would see ﬁrst a single point, then a circle gradually growing larger, then smaller again until it shrank
to a point and then disappeared. Similarly, if a four-dimensional object passed through three dimensions, we would
see a three-dimensional cross-section of the four-dimensional object—for example, a hypersphere would appear ﬁrst
as a point, then as a growing sphere, with the sphere then shrinking to a single point and then disappearing.[17] This
means of visualizing aspects of the fourth dimension was used in the novel Flatland and also in several works of
Charles Howard Hinton.[18]

19.6.2

Projections

A useful application of dimensional analogy in visualizing the fourth dimension is in projection. A projection is a way
for representing an n-dimensional object in n − 1 dimensions. For instance, computer screens are two-dimensional,
and all the photographs of three-dimensional people, places and things are represented in two dimensions by projecting
the objects onto a ﬂat surface. When this is done, depth is removed and replaced with indirect information. The retina
of the eye is also a two-dimensional array of receptors but the brain is able to perceive the nature of three-dimensional
objects by inference from indirect information (such as shading, foreshortening, binocular vision, etc.). Artists often
use perspective to give an illusion of three-dimensional depth to two-dimensional pictures.

19.6. DIMENSIONAL ANALOGY

79

Similarly, objects in the fourth dimension can be mathematically projected to the familiar 3 dimensions, where they
can be more conveniently examined. In this case, the 'retina' of the four-dimensional eye is a three-dimensional
array of receptors. A hypothetical being with such an eye would perceive the nature of four-dimensional objects by
inferring four-dimensional depth from indirect information in the three-dimensional images in its retina.
The perspective projection of three-dimensional objects into the retina of the eye introduces artifacts such as foreshortening, which the brain interprets as depth in the third dimension. In the same way, perspective projection
from four dimensions produces similar foreshortening eﬀects. By applying dimensional analogy, one may infer fourdimensional “depth” from these eﬀects.
As an illustration of this principle, the following sequence of images compares various views of the 3-dimensional
cube with analogous projections of the 4-dimensional tesseract into three-dimensional space.

19.6.3

Shadows

A concept closely related to projection is the casting of shadows.

If a light is shone on a three dimensional object, a two-dimensional shadow is cast. By dimensional analogy, light
shone on a two-dimensional object in a two-dimensional world would cast a one-dimensional shadow, and light on
a one-dimensional object in a one-dimensional world would cast a zero-dimensional shadow, that is, a point of nonlight. Going the other way, one may infer that light shone on a four-dimensional object in a four-dimensional world

80

CHAPTER 19. FOUR-DIMENSIONAL SPACE

would cast a three-dimensional shadow.
If the wireframe of a cube is lit from above, the resulting shadow is a square within a square with the corresponding
corners connected. Similarly, if the wireframe of a tesseract were lit from “above” (in the fourth dimension), its
shadow would be that of a three-dimensional cube within another three-dimensional cube. (Note that, technically,
the visual representation shown here is actually a two-dimensional image of the three-dimensional shadow of the
four-dimensional wireframe ﬁgure.)

19.6.4

Bounding volumes

Dimensional analogy also helps in inferring basic properties of objects in higher dimensions. For example, twodimensional objects are bounded by one-dimensional boundaries: a square is bounded by four edges. Three-dimensional
objects are bounded by two-dimensional surfaces: a cube is bounded by 6 square faces. By applying dimensional
analogy, one may infer that a four-dimensional cube, known as a tesseract, is bounded by three-dimensional volumes.
And indeed, this is the case: mathematics shows that the tesseract is bounded by 8 cubes. Knowing this is key to understanding how to interpret a three-dimensional projection of the tesseract. The boundaries of the tesseract project
to volumes in the image, not merely two-dimensional surfaces.

19.6.5

Visual scope

Being three-dimensional, we are only able to see the world with our eyes in two dimensions. A four-dimensional
being would be able to see the world in three dimensions. For example, it would be able to see all six sides of an
opaque box simultaneously, and in fact, what is inside the box at the same time, just as we can see the interior of a
square on a piece of paper. It would be able to see all points in 3-dimensional space simultaneously, including the
inner structure of solid objects and things obscured from our three-dimensional viewpoint. Our brains receive images
in the second dimension and use reasoning to help us “picture” three-dimensional objects.

19.6.6

Limitations

Reasoning by analogy from familiar lower dimensions can be an excellent intuitive guide, but care must be exercised
not to accept results that are not more rigorously tested. For example, consider the formulas for the circumference of
a circle C = 2πr and the surface area of a sphere: A = 4πr2 . One might be tempted to suppose that the surface
volume of a hypersphere is V = 6πr3 , or perhaps V = 8πr3 , but either of these would be wrong. The correct
formula is V = 2π 2 r3 .[10]

19.7 See also
• Euclidean space
• Euclidean geometry
• 4-manifold
• Exotic R4
• Fourth dimension in art
• Dimension
• Four-dimensionalism
• Fifth dimension
• Sixth dimension
• 4-polytope
• Polytope

19.8. REFERENCES

81

• List of geometry topics
• Block Theory of the Universe
• Flatland, a book by Edwin A. Abbott about two- and three-dimensional spaces, to understand the concept of
four dimensions
• Sphereland, an unoﬃcial sequel to Flatland
• Charles Howard Hinton
• Dimensions, a set of ﬁlms about two-, three- and four-dimensional polytopes
• List of four-dimensional games

19.8 References
[1] Bell, E.T. (1937). Men of Mathematics, Simon and Schuster, p. 154.
[2] Coxeter, H. S. M. (1973). Regular Polytopes, Dover Publications, Inc., p. 141.
[3] Coxeter, H. S. M. (1973). Regular Polytopes, Dover Publications, Inc., pp. 142–143.
[4] Rudolf v.B. Rucker, editor Speculations on the Fourth Dimension: Selected Writings of Charles H. Hinton, p. vii, Dover
Publications Inc., 1980 ISBN 0-486-23916-0
[5] Hinton, Charles Howard (1904). Fourth Dimension. ISBN 1-5645-9708-3.
[6] Gardner, Martin (1975). Mathematical Carnival. Knopf Publishing. pp. 42, 52–53. ISBN 0-394-49406-7.
[7] Victor Schlegel (1886) Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper, Waren
[8] Minkowski, Hermann (1909), "Raum und Zeit", Physikalische Zeitschrift 10: 75–88
• Various English translations on Wikisource: Space and Time
[9] C Møller (1952). The Theory of Relativity. Oxford UK: Clarendon Press. p. 93. ISBN 0-19-851256-2.
[10] Coxeter, H. S. M. (1973). Regular Polytopes, Dover Publications, Inc., p. 119.
[11] J. Scott Carter, Masahico Saito Knotted Surfaces and Their Diagrams
[12] Ray d'Inverno (1992), Introducing Einstein’s Relativity, Clarendon Press, chp. 22.8 Geometry of 3-spaces of constant curvature, p.319ﬀ, ISBN 0-19-859653-7
[13] Ambinder MS, Wang RF, Crowell JA, Francis GK, Brinkmann P. (2009). Human four-dimensional spatial intuition in
virtual reality. Psychon Bull Rev. 16(5):818-23. doi:10.3758/PBR.16.5.818 PMID 19815783 online supplementary material
[14] Aﬂalo TN, Graziano MS (2008). Four-Dimensional Spatial Reasoning in Humans. Journal of Experimental Psychology:
Human Perception and Performance 34(5):1066-1077. doi:10.1037/0096-1523.34.5.1066 Preprint
[15] John McIntosh’s four dimensional maze game. Free software
[16] Michio Kaku (1994). Hyperspace: A Scientiﬁc Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension,
Part I, chapter 3, The Man Who “Saw” the Fourth Dimension (about tesseracts in years 1870–1910). ISBN 0-19-286189-1.
[17] Rucker, Rudy (1984), The Fourth Dimension /A Guided Tour of the Higher Universes, Houghton Miﬄin, p. 18, ISBN
0-395-39388-4
[18] In particular, Hinton, Charles Howard (1904). Fourth Dimension. pp. 11–14. ISBN 1-5645-9708-3.

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CHAPTER 19. FOUR-DIMENSIONAL SPACE

19.9 Further reading
• Andrew Forsyth (1930) Geometry of Four Dimensions, link from Internet Archive.
• Gamow, George (1988). One Two Three . . . Inﬁnity: Facts and Speculations of Science (3rd ed.). Courier
Dover Publications. p. 68. ISBN 0-486-25664-2., Extract of page 68
• E. H. Neville (1921) The Fourth Dimension, Cambridge University Press, link from University of Michigan
Historical Math Collection.

19.10 External links
• “Dimensions” videos, showing several diﬀerent ways to visualize four dimensional objects
• Science News article summarizing the “Dimensions” videos, with clips
• Garrett Jones’ tetraspace page
• Flatland: a Romance of Many Dimensions (second edition)
• TeV scale gravity, mirror universe, and ... dinosaurs Article from Acta Physica Polonica B by Z.K. Silagadze.
• Exploring Hyperspace with the Geometric Product
• 4D Euclidean space
• 4D Building Blocks - Interactive game to explore 4D space
• 4DNav - A small tool to view a 4D space as four 3D space uses ADSODA algorithm
• MagicCube 4D A 4-dimensional analog of traditional Rubik’s Cube.
• Frame-by-frame animations of 4D - 3D analogies

Chapter 20

Fourth dimension in art
New possibilities opened up by the concept of four-dimensional space (and diﬃculties involved in trying to visualize
it) helped inspire many modern artists in the ﬁrst half of the twentieth century. Early Cubists, Surrealists, Futurists,
and abstract artists took ideas from higher-dimensional mathematics and used them to radically advance their work.[1]

20.1 Early inﬂuence
Further information: Proto-Cubism and Mathematics and art
French mathematician Maurice Princet was known as “le mathématicien du cubisme” (“the mathematician of cubism”).[2]
An associate of the School of Paris, a group of avant-gardists including Pablo Picasso, Guillaume Apollinaire, Max
Jacob, Jean Metzinger, and Marcel Duchamp, Princet is credited with introducing the work of Henri Poincaré and the
concept of the "fourth dimension" to the cubists at the Bateau-Lavoir during the ﬁrst decade of the 20th century.[3]
Princet introduced Picasso to Esprit Jouﬀret's Traité élémentaire de géométrie à quatre dimensions (Elementary Treatise on the Geometry of Four Dimensions, 1903),[4] a popularization of Poincaré's Science and Hypothesis in which
Jouﬀret described hypercubes and other complex polyhedra in four dimensions and projected them onto the twodimensional page. Picasso’s Portrait of Daniel-Henry Kahnweiler in 1910 was an important work for the artist, who
spent many months shaping it.[5] The portrait bears similarities to Jouﬀret’s work and shows a distinct movement
away from the Proto-Cubist fauvism displayed in Les Demoiselles d'Avignon, to a more considered analysis of space
and form.[6]
Early cubist Max Weber wrote an article entitled “In The Fourth Dimension from a Plastic Point of View”, for Alfred
Stieglitz's July 1910 issue of Camera Work. In the piece, Weber states, “In plastic art, I believe, there is a fourth
dimension which may be described as the consciousness of a great and overwhelming sense of space-magnitude in
all directions at one time, and is brought into existence through the three known measurements.”[7]
Another inﬂuence on the School of Paris was that of Jean Metzinger and Albert Gleizes, both painters and theoreticians. The ﬁrst major treatise written on the subject of Cubism was their 1912 collaboration Du “Cubisme”, which
says that:
“If we wished to relate the space of the [Cubist] painters to geometry, we should have to refer it
to the non-Euclidian mathematicians; we should have to study, at some length, certain of Riemann’s
theorems.”[8]
In a review of the 1913 Armory Show for the Philadelphia Enquirer, the inﬂuence of the fourth dimension on avantegarde painting was discussed; the paper’s art-critic describing how the artists’ employed "..harmonic use of what may
arbitrarily be called volume”.[9]

20.2 Dimensionist manifesto
In 1936 in Paris, Charles Tamkó Sirató published his Manifeste Dimensioniste,[10] which described how
83

84

CHAPTER 20. FOURTH DIMENSION IN ART

An illustration from Jouﬀret’s Traité élémentaire de géométrie à quatre dimensions. The book, which inﬂuenced Picasso, was given
to him by Princet.

the Dimensionist tendency has led to:
1. Literature leaving the line and entering the plane.
2. Painting leaving the plane and entering space.

20.2. DIMENSIONIST MANIFESTO

85

Picasso’s Portrait of Daniel-Henry Kahnweiler, 1910, The Art Institute of Chicago

3. Sculpture stepping out of closed, immobile forms.
4. …The artistic conquest of four-dimensional space, which to date has been completely art-free.
The manifesto was signed by many prominent modern artists worldwide. Hans Arp, Francis Picabia, Kandinsky,
Robert Delaunay and Marcel Duchamp amongst others added their names in Paris, then a short while later it was

86

CHAPTER 20. FOURTH DIMENSION IN ART

Jean Metzinger, 1912-1913, L'Oiseau bleu, (The Blue Bird), oil on canvas, 230 x 196 cm, Musée d'Art Moderne de la Ville de Paris

endorsed by artists abroad including László Moholy-Nagy, Joan Miró, David Kakabadze, Alexander Calder, and Ben
Nicholson.[10]

20.3 Cruciﬁxion (Corpus Hypercubus)
In 1953, the surrealist Salvador Dalí proclaimed his intention to paint “an explosive, nuclear and hypercubic” cruciﬁxion scene.[11][12] He said that, “This picture will be the great metaphysical work of my summer”.[13] Completed
the next year, Cruciﬁxion (Corpus Hypercubus) depicts Jesus Christ upon the net of a hypercube, also known as a
tesseract. The unfolding of a tesseract into eight cubes is analogous to unfolding the sides of a cube into six squares.
The Metropolitan Museum of Art describes the painting as a “new interpretation of an oft-depicted subject. ..[show-

20.4. ABSTRACT ART

87

ing] Christ’s spiritual triumph over corporeal harm.”[14]

20.4 Abstract art
Some of Piet Mondrian's (1872–1944) abstractions and his practice of Neoplasticism are said to be rooted in his
view of a utopian universe, with perpendiculars visually extending into another dimension.[15]

20.5 Other forms of art
Main article: Fourth dimension in literature
The Fourth dimension has been the subject of numerous ﬁctional stories.[16]

20.6 See also
• De Stijl
• Five-dimensional space
• Four-dimensional space
• Duration (philosophy)
• Philosophy of space and time
• Octacube

20.7 References
[1] Henderson, Linda Dalrymple. “Overview of The Fourth Dimension And Non-Euclidean Geometry In Modern Art, Revised
Edition". MIT Press. Retrieved 24 March 2013.
[2] Décimo, Marc (2007). Maurice Princet, Le Mathématicien du Cubisme (in French). Paris: Éditions L'Echoppe. ISBN
2-84068-191-9.
[3] Miller, Arthur I. (2001). Einstein, Picasso: space, time, and beauty that causes havoc (Print). New York: Basic Books. p.
101. ISBN 0-465-01859-9.
[4] Jouﬀret, Esprit (1903). Traité élémentaire de géométrie à quatre dimensions et introduction à la géométrie à n dimensions
(in French). Paris: Gauthier-Villars. OCLC 1445172. Retrieved 2008-02-06.
[5] Robbin, Tony (2006). Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought (Print). New
Haven: Yale University Press. p. 28. ISBN 9780300110395.
[6] Robbin, Tony (2006). Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought (Print). New
Haven: Yale University Press. pp. 28–30. ISBN 9780300110395.
[7] Weber, Max (1910). “In The Fourth Dimension from a Plastic Point of View”. Camera Work 31 (July 1910).
[8] Gleizes, Albert; Metzinger, Jean (1912). Du “Cubisme” (Edition Figuière). Paris.
[9] Oja, Carol J. (2000, February 2003). Making Music Modern: New York in the 1920s. Oxford University Press, USA. p.
84. Check date values in: |date= (help)ISBN 9780195162578
[10] Sirató, Charles Tamkó (1936). “Dimensionist Manifesto” (PDF). Paris. Retrieved 24 March 2013.
[11] Dalí, Salvador; Gómez de la Serna, Ramón (2001) [1988]. Dali. Secaucus, NJ: Wellﬂeet Press. p. 41. ISBN 1555213421.
[12] “Salvador Dalí (1904–1989)". SpanishArts. 2013. Retrieved 24 March 2013.

88

CHAPTER 20. FOURTH DIMENSION IN ART

[13] “Cruciﬁxion ('Corpus Hypercubus’), 1954”. Dalí gallery website. Retrieved 25 March 2013.
[14] "Cruciﬁxion (Corpus Hypercubus)". The Metropolitan Museum of Art. Retrieved 24 March 2013.
[15] Kruger, Runette (Summer 2007). “Art in the Fourth Dimension: Giving Form to Form – The Abstract Paintings of Piet
Mondrian” (PDF) (5). Spaces of Utopia: An Electronic Journal. pp. 23–35. ISSN 1646-4729.
[16] Clair, Bryan (16 September 2002). “Spirits, Art, and the Fourth Dimension”. Strange Horizons. Retrieved 25 March
2012.

20.8 Sources
• Clair, Bryan (16 September 2002). “Spirits, Art, and the Fourth Dimension”. Strange Horizons. Retrieved 25
March 2012.
• Dalí, Salvador; Gómez de la Serna, Ramón (2001) [1988]. Dali (Print). Secaucus, NJ: Wellﬂeet Press. ISBN
1555213421.
• Décimo, Marc (2007). Maurice Princet, Le Mathématicien du Cubisme (in French). Paris: Éditions L'Echoppe.
ISBN 2-84068-191-9.
• Gleizes, Albert; Metzinger, Jean (1912). Du “Cubisme” (Edition Figuière). Paris.
• Henderson, Linda Dalrymple. “Overview of The Fourth Dimension And Non-Euclidean Geometry In Modern
Art, Revised Edition". MIT Press. Retrieved 24 March 2013.
• Jouﬀret, Esprit (1903). Traité élémentaire de géométrie à quatre dimensions et introduction à la géométrie à n
dimensions (in French). Paris: Gauthier-Villars. OCLC 1445172. Retrieved 2008-02-06.
• Kruger, Runette (Summer 2007). “Art in the Fourth Dimension: Giving Form to Form – The Abstract Paintings of Piet Mondrian” (PDF) (5). Spaces of Utopia: An Electronic Journal. pp. 23–35. ISSN 1646-4729.
Retrieved 25 March 2012.
• Miller, Arthur I. (2001). Einstein, Picasso: space, time, and beauty that causes havoc (Print). New York: Basic
Books. ISBN 0-465-01859-9.
• Oja, Carol J. (2000, February 2003). Making Music Modern: New York in the 1920s. Oxford University Press,
USA. p. 84. Check date values in: |date= (help)ISBN 9780195162578
• Robbin, Tony (2006). Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought
(Print). New Haven: Yale University Press. pp. 28–30. ISBN 9780300110395.
• Sirató, Charles Tamkó (1936). “Dimensionist Manifesto” (PDF). Paris. Retrieved 24 March 2013.
• Weber, Max (1910). “In The Fourth Dimension from a Plastic Point of View”. Camera Work 31 (July 1910).

20.9 Further reading
• Henderson, Linda Dalrymple (2013). The Fourth Dimension And Non-Euclidean Geometry In Modern Art
(Revised ed.). Cambridge, Massachusetts: The MIT Press. ISBN 0262582449.
• Henderson, Linda Dalrymple (1998). Duchamp in Context: Science and Technology in the Large Glass and
Related Works. Princeton, N.J: Princeton University Press. ISBN 9780691123868.

20.10 External links
• Talk at the Dali museum on his 4th dimension art

20.10. EXTERNAL LINKS

Dalí's 1954 painting Cruciﬁxion (Corpus Hypercubus)

89

Chapter 21

Gelfand–Kirillov dimension
In algebra, the Gelfand–Kirillov dimension (or GK dimension) of a right module M over a k-algebra A is:

GKdim = sup lim sup logn dimk M0 V n
V,M0 n→∞

where the sup is taken over all ﬁnite-dimensional subspaces V ⊂ A and M0 ⊂ M .
An algebra is said to have polynomial growth if its Gelfand–Kirillov dimension is ﬁnite.

21.1 Basic facts
• The Gelfand–Kirillov dimension of a ﬁnitely generated commutative algebra A over a ﬁeld is the Krull dimension of A (or equivalently the transcendence degree of the ﬁeld of fractions of A over the base ﬁeld.)
• In particular, the GK dimension of the polynomial ring k[x1 , . . . , xn ] Is n.
• (Warﬁeld) For any real number r ≥ 2, there exists a ﬁnitely generated algebra whose GK dimension is r.[1]

21.2 In the theory of D-Modules
Given a right module M over the Weyl algebra An , the Gelfand–Kirillov dimension of M over the Weyl algebra
coincides with the dimension of M, which is by deﬁnition the degree of the Hilbert polynomial of M. This enables to
prove additivity in short exact sequences for the Gelfand–Kirillov dimension and ﬁnally to prove Bernstein’s inequality,
which states that the dimension of M must be at least n. This leads to the deﬁnition of holonomic D-Modules as those
with the minimal dimension n, and these modules play a great role in the geometric Langlands program.

21.3 References
[1] Artin 1999, Theorem VI.2.1.

• Smith, S. Paul; Zhang, James J. (1998). “A remark on Gelfand–Kirillov dimension” (PDF). Proceedings of the
American Mathematical Society 126 (2): 349–352.
• Coutinho: A primer of algebraic D-modules. Cambridge, 1995

21.4 Further reading
• Artin, Michael (1999). “Noncommutative Rings” (PDF). Chapter VI.

90

Chapter 22

Global dimension
In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just
called homological dimension) of a ring A denoted gl dim A, is a non-negative integer or inﬁnity which is a homological invariant of the ring. It is deﬁned to be the supremum of the set of projective dimensions of all A-modules.
Global dimension is an important technical notion in the dimension theory of Noetherian rings. By a theorem of
Jean-Pierre Serre, global dimension can be used to characterize within the class of commutative Noetherian local
rings those rings which are regular. Their global dimension coincides with the Krull dimension, whose deﬁnition is
module-theoretic.
When the ring A is noncommutative, one initially has to consider two versions of this notion, right global dimension
that arises from consideration of the right A-modules, and left global dimension that arises from consideration of the
left A-modules. For an arbitrary ring A the right and left global dimensions may diﬀer. However, if A is a Noetherian
ring, both of these dimensions turn out to be equal to weak global dimension, whose deﬁnition is left-right symmetric.
Therefore, for noncommutative Noetherian rings, these two versions coincide and one is justiﬁed in talking about the
global dimension.

22.1 Examples
Let A = K[x1,...,xn] be the ring of polynomials in n variables over a ﬁeld K. Then the global dimension of A is equal
to n. This statement goes back to David Hilbert's foundational work on homological properties of polynomial rings,
see Hilbert’s syzygy theorem. More generally, if R is a Noetherian ring of ﬁnite global dimension k and A = R[x] is
a ring of polynomials in one variable over R then the global dimension of A is equal to k + 1.
The ﬁrst Weyl algebra A1 is a noncommutative Noetherian domain of global dimension one.
A ring has global dimension zero if and only if it is semisimple. The global dimension of a ring A is less than or
equal to one if and only if A is hereditary. In particular, a commutative principal ideal domain which is not a ﬁeld
has global dimension one.

22.2 Alternative characterizations
The right global dimension of a ring A can be alternatively deﬁned as:
• the supremum of the set of projective dimensions of all cyclic right A-modules;
• the supremum of the set of projective dimensions of all ﬁnite right A-modules;
• the supremum of the injective dimensions of all right A-modules;
• when A is a commutative Noetherian local ring with maximal ideal m, the projective dimension of the residue
ﬁeld A/m.
91

92

CHAPTER 22. GLOBAL DIMENSION

The left global dimension of A has analogous characterizations obtained by replacing “right” with “left” in the above
list.
Serre proved that a commutative Noetherian local ring A is regular if and only if it has ﬁnite global dimension, in
which case the global dimension coincides with the Krull dimension of A. This theorem opened the door to application
of homological methods to commutative algebra.

22.3 References
• Eisenbud, David (1999), Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in
Mathematics 150 (3rd ed.), Springer-Verlag, ISBN 0-387-94268-8.
• Kaplansky, Irving (1972), Fields and Rings, Chicago Lectures in Mathematics (2nd ed.), University Of Chicago
Press, ISBN 0-226-42451-0, Zbl 1001.16500
• Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8,
Cambridge University Press, ISBN 0-521-36764-6.
• McConnell, J. C.; Robson, J. C.; Small, Lance W. (2001), Revised, ed., Noncommutative Noetherian Rings,
Graduate Studies in Mathematics 30, American Mathematical Society, ISBN 0-8218-2169-5.

Chapter 23

Interdimensional
Interdimensional may refer to:
• Interdimensional hypothesis
• Interdimensional doorway
• Interdimensional travel
• Interdimensional being
• Interdimensional (song by Darkest Horizon) [1]

23.1 References
[1] http://www.darkesthorizon.com

93

Chapter 24

Isoperimetric dimension
In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the
large-scale behavior of the manifold resembles that of a Euclidean space (unlike the topological dimension or the
Hausdorﬀ dimension which compare diﬀerent local behaviors against those of the Euclidean space).
In the Euclidean space, the isoperimetric inequality says that of all bodies with the same volume, the ball has the
smallest surface area. In other manifolds it is usually very diﬃcult to ﬁnd the precise body minimizing the surface
area, and this is not what the isoperimetric dimension is about. The question we will ask is, what is approximately the
minimal surface area, whatever the body realizing it might be.

24.1 Formal deﬁnition
We say about a diﬀerentiable manifold M that it satisﬁes a d-dimensional isoperimetric inequality if for any open
set D in M with a smooth boundary one has

area (∂D) ≥ C vol (D)(d−1)/d .
The notations vol and area refer to the regular notions of volume and surface area on the manifold, or more precisely,
if the manifold has n topological dimensions then vol refers to n-dimensional volume and area refers to (n − 1)dimensional volume. C here refers to some constant, which does not depend on D (it may depend on the manifold
and on d).
The isoperimetric dimension of M is the supremum of all values of d such that M satisﬁes a d-dimensional isoperimetric inequality.

24.2 Examples
A d-dimensional Euclidean space has isoperimetric dimension d. This is the well known isoperimetric problem —
as discussed above, for the Euclidean space the constant C is known precisely since the minimum is achieved for the
ball.
An inﬁnite cylinder (i.e. a product of the circle and the line) has topological dimension 2 but isoperimetric dimension
1. Indeed, multiplying any manifold with a compact manifold does not change the isoperimetric dimension (it only
changes the value of the constant C). Any compact manifold has isoperimetric dimension 0.
It is also possible for the isoperimetric dimension to be larger than the topological dimension. The simplest example
is the inﬁnite jungle gym, which has topological dimension 2 and isoperimetric dimension 3. See for pictures and
Mathematica code.
The hyperbolic plane has topological dimension 2 and isoperimetric dimension inﬁnity. In fact the hyperbolic plane
has positive Cheeger constant. This means that it satisﬁes the inequality
94

24.3. ISOPERIMETRIC DIMENSION OF GRAPHS

95

area (∂D) ≥ C vol (D),
which obviously implies inﬁnite isoperimetric dimension.

24.3 Isoperimetric dimension of graphs
Main article: Expander graph
The isoperimetric dimension of graphs can be deﬁned in a similar fashion. A precise deﬁnition is given in Chung’s
survey.[1] Area and volume are measured by set sizes. For every subset A of the graph G one deﬁnes ∂A as the set of
vertices in G \ A with a neighbor in A. A d-dimensional isoperimetric inequality is now deﬁned by

|∂A| ≥ C (min (|A|, |G \ A|))

(d−1)/d

.

(This MathOverﬂow question provides more details.) The graph analogs of all the examples above hold but the
deﬁnition is slightly diﬀerent in order to avoid that the isoperimetric dimension of any ﬁnite graph is 0: In the above
formula the volume of A is replaced by min(|A|, |G \ A|) (see Chung’s survey, section 7).
The isoperimetric dimension of a d-dimensional grid is d. In general, the isoperimetric dimension is preserved by
quasi isometries, both by quasi-isometries between manifolds, between graphs, and even by quasi isometries carrying
manifolds to graphs, with the respective deﬁnitions. In rough terms, this means that a graph “mimicking” a given
manifold (as the grid mimics the Euclidean space) would have the same isoperimetric dimension as the manifold. An
inﬁnite complete binary tree has isoperimetric dimension ∞.

24.4 Consequences of isoperimetry
A simple integration over r (or sum in the case of graphs) shows that a d-dimensional isoperimetric inequality implies
a d-dimensional volume growth, namely

vol B(x, r) ≥ Crd
where B(x,r) denotes the ball of radius r around the point x in the Riemannian distance or in the graph distance.
In general, the opposite is not true, i.e. even uniformly exponential volume growth does not imply any kind of
isoperimetric inequality. A simple example can be had by taking the graph Z (i.e. all the integers with edges between
n and n + 1) and connecting to the vertex n a complete binary tree of height |n|. Both properties (exponential growth
and 0 isoperimetric dimension) are easy to verify.
An interesting exception is the case of groups. It turns out that a group with polynomial growth of order d has
isoperimetric dimension d. This holds both for the case of Lie groups and for the Cayley graph of a ﬁnitely generated
group.
A theorem of Varopoulos connects the isoperimetric dimension of a graph to the rate of escape of random walk on
the graph. The result states
Varopoulos’ theorem: If G is a graph satisfying a d-dimensional isoperimetric inequality then

pn (x, y) ≤ Cn−d/2
where pn (x,y) is the probability that a random walk on G starting from x will be in y after n steps, and C is some
constant.

96

CHAPTER 24. ISOPERIMETRIC DIMENSION

24.5 References
[1] Chung, Fan. “Discrete Isoperimetric Inequalities” (PDF).

• Isaac Chavel, Isoperimetric Inequalities: Diﬀerential geometric and analytic persepectives, Cambridge university
press, Cambridge, UK (2001), ISBN 0-521-80267-9
Discusses the topic in the context of manifolds, no mention of graphs.
• N. Th. Varopoulos, Isoperimetric inequalities and Markov chains, J. Funct. Anal. 63:2 (1985), 215–239.
• Thierry Coulhon and Laurent Saloﬀ-Coste, Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamericana 9:2 (1993), 293–314.
This paper contains the result that on groups of polynomial growth, volume growth and isoperimetric
inequalities are equivalent. In French.
• Fan Chung, Discrete Isoperimetric Inequalities. Surveys in Diﬀerential Geometry IX, International Press, (2004),
53–82. http://math.ucsd.edu/~{}fan/wp/iso.pdf.
This paper contains a precise deﬁnition of the isoperimetric dimension of a graph, and establishes many
of its properties.

Chapter 25

Kaplan–Yorke conjecture
In applied mathematics, the Kaplan–Yorke conjecture concerns the dimension of an attractor, using Lyapunov
exponents.[1][2] By arranging the Lyapunov exponents in order from largest to smallest λ1 ≥ λ2 ≥ · · · ≥ λn , let j
be the index for which

j
∑

λi > 0

i=1

and

j+1
∑

λi < 0.

i=1

Then the conjecture is that the dimension of the attractor is
∑j
D=j+

λi
.
|λj+1 |
i=1

25.1 Examples
Especially for chaotic systems, the Kaplan–Yorke conjecture is a useful tool in order to determine the fractal dimension
of the corresponding attractor.[3]
• The Hénon map with parameters a = 1.4 and b = 0.3 has the ordered Lyapunov exponents λ1 = 0.603 and
λ2 = −2.34 . In this case, we ﬁnd j = 1 and the dimension formula reduces to

D=j+

0.603
λ1
=1+
= 1.26.
|λ2 |
| − 2.34|

• The Lorenz system shows chaotic behavior at the parameter values σ = 16 , ρ = 45.92 and β = 4.0 . The
resulting Lyapunov exponents are {2.16, 0.00, −32.4}. Noting that j = 2, we ﬁnd

D =2+

2.16 + 0.00
= 2.07.
| − 32.4|
97

98

CHAPTER 25. KAPLAN–YORKE CONJECTURE

25.2 References
[1] J. Kaplan and J. Yorke, “Chaotic behavior of multidimensional diﬀerence equations,” in: Functional Diﬀerential Equations
and the Approximation of Fixed Points, Lecture Notes in Mathematics, vol. 730, H.O. Peitgen and H.O. Walther, eds.
(Springer, Berlin), p. 228.
[2] P. Frederickson, J. Kaplan, E. Yorke and J. Yorke, “The Lyapunov Dimension of Strange Attractors,” J. Diﬀ. Eqs. 49
(1983) 185.
[3] A. Wolf, A. Swift, B. Jack, H. L. Swinney and J.A. Vastano “Determining Lyapunov Exponents from a Time Series,”
Physica 16D, 1985, 16, pp. 285–317.

Chapter 26

Kodaira dimension
In algebraic geometry, the Kodaira dimension κ(X) (or canonical dimension) measures the size of the canonical
model of a projective variety X.
Igor Shafarevich introduced an important numerical invariant of surfaces with the notation κ in the seminar Shafarevich
1965. Shigeru Iitaka (1970) extended it and deﬁned the Kodaira dimension for higher dimensional varieties (under
the name of canonical dimension), and later named it after Kunihiko Kodaira in Iitaka (1971).

26.1 The plurigenera
The canonical bundle of a smooth algebraic variety X of dimension n over a ﬁeld is the line bundle of n-forms,

KX =

n
∧

Ω1X ,

which is the nth exterior power of the cotangent bundle of X. For an integer d, the dth tensor power of KX is again a
line bundle. For d ≥ 0, the vector space of global sections H0 (X,KXd ) has the remarkable property that it is a birational
invariant of smooth projective varieties X. That is, this vector space is canonically identiﬁed with the corresponding
space for any smooth projective variety which is isomorphic to X outside lower-dimensional subsets.
For d ≥ 0, the dth plurigenus of X is deﬁned as the dimension of the vector space of global sections of KXd :

d
d
Pd = h0 (X, KX
) = dim H 0 (X, KX
).

The plurigenera are important birational invariants of an algebraic variety. In particular, the simplest way to prove
that a variety is not rational (that is, not birational to projective space) is to show that some plurigenus Pd with d > 0
is not zero. If the space of sections of KXd is nonzero, then there is a natural rational map from X to the projective
space
d
P(H 0 (X, KX
)) = PPd −1

called the d-canonical map. The canonical ring R(KX) of a variety X is the graded ring

R(KX ) :=

⊕

d
H 0 (X, KX
).

d≥0

Also see geometric genus and arithmetic genus.
The Kodaira dimension of X is deﬁned to be −∞ if the plurigenera Pd are zero for all d > 0; otherwise, it is the
minimum κ such that Pd/dκ is bounded. The Kodaira dimension of an n-dimensional variety is either −∞ or an integer
in the range from 0 to n.
99

100

26.1.1

CHAPTER 26. KODAIRA DIMENSION

Interpretations of the Kodaira dimension

The following integers are equal if they are non-negative. A good reference is Lazarsfeld (2004), Theorem 2.1.33.
• The dimension of the Proj construction Proj R(KX) (this variety is called the canonical model of X; it only
depends on the birational equivalence class of X).
• The dimension of the image of the d-canonical mapping for all positive multiples d of some positive integer
d0 .
• The transcendence degree of R, minus one, i.e. t − 1, where t is the number of algebraically independent
generators one can ﬁnd.
• The rate of growth of the plurigenera: that is, the smallest number κ such that Pd/dκ is bounded. In Big O
notation, it is the minimal κ such that Pd = O(dκ ).
When one of these numbers is undeﬁned or negative, then all of them are. In this case, the Kodaira dimension is said
to be negative or to be −∞. Some historical references deﬁne it to be −1, but then the formula κ(X × Y) = κ(X) +
κ(Y) does not always hold, and the statement of the Iitaka conjecture becomes more complicated. For example, the
Kodaira dimension of P1 × X is −∞ for all varieties X.

26.1.2

Application

The Kodaira dimension gives a useful rough division of all algebraic varieties into several classes.
Varieties with low Kodaira dimension can be considered special, while varieties of maximal Kodaira dimension are
said to be of general type.
Geometrically, there is a very rough correspondence between Kodaira dimension and curvature: negative Kodaira
dimension corresponds to positive curvature, zero Kodaira dimension corresponds to ﬂatness, and maximum Kodaira
dimension (general type) corresponds to negative curvature.
The specialness of varieties of low Kodaira dimension is analogous to the specialness of Riemannian manifolds of
positive curvature (and general type corresponds to the genericity of non-positive curvature); see classical theorems,
especially on Pinched sectional curvature and Positive curvature.
These statements are made more precise below.

26.1.3

Dimension 1

Smooth projective curves are discretely classiﬁed by genus, which can be any natural number g = 0, 1, ....
By “discretely classiﬁed”, we mean that for a given genus, there is a connected, irreducible moduli space of curves of
that genus.
The Kodaira dimension of a curve X is:
• κ = −∞: genus 0 (the projective line P1 ): KX is not eﬀective, Pd = 0 for all d > 0.
• κ = 0: genus 1 (elliptic curves): KX is a trivial bundle, Pd = 1 for all d ≥ 0.
• κ = 1: genus g ≥ 2: KX is ample, Pd = (2d−1)(g−1) for all d ≥ 2.
Compare with the Uniformization theorem for surfaces (real surfaces, since a complex curve has real dimension
2): Kodaira dimension −∞ corresponds to positive curvature, Kodaira dimension 0 corresponds to ﬂatness, Kodaira
dimension 1 corresponds to negative curvature. Note that most algebraic curves are of general type: in the moduli
space of curves, two connected components correspond to curves not of general type, while all the other components
correspond to curves of general type. Further, the space of curves of genus 0 is a point, the space of curves of genus
1 has (complex) dimension 1, and the space of curves of genus g ≥ 2 has dimension 3g−3.

26.2. GENERAL TYPE

26.1.4

101

Dimension 2

The Enriques–Kodaira classiﬁcation classiﬁes algebraic surfaces: coarsely by Kodaira dimension, then in more detail
within a given Kodaira dimension. To give some simple examples: the product P1 × X has Kodaira dimension −∞
for any curve X; the product of two curves of genus 1 (an abelian surface) has Kodaira dimension 0; the product of
a curve of genus 1 with a curve of genus at least 2 (an elliptic surface) has Kodaira dimension 1; and the product of
two curves of genus at least 2 has Kodaira dimension 2 and hence is of general type.

For a surface X of general type, the image of the d-canonical map is birational to X if d ≥ 5.

26.1.5

Any dimension

Rational varieties (varieties birational to projective space) have Kodaira dimension −∞. Abelian varieties (the compact
complex tori that are projective) have Kodaira dimension zero. More generally, Calabi–Yau manifolds (in dimension
1, elliptic curves; in dimension 2, abelian surfaces, K3 surfaces, and quotients of those varieties by ﬁnite groups) have
Kodaira dimension zero (corresponding to admitting Ricci ﬂat metrics).
Any variety in characteristic zero that is covered by rational curves (nonconstant maps from P1 ), called a uniruled
variety, has Kodaira dimension −∞. Conversely, the main conjectures of minimal model theory (notably the abundance conjecture) would imply that every variety of Kodaira dimension −∞ is uniruled. This converse is known for
varieties of dimension at most 3.
Siu (2002) proved the invariance of plurigenera under deformations for all smooth complex projective varieties. In
particular, the Kodaira dimension does not change when the complex structure of the manifold is changed continuously.

A ﬁbration of normal projective varieties X → Y means a surjective morphism with connected ﬁbers.
For a 3-fold X of general type, the image of the d-canonical map is birational to X if d ≥ 61.[1]

26.2 General type
A variety of general type X is one of maximal Kodaira dimension (Kodaira dimension equal to its dimension):

κ(X) = dim X.
Equivalent conditions are that the line bundle KX is big, or that the d-canonical map is generically injective (that is,
a birational map to its image) for d suﬃciently large.
For example, a variety with ample canonical bundle is of general type.
In some sense, most algebraic varieties are of general type. For example, a smooth hypersurface of degree d in the ndimensional projective space is of general type if and only if d > n+1. So we can say that most smooth hypersurfaces
in projective space are of general type.
Varieties of general type seem too complicated to classify explicitly, even for surfaces. Nonetheless, there are some
strong positive results about varieties of general type. For example, Bombieri showed in 1973 that the d-canonical map
of any complex surface of general type is birational for every d ≥ 5. More generally, Hacon-McKernan, Takayama,
and Tsuji showed in 2006 that for every positive integer n, there is a constant c(n) such that the d-canonical map of
any complex n-dimensional variety of general type is birational when d ≥ c(n).
The birational automorphism group of a variety of general type is ﬁnite.

102

CHAPTER 26. KODAIRA DIMENSION

26.3 Application to classiﬁcation
Let X be a variety of nonnegative Kodaira dimension over a ﬁeld of characteristic zero, and let B be the canonical
model of X, B = Proj R(X, KX); the dimension of B is equal to the Kodaira dimension of X. There is a natural rational
map X – → B; any morphism obtained from it by blowing up X and B is called the Iitaka ﬁbration. The minimal
model and abundance conjectures would imply that the general ﬁber of the Iitaka ﬁbration can be arranged to be a
Calabi-Yau variety, which in particular has Kodaira dimension zero. Moreover, there is an eﬀective Q-divisor Δ on B
(not unique) such that the pair (B, Δ) is klt, KB + Δ is ample, and the canonical ring of X is the same as the canonical
ring of (B, Δ) in degrees a multiple of some d > 0.[2] In this sense, X is decomposed into a family of varieties of
Kodaira dimension zero over a base (B, Δ) of general type. (Note that the variety B by itself need not be of general
type. For example, there are surfaces of Kodaira dimension 1 for which the Iitaka ﬁbration is an elliptic ﬁbration over
P1 .)
Given the conjectures mentioned, the classiﬁcation of algebraic varieties would largely reduce to the cases of Kodaira
dimension −∞, 0 and general type. For Kodaira dimension −∞ and 0, there are some approaches to classiﬁcation.
The minimal model and abundance conjectures would imply that every variety of Kodaira dimension −∞ is uniruled,
and it is known that every uniruled variety in characteristic zero is birational to a Fano ﬁber space. The minimal model
and abundance conjectures would imply that every variety of Kodaira dimension 0 is birational to a Calabi-Yau variety
with terminal singularities.
The Iitaka conjecture states that the Kodaira dimension of a ﬁbration is at least the sum of the Kodaira dimension of
the base and the Kodaira dimension of a general ﬁber; see Mori (1987) for a survey. The Iitaka conjecture helped to
inspire the development of minimal model theory in the 1970s and 1980s. It is now known in many cases, and would
follow in general from the minimal model and abundance conjectures.

26.4 The relationship to Moishezon manifolds
Nakamura and Ueno proved the following additivity formula for complex manifolds (Ueno (1975)). Although the
base space is not required to be algebraic, the assumption that all the ﬁbers are isomorphic is very special. Even with
this assumption, the formula can fail when the ﬁber is not Moishezon.
Let π: V → W be an analytic ﬁber bundle of compact complex manifolds, meaning that π is locally a
product (and so all ﬁbers are isomorphic as complex manifolds). Suppose that the ﬁber F is a Moishezon
manifold. Then
κ(V ) = κ(F ) + κ(W ).

26.5 Notes
[1] J. A. Chen and M. Chen, Explicit birational geometry of 3-folds and 4-folds of general type III, Theorem 1.4.
[2] O. Fujino and S. Mori, J. Diﬀ. Geom. 56 (2000), 167-188. Theorems 5.2 and 5.4.

26.6 References
• Chen, Jungkai A.; Chen, Meng (2013), Explicit birational geometry of 3-folds and 4-folds of general type, III,
arXiv:1302.0374, Bibcode:2013arXiv1302.0374M
• Dolgachev, I, (2001), “Kodaira dimension”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,
ISBN 978-1-55608-010-4
• Fujino, Osamu; Mori, Shigefumi (2000), “A canonical bundle formula”, Journal of Diﬀerential Geometry 56
(1): 167–188, MR 1863025
• Iitaka, Shigeru (1970), “On D-dimensions of algebraic varieties”, Proc. Japan Acad. 46: 487–489, doi:10.3792/pja/1195520260,
MR 0285532

26.6. REFERENCES

103

• Iitaka, Shigeru (1971), “On D-dimensions of algebraic varieties.”, J. Math. Soc. Japan 23: 356–373, doi:10.2969/jmsj/02320356,
MR 0285531
• Lazarsfeld, Robert (2004), Positivity in algebraic geometry 1, Berlin: Springer-Verlag, ISBN 3-540-22533-1,
MR 2095471
• Mori, Shigefumi (1987), “Classiﬁcation of higher-dimensional varieties”, Algebraic geometry (Bowdoin, 1985),
Proceedings of Symposia in Pure Mathematics, 46, Part 1, American Mathematical Society, pp. 269–331, MR
0927961
• Shafarevich, Igor R.; Averbuh, B. G.; Vaĭnberg, Ju. R.; Zhizhchenko, A. B.; Manin, Ju. I.; Moĭshezon, B. G.;
Tjurina, G. N.; Tjurin, A. N. (1965), “Algebraic surfaces”, Akademiya Nauk SSSR. Trudy Matematicheskogo
Instituta imeni V. A. Steklova 75: 1–215, ISSN 0371-9685, MR 0190143, Zbl 0154.21001
• Siu, Y.-T. (2002), “Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance
of semi-positively twisted plurigenera for manifolds not necessarily of general type”, Complex geometry (Gottingen, 2000), Berlin: Springer-Verlag, pp. 223–277, MR 1922108
• Ueno, Kenji (1975), Classiﬁcation theory of algebraic varieties and compact complex spaces, Lecture Notes in
Mathematics 439, Springer-Verlag, MR 0506253

Chapter 27

Krull dimension
In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum
of the lengths of all chains of prime ideals. The Krull dimension need not be ﬁnite even for a Noetherian ring. More
generally the Krull dimension can be deﬁned for modules over possibly non-commutative rings as the deviation of
the poset of submodules.
The Krull dimension has been introduced to provide an algebraic deﬁnition of the dimension of an algebraic variety:
the dimension of the aﬃne variety deﬁned by an ideal I in a polynomial ring R is the Krull dimension of R/I.
A ﬁeld k has Krull dimension 0; more generally, k[x1 , ..., xn] has Krull dimension n. A principal ideal domain that
is not a ﬁeld has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal
ideal is nilpotent.

27.1 Explanation
We say that a chain of prime ideals of the form p0 ⊊ p1 ⊊ . . . ⊊ pn has length n. That is, the length is the number
of strict inclusions, not the number of primes; these diﬀer by 1. We deﬁne the Krull dimension of R to be the
supremum of the lengths of all chains of prime ideals in R .
Given a prime p in R, we deﬁne the height of p , written ht(p) , to be the supremum of the lengths of all chains of
prime ideals contained in p , meaning that p0 ⊊ p1 ⊊ . . . ⊊ pn = p .[1] In other words, the height of p is the Krull
dimension of the localization of R at p . A prime ideal has height zero if and only if it is a minimal prime ideal. The
Krull dimension of a ring is the supremum of the heights of all maximal ideals, or those of all prime ideals.
In a Noetherian ring, every prime ideal has ﬁnite height. Nonetheless, Nagata gave an example of a Noetherian ring
of inﬁnite Krull dimension.[2] A ring is called catenary if any inclusion p ⊂ q of prime ideals can be extended to a
maximal chain of prime ideals between p and q , and any two maximal chains between p and q have the same length.
A ring is called universally catenary if any ﬁnitely generated algebra over it is catenary. Nagata gave an example of
a Noetherian ring which is not catenary.[3]
In a Noetherian ring, Krull’s height theorem says that the height of an ideal generated by n elements is no greater than
n.
More generally, the height of an ideal I is the inﬁmum of the heights of all prime ideals containing I. In the language
of algebraic geometry, this is the codimension of the subvariety of Spec( R ) corresponding to I.[4]

27.2 Krull dimension and schemes
It follows readily from the deﬁnition of the spectrum of a ring Spec(R), the space of prime ideals of R equipped with
the Zariski topology, that the Krull dimension of R is equal to the dimension of its spectrum as a topological space,
meaning the supremum of the lengths of all chains of irreducible closed subsets. This follows immediately from the
Galois connection between ideals of R and closed subsets of Spec(R) and the observation that, by the deﬁnition of
Spec(R), each prime ideal p of R corresponds to a generic point of the closed subset associated to p by the Galois
connection.
104

27.3. EXAMPLES

105

27.3 Examples
• The dimension of a polynomial ring over a ﬁeld k[x1 , ..., xn] is the number of variables n. In the language of
algebraic geometry, this says that the aﬃne space of dimension n over a ﬁeld has dimension n, as expected.
In general, if R is a Noetherian ring of dimension n, then the dimension of R[x] is n + 1. If the Noetherian
hypothesis is dropped, then R[x] can have dimension anywhere between n + 1 and 2n + 1.
• The ring of integers Z has dimension 1. More generally, any principal ideal domain that is not a ﬁeld has
dimension 1.
• An integral domain is a ﬁeld if and only if its Krull dimension is zero. Dedekind domains that are not ﬁelds
(for example, discrete valuation rings) have dimension one.
• The Krull dimension of the zero ring is typically deﬁned to be either −∞ or −1 . The zero ring is the only
ring with a negative dimension.
• A ring is Artinian if and only if it is Noetherian and its Krull dimension is ≤0.
• An integral extension of a ring has the same dimension as the ring does.
• Let R be an algebra over a ﬁeld k that is an integral domain. Then the Krull dimension of R is less than or equal
to the transcendence degree of the ﬁeld of fractions of R over k.[5] The equality holds if R is ﬁnitely generated
as algebra (for instance by the noether normalization lemma).
k
k+1
be the associated graded ring (geometers
• Let R be a Noetherian ring, I an ideal and grI (R) = ⊕∞
0 I /I
call it the ring of the normal cone of I.) Then dim grI (R) is the supremum of the heights of maximal ideals of
R containing I.[6]

• A commutative Noetherian ring of Krull dimension zero is a direct product of a ﬁnite number (possibly one)
of local rings of Krull dimension zero.
• A Noetherian local ring is called a Cohen–Macaulay ring if its dimension is equal to its depth. A regular local
ring is an example of such a ring.
• A Noetherian integral domain is a unique factorization domain if and only if every height 1 prime ideal is
principal.[7]
• For a commutative Noetherian ring the three following conditions are equivalent: being a reduced ring of Krull
dimension zero, being a ﬁeld or a direct product of a ﬁnite number of ﬁelds, being von Neumann regular.

27.4 Krull dimension of a module
If R is a commutative ring, and M is an R-module, we deﬁne the Krull dimension of M to be the Krull dimension of
the quotient of R making M a faithful module. That is, we deﬁne it by the formula:

dimR M := dim(R/ AnnR (M ))
where AnnR(M), the annihilator, is the kernel of the natural map R → EndR(M) of R into the ring of R-linear
endomorphisms of M.
In the language of schemes, ﬁnitely generated modules are interpreted as coherent sheaves, or generalized ﬁnite rank
vector bundles.

106

CHAPTER 27. KRULL DIMENSION

27.5 Krull dimension for non-commutative rings
The Krull dimension of a module over a possibly non-commutative ring is deﬁned as the deviation of the poset of
submodules ordered by inclusion. For commutative Noetherian rings, this is the same as the deﬁnition using chains
of prime ideals.[8] The two deﬁnitions can be diﬀerent for commutative rings which are not Noetherian.

27.6 See also
• Dimension theory (algebra)
• Regular local ring
• Hilbert function
• Krull’s principal ideal theorem
• Gelfand–Kirillov dimension
• Homological conjectures in commutative algebra

27.7 Notes
[1] Matsumura, Hideyuki: “Commutative Ring Theory”, page 30–31, 1989
[2] Eisenbud, D. Commutative Algebra (1995). Springer, Berlin. Exercise 9.6.
[3] Matsumura, H. Commutative Algebra (1970). Benjamin, New York. Example 14.E.
[4] Matsumura, Hideyuki: “Commutative Ring Theory”, page 30–31, 1989
[5] http://mathoverflow.net/questions/79959/krull-dimension-transcendence-degree
[6] Eisenbud 2004, Exercise 13.8
[7] Hartshorne,Robin:"Algebraic Geometry”, page 7,1977
[8] McConnell, J.C. and Robson, J.C. Noncommutative Noetherian Rings (2001). Amer. Math. Soc., Providence. Corollary
6.4.8.

27.8 Bibliography
• Irving Kaplansky, Commutative rings (revised ed.), University of Chicago Press, 1974, ISBN 0-226-42454-5.
Page 32.
• L.A. Bokhut'; I.V. L'vov; V.K. Kharchenko (1991). “I. Noncommuative rings”. In Kostrikin, A.I.; Shafarevich,
I.R. Algebra II. Encyclopaedia of Mathematical Sciences 18. Springer-Verlag. ISBN 3-540-18177-6. Sect.4.7.
• Eisenbud, David (1995), Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1, MR 1322960
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics 52, New York: SpringerVerlag, ISBN 978-0-387-90244-9, MR 0463157
• Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics (2nd
ed.), Cambridge University Press, ISBN 978-0-521-36764-6

Chapter 28

Matroid rank
In the mathematical theory of matroids, the rank of a matroid is the maximum size of an independent set in the
matroid. The rank of a subset S of elements of the matroid is, similarly, the maximum size of an independent subset
of S, and the rank function of the matroid maps sets of elements to their ranks.
The rank function is one of the fundamental concepts of matroid theory via which matroids may be axiomatized.
The rank functions of matroids form an important subclass of the submodular set functions, and the rank functions of
the matroids deﬁned from certain other types of mathematical object such as undirected graphs, matrices, and ﬁeld
extensions are important within the study of those objects.

28.1 Properties and axiomatization
The rank function of a matroid obeys the following properties.
• The value of the rank function is always a non-negative integer.
• For any two subsets A and B of E , r(A ∪ B) + r(A ∩ B) ≤ r(A) + r(B) . That is, the rank is a submodular
function.
• For any set A and element x , r(A) ≤ r(A ∪ {x}) ≤ r(A) + 1 . From the ﬁrst of these two inequalities it
follows more generally that, if A ⊂ B ⊂ E , then r(A) ≤ r(B) ≤ r(E) . That is, the rank is a monotonic
function.
These properties may be used as axioms to characterize the rank function of matroids: every integer-valued submodular function on the subsets of a ﬁnite set that obeys the inequalities r(A) ≤ r(A ∪ {x}) ≤ r(A) + 1 for all A and
x is the rank function of a matroid.[1][2]

28.2 Other matroid properties from rank
The rank function may be used to determine the other important properties of a matroid:
• A set is independent if and only if its rank equals its cardinality, and dependent if and only if it has greater
cardinality than rank.[3]
• A nonempty set is a circuit if its rank equals one plus its cardinality and every subset formed by removing one
element from the set has equal rank.[3]
• A set is a basis if its rank equals both its cardinality and the rank of the matroid.[3]
• A set is closed if it is maximal for its rank, in the sense that there does not exist another element that can be
added to it while maintaining the same rank.
• The diﬀerence |A| − r(A) is called the nullity or corank of the subset A . It is the minimum number of
elements that must be removed from A to obtain an independent set.[4]
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28.3 Ranks of special matroids
In graph theory, the circuit rank (or cyclomatic number) of a graph is the corank of the associated graphic matroid;
it measures the minimum number of edges that must be removed from the graph to make the remaining edges form
a forest.[5] Several authors have studied the parameterized complexity of graph algorithms parameterized by this
number.[6][7]
In linear algebra, the rank of a linear matroid deﬁned by linear independence from the columns of a matrix is the
rank of the matrix,[8] and it is also the dimension of the vector space spanned by the columns.
In abstract algebra, the rank of a matroid deﬁned from sets of elements in a ﬁeld extension L/K by algebraic independence is known as the transcendence degree.[9]

28.4 See also
• Rank oracle

28.5 References
[1] Shikare, M. M.; Waphare, B. N. (2004), Combinatorial Optimization, Alpha Science Int'l Ltd., p. 155, ISBN 9788173195600.
[2] Welsh, D. J. A. (2010), Matroid Theory, Courier Dover Publications, p. 8, ISBN 9780486474397.
[3] Oxley (2006), p. 25.
[4] Oxley (2006), p. 34.
[5] Berge, Claude (2001), “Cyclomatic number”, The Theory of Graphs, Courier Dover Publications, pp. 27–30, ISBN
9780486419756.
[6] Coppersmith, Don; Vishkin, Uzi (1985), “Solving NP-hard problems in 'almost trees’: Vertex cover”, Discrete Applied
Mathematics 10 (1): 27–45, doi:10.1016/0166-218X(85)90057-5, Zbl 0573.68017.
[7] Fiala, Jiří; Kloks, Ton; Kratochvíl, Jan (2001), “Fixed-parameter complexity of λ-labelings”, Discrete Applied Mathematics
113 (1): 59–72, doi:10.1016/S0166-218X(00)00387-5, Zbl 0982.05085.
[8] Oxley, James G. (2006), Matroid Theory, Oxford Graduate Texts in Mathematics 3, Oxford University Press, p. 81, ISBN
9780199202508.
[9] Lindström, B. (1988), “Matroids, algebraic and non-algebraic”, Algebraic, extremal and metric combinatorics, 1986 (Montreal, PQ, 1986), London Math. Soc. Lecture Note Ser. 131, Cambridge: Cambridge Univ. Press, pp. 166–174, MR
1052666.

Chapter 29

Negative-dimensional space
In topology, a discipline within mathematics, a negative-dimensional space is an extension of the usual notion of
space allowing for negative dimensions.[1]

29.1 Deﬁnition
Suppose that Mt 0 is a compact space of Hausdorﬀ dimension t 0 , which is an element of a scale of compact spaces
embedded in each other and parametrized by t (0 < t < ∞). Such scales are considered equivalent with respect to Mt 0
if the compact spaces constituting them coincide for t ≥ t 0 . It is said that the compact space Mt 0 is the hole in this
equivalent set of scales, and −t 0 is the negative dimension of the corresponding equivalence class.[2]

29.2 History
By the 1940s, the science of topology had developed and studied a thorough basic theory of topological spaces of
positive dimension. Motivated by computations, and to some extent aesthetics, topologists searched for mathematical
frameworks that extended our notion of space to allow for negative dimensions. Such dimensions, as well as the
fourth and higher dimensions, are hard to imagine since we are not able to directly observe them. It wasn’t until the
1960s that a special topological framework was constructed—the category of spectra. A spectrum is a generalization
of space that allows for negative dimensions. The concept of negative-dimensional spaces is applied, for example, to
analyze linguistic statistics.[3]

29.3 See also
• Cone (topology)
• Equidimensionality
• Join (topology)
• Suspension/desuspension
• Spectrum (topology)

29.4 References
[1] Wolcott, Luke; McTernan, Elizabeth (2012). “Imagining Negative-Dimensional Space” (PDF). In Bosch, Robert; McKenna,
Douglas; Sarhangi, Reza. Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture. Phoenix, Arizona,
USA: Tessellations Publishing. pp. 637–642. ISBN 978-1-938664-00-7. ISSN 1099-6702. Retrieved 25 June 2015.

109

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CHAPTER 29. NEGATIVE-DIMENSIONAL SPACE

[2] Maslov, V.P. “General Notion of a Topological Space of Negative Dimension and Quantization of Its Density”. springer.com.
Retrieved 2015-06-23.
[3] Maslov, V.P. “Negative Dimension in General and Asymptotic Topology”. arxiv.org. Retrieved 2015-06-25.

29.5 External links
• Отрицательная асимптотическая топологическая размерность, новый конденсат и их связь с квантованным
законом Ципфа. For a translation into English, see Maslov, V.P. (November 2006). “Negative asymptotic
topological dimension, a new condensate, and their relation to the quantized Zipf law”. Mathematical Notes 80
(5-6): 806–813. Retrieved 30 June 2015.

Chapter 30

Nonlinear dimensionality reduction
High-dimensional data, meaning data that requires more than two or three dimensions to represent, can be diﬃcult to
interpret. One approach to simpliﬁcation is to assume that the data of interest lie on an embedded non-linear manifold
within the higher-dimensional space. If the manifold is of low enough dimension, the data can be visualised in the
low-dimensional space.

Top-left: a 3D dataset of 1000 points in a spiraling band (a.k.a. the Swiss roll) with a rectangular hole in the middle. Top-right:
the original 2D manifold used to generate the 3D dataset. Bottom left and right: 2D recoveries of the manifold respectively using the
LLE and Hessian LLE algorithms as implemented by the Modular Data Processing toolkit.

Below is a summary of some of the important algorithms from the history of manifold learning and nonlinear
dimensionality reduction (NLDR).[1] Many of these non-linear dimensionality reduction methods are related to
the linear methods listed below. Non-linear methods can be broadly classiﬁed into two groups: those that provide a
mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa), and those that
just give a visualisation. In the context of machine learning, mapping methods may be viewed as a preliminary feature
111

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CHAPTER 30. NONLINEAR DIMENSIONALITY REDUCTION

extraction step, after which pattern recognition algorithms are applied. Typically those that just give a visualisation
are based on proximity data – that is, distance measurements.

30.1 Linear methods
• Independent component analysis (ICA).
• Principal component analysis (PCA) (also called Karhunen–Loève transform — KLT).
• Singular value decomposition (SVD).
• Factor analysis.

30.2 Uses for NLDR
Consider a dataset represented as a matrix (or a database table), such that each row represents a set of attributes (or
features or dimensions) that describe a particular instance of something. If the number of attributes is large, then the
space of unique possible rows is exponentially large. Thus, the larger the dimensionality, the more diﬃcult it becomes
to sample the space. This causes many problems. Algorithms that operate on high-dimensional data tend to have a
very high time complexity. Many machine learning algorithms, for example, struggle with high-dimensional data.
This has become known as the curse of dimensionality. Reducing data into fewer dimensions often makes analysis
algorithms more eﬃcient, and can help machine learning algorithms make more accurate predictions.
Humans often have diﬃculty comprehending data in many dimensions. Thus, reducing data to a small number of
dimensions is useful for visualization purposes.

Plot of the two-dimensional points that results from using a NLDR algorithm. In this case, Manifold Sculpting used to reduce the
data into just two dimensions (rotation and scale).

The reduced-dimensional representations of data are often referred to as “intrinsic variables”. This description implies
that these are the values from which the data was produced. For example, consider a dataset that contains images of
a letter 'A', which has been scaled and rotated by varying amounts. Each image has 32x32 pixels. Each image can be
represented as a vector of 1024 pixel values. Each row is a sample on a two-dimensional manifold in 1024-dimensional
space (a Hamming space). The intrinsic dimensionality is two, because two variables (rotation and scale) were varied
in order to produce the data. Information about the shape or look of a letter 'A' is not part of the intrinsic variables
because it is the same in every instance. Nonlinear dimensionality reduction will discard the correlated information
(the letter 'A') and recover only the varying information (rotation and scale). The image to the right shows sample
images from this dataset (to save space, not all input images are shown), and a plot of the two-dimensional points
that results from using a NLDR algorithm (in this case, Manifold Sculpting was used) to reduce the data into just two
dimensions.

30.3. MANIFOLD LEARNING ALGORITHMS

113

PCA (a linear dimensionality reduction algorithm) is used to reduce this same dataset into two dimensions, the resulting values are
not so well organized.

By comparison, if PCA (a linear dimensionality reduction algorithm) is used to reduce this same dataset into two
dimensions, the resulting values are not so well organized. This demonstrates that the high-dimensional vectors (each
representing a letter 'A') that sample this manifold vary in a non-linear manner.
It should be apparent, therefore, that NLDR has several applications in the ﬁeld of computer-vision. For example,
consider a robot that uses a camera to navigate in a closed static environment. The images obtained by that camera
can be considered to be samples on a manifold in high-dimensional space, and the intrinsic variables of that manifold
will represent the robot’s position and orientation. This utility is not limited to robots. Dynamical systems, a more
general class of systems, which includes robots, are deﬁned in terms of a manifold. Active research in NLDR seeks to
unfold the observation manifolds associated with dynamical systems to develop techniques for modeling such systems
and enable them to operate autonomously.[2]

30.3 Manifold learning algorithms
Some of the more prominent manifold learning algorithms are listed below (in approximately chronological order).
An algorithm may learn an internal model of the data, which can be used to map points unavailable at training time
into the embedding in a process often called out-of-sample extension.

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30.3.1

CHAPTER 30. NONLINEAR DIMENSIONALITY REDUCTION

Sammon’s mapping

Sammon’s mapping is one of the ﬁrst and most popular NLDR techniques.

Approximation of a principal curve by one-dimensional SOM (a broken line with red squares, 20 nodes). The ﬁrst principal component is presented by a blue straight line. Data points are the small grey circles. For PCA, the Fraction of variance unexplained in
this example is 23.23%, for SOM it is 6.86%.[3]

30.3.2

Self-organizing map

The self-organizing map (SOM, also called Kohonen map) and its probabilistic variant generative topographic mapping (GTM) use a point representation in the embedded space to form a latent variable model based on a non-linear
mapping from the embedded space to the high-dimensional space.[4] These techniques are related to work on density
networks, which also are based around the same probabilistic model.

30.3.3

Principal curves and manifolds

Principal curves and manifolds give the natural geometric framework for nonlinear dimensionality reduction and
extend the geometric interpretation of PCA by explicitly constructing an embedded manifold, and by encoding using standard geometric projection onto the manifold. This approach was proposed by Trevor Hastie in his thesis
(1984)[8] and developed further by many authors.[9] How to deﬁne the “simplicity” of the manifold is problemdependent, however, it is commonly measured by the intrinsic dimensionality and/or the smoothness of the manifold.
Usually, the principal manifold is deﬁned as a solution to an optimization problem. The objective function includes
a quality of data approximation and some penalty terms for the bending of the manifold. The popular initial approximations are generated by linear PCA, Kohonen’s SOM or autoencoders. The elastic map method provides the
expectation-maximization algorithm for principal manifold learning with minimization of quadratic energy functional
at the “maximization” step.

30.3. MANIFOLD LEARNING ALGORITHMS

115

Application of principal curves: Nonlinear quality of life index.[5] Points represent data of the UN 171 countries in 4-dimensional
space formed by the values of 4 indicators: gross product per capita, life expectancy, infant mortality, tuberculosis incidence. Diﬀerent
forms and colors correspond to various geographical locations. Red bold line represents the principal curve, approximating the
dataset. This principal curve was produced by the method of elastic map. Software is available for free non-commercial use.[6][7]

30.3.4

Autoencoders

An autoencoder is a feed-forward neural network which is trained to approximate the identity function. That is, it is
trained to map from a vector of values to the same vector. When used for dimensionality reduction purposes, one of
the hidden layers in the network is limited to contain only a small number of network units. Thus, the network must
learn to encode the vector into a small number of dimensions and then decode it back into the original space. Thus,
the ﬁrst half of the network is a model which maps from high to low-dimensional space, and the second half maps
from low to high-dimensional space. Although the idea of autoencoders is quite old, training of deep autoencoders has
only recently become possible through the use of restricted Boltzmann machines and stacked denoising autoencoders.
Related to autoencoders is the NeuroScale algorithm, which uses stress functions inspired by multidimensional scaling
and Sammon mappings (see below) to learn a non-linear mapping from the high-dimensional to the embedded space.
The mappings in NeuroScale are based on radial basis function networks.

30.3.5

Gaussian process latent variable models

Gaussian process latent variable models (GPLVM)[10] are probabilistic dimensionality reduction methods that use
Gaussian Processes (GPs) to ﬁnd a lower dimensional non-linear embedding of high dimensional data. They are an
extension of the Probabilistic formulation of PCA. The model is deﬁned probabilistically and the latent variables are
then marginalized and parameters are obtained by maximizing the likelihood. Like kernel PCA they use a kernel
function to form a non linear mapping (in the form of a Gaussian process). However in the GPLVM the mapping is
from the embedded(latent) space to the data space (like density networks and GTM) whereas in kernel PCA it is in
the opposite direction. It was originally proposed for visualization of high dimensional data but has been extended to
construct a shared manifold model between two observation spaces.

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CHAPTER 30. NONLINEAR DIMENSIONALITY REDUCTION

30.3.6

Curvilinear component analysis

Curvilinear component analysis (CCA)[11] looks for the conﬁguration of points in the output space that preserves
original distances as much as possible while focusing on small distances in the output space (conversely to Sammon’s
mapping which focus on small distances in original space).
It should be noticed that CCA, as an iterative learning algorithm, actually starts with focus on large distances (like the
Sammon algorithm), then gradually change focus to small distances. The small distance information will overwrite
the large distance information, if compromises between the two have to be made.
The stress function of CCA is related to a sum of right Bregman divergences[12]

30.3.7

Curvilinear distance analysis

CDA[11] trains a self-organizing neural network to ﬁt the manifold and seeks to preserve geodesic distances in its
embedding. It is based on Curvilinear Component Analysis (which extended Sammon’s mapping), but uses geodesic
distances instead.

30.3.8

Diﬀeomorphic dimensionality reduction

Diﬀeomorphic Dimensionality Reduction or Diﬀeomap[13] learns a smooth diﬀeomorphic mapping which transports
the data onto a lower-dimensional linear subspace. The methods solves for a smooth time indexed vector ﬁeld such that
ﬂows along the ﬁeld which start at the data points will end at a lower-dimensional linear subspace, thereby attempting
to preserve pairwise diﬀerences under both the forward and inverse mapping.

30.3.9

Kernel principal component analysis

Perhaps the most widely used algorithm for manifold learning is kernel PCA.[14] It is a combination of Principal
component analysis and the kernel trick. PCA begins by computing the covariance matrix of the m × n matrix X
1 ∑
xi xT
i.
m i=1
m

C=

It then projects the data onto the ﬁrst k eigenvectors of that matrix. By comparison, KPCA begins by computing the
covariance matrix of the data after being transformed into a higher-dimensional space,
1 ∑
Φ(xi )Φ(xi )T .
m i=1
m

C=

It then projects the transformed data onto the ﬁrst k eigenvectors of that matrix, just like PCA. It uses the kernel trick
to factor away much of the computation, such that the entire process can be performed without actually computing
Φ(x) . Of course Φ must be chosen such that it has a known corresponding kernel. Unfortunately, it is not trivial to
ﬁnd a good kernel for a given problem, so KPCA does not yield good results with some problems when using standard
kernels. For example, it is known to perform poorly with these kernels on the Swiss roll manifold. However, one can
view certain other methods that perform well in such settings (e.g., Laplacian Eigenmaps, LLE) as special cases of
kernel PCA by constructing a data-dependent kernel matrix.[15]
KPCA has an internal model, so it can be used to map points onto its embedding that were not available at training
time.

30.3.10

Isomap

Isomap[16] is a combination of the Floyd–Warshall algorithm with classic Multidimensional Scaling. Classic Multidimensional Scaling (MDS) takes a matrix of pair-wise distances between all points, and computes a position for
each point. Isomap assumes that the pair-wise distances are only known between neighboring points, and uses the

30.3. MANIFOLD LEARNING ALGORITHMS

117

Floyd–Warshall algorithm to compute the pair-wise distances between all other points. This eﬀectively estimates the
full matrix of pair-wise geodesic distances between all of the points. Isomap then uses classic MDS to compute the
reduced-dimensional positions of all the points.
Landmark-Isomap is a variant of this algorithm that uses landmarks to increase speed, at the cost of some accuracy.

30.3.11

Locally-linear embedding

Locally-Linear Embedding (LLE)[17] was presented at approximately the same time as Isomap. It has several advantages over Isomap, including faster optimization when implemented to take advantage of sparse matrix algorithms,
and better results with many problems. LLE also begins by ﬁnding a set of the nearest neighbors of each point. It
then computes a set of weights for each point that best describe the point as a linear combination of its neighbors.
Finally, it uses an eigenvector-based optimization technique to ﬁnd the low-dimensional embedding of points, such
that each point is still described with the same linear combination of its neighbors. LLE tends to handle non-uniform
sample densities poorly because there is no ﬁxed unit to prevent the weights from drifting as various regions diﬀer in
sample densities. LLE has no internal model.
LLE computes the barycentric coordinates of a point Xi based on its neighbors Xj. The original point is reconstructed
by a linear combination, given by the weight matrix Wij, of its neighbors. The reconstruction error is given by the
cost function E(W).

E(W ) =

∑

|Xi −

i

∑

Wij Xj |

2

j

The weights Wij refer to the amount of contribution the point Xj has while reconstructing the point Xi. The cost
function is minimized under two constraints: (a) Each data point Xi is reconstructed only from its neighbors, thus
enforcing Wij to be zero if point Xj is not a neighbor of the point Xi and (b) The sum of every row of the weight
matrix equals 1.
∑

Wij = 1

j

The original data points are collected in a D dimensional space and the goal of the algorithm is to reduce the dimensionality to d such that D >> d. The same weights Wij that reconstructs the ith data point in the D dimensional
space will be used to reconstruct the same point in the lower d dimensional space. A neighborhood preserving map is
created based on this idea. Each point Xᵢ in the D dimensional space is mapped onto a point Yᵢ in the d dimensional
space by minimizing the cost function

C(Y ) =

∑
i

|Yi −

∑

Wij Yj |

2

j

In this cost function, unlike the previous one, the weights Wᵢ are kept ﬁxed and the minimization is done on the
points Yᵢ to optimize the coordinates. This minimization problem can be solved by solving a sparse N X N eigen
value problem (N being the number of data points), whose bottom d nonzero eigen vectors provide an orthogonal
set of coordinates. Generally the data points are reconstructed from K nearest neighbors, as measured by Euclidean
distance. For such an implementation the algorithm has only one free parameter K, which can be chosen by cross
validation.

30.3.12

Laplacian eigenmaps

Laplacian Eigenmaps[18] uses spectral techniques to perform dimensionality reduction. This technique relies on the
basic assumption that the data lies in a low-dimensional manifold in a high-dimensional space.[19] This algorithm
cannot embed out of sample points, but techniques based on Reproducing kernel Hilbert space regularization exist
for adding this capability.[20] Such techniques can be applied to other nonlinear dimensionality reduction algorithms
as well.

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CHAPTER 30. NONLINEAR DIMENSIONALITY REDUCTION

Traditional techniques like principal component analysis do not consider the intrinsic geometry of the data. Laplacian
eigenmaps builds a graph from neighborhood information of the data set. Each data point serves as a node on the
graph and connectivity between nodes is governed by the proximity of neighboring points (using e.g. the k-nearest
neighbor algorithm). The graph thus generated can be considered as a discrete approximation of the low-dimensional
manifold in the high-dimensional space. Minimization of a cost function based on the graph ensures that points close
to each other on the manifold are mapped close to each other in the low-dimensional space, preserving local distances.
The eigenfunctions of the Laplace–Beltrami operator on the manifold serve as the embedding dimensions, since under
mild conditions this operator has a countable spectrum that is a basis for square integrable functions on the manifold
(compare to Fourier series on the unit circle manifold). Attempts to place Laplacian eigenmaps on solid theoretical
ground have met with some success, as under certain nonrestrictive assumptions, the graph Laplacian matrix has
been shown to converge to the Laplace–Beltrami operator as the number of points goes to inﬁnity.[21] Matlab code
for Laplacian Eigenmaps can be found in algorithms[22] and the PhD thesis of Belkin can be found at the Ohio State
University.[23]
In classiﬁcation applications, low dimension manifolds can be used to model data classes which can be deﬁned from
sets of observed instances. Each observed instance can be described by two independent factors termed ’content’ and
’style’, where ’content’ is the invariant factor related to the essence of the class and ’style’ expresses variations in that
class between instances.[24] Unfortunately, Laplacian Eigenmaps may fail to produce a coherent representation of a
class of interest when training data consist of instances varying signiﬁcantly in terms of style.[25] In the case of classes
which are represented by multivariate sequences, Structural Laplacian Eigenmaps has been proposed to overcome
this issue by adding additional constraints within the Laplacian Eigenmaps neighborhood information graph to better
reﬂect the intrinsic structure of the class.[26] More speciﬁcally, the graph is used to encode both the sequential structure of the multivariate sequences and, to minimise stylistic variations, proximity between data points of diﬀerent
sequences or even within a sequence, if it contains repetitions. Using dynamic time warping, proximity is detected
by ﬁnding correspondences between and within sections of the multivariate sequences that exhibit high similarity.
Experiments conducted on vision-based activity recognition, object orientation classiﬁcation and human 3D pose
recovery applications have demonstrate the added value of Structural Laplacian Eigenmaps when dealing with multivariate sequence data.[26] An extension of Structural Laplacian Eigenmaps, Generalized Laplacian Eigenmaps led to
the generation of manifolds where one of the dimensions speciﬁcally represents variations in style. This has proved
particularly valuable in applications such as tracking of the human articulated body and silhouette extraction.[27]

30.3.13

Manifold alignment

Manifold alignment takes advantage of the assumption that disparate data sets produced by similar generating processes will share a similar underlying manifold representation. By learning projections from each original space to
the shared manifold, correspondences are recovered and knowledge from one domain can be transferred to another.
Most manifold alignment techniques consider only two data sets, but the concept extends to arbitrarily many initial
data sets.[28]

30.3.14

Diﬀusion maps

Diﬀusion maps leverages the relationship between heat diﬀusion and a random walk (Markov Chain); an analogy is
drawn between the diﬀusion operator on a manifold and a Markov transition matrix operating on functions deﬁned
on the graph whose nodes were sampled from the manifold.[29] In particular let a data set be represented by X =
[x1 , x2 , . . . , xn ] ∈ Ω ⊂ RD . The underlying assumption of diﬀusion map is that the data although high-dimensional,
lies on a low-dimensional manifold of dimensions d .X represents the data set and let µ represent the distribution of
the data points on X. In addition to this lets deﬁne a kernel which represents some notion of aﬃnity of the points in
X. The kernel k has the following properties[30]

k(x, y) = k(y, x),
k is symmetric

k(x, y) ≥ 0

∀x, y, k

k is positivity preserving

30.3. MANIFOLD LEARNING ALGORITHMS

119

Thus one can think of the individual data points as the nodes of a graph and the kernel k deﬁning some sort of aﬃnity
on that graph. The graph is symmetric by construction since the kernel is symmetric. It is easy to see here that from
the tuple {X,k} one can construct a reversible Markov Chain. This technique is fairly popular in a variety of ﬁelds
and is known as the graph laplacian.
The graph K = (X,E) can be constructed for example using a Gaussian kernel.
{
Kij =

e−∥xi −xj ∥2 /σ
0
2

2

ifxi ∼ xj
otherwise

In this above equation xi ∼ xj denotes that xi is a nearest neighbor of xj . In reality Geodesic distance should be
used to actually measure distances on the manifold. Since the exact structure of the manifold is not available, the
geodesic distance is approximated by euclidean distances with only nearest neighbors. The choice σ modulates our
notion of proximity in the sense that if ∥xi − xj ∥2 ≫ σ then Kij = 0 and if ∥xi − xj ∥2 ≪ σ then Kij = 1 .
The former means that very little diﬀusion has taken place while the latter implies that the diﬀusion process is nearly
complete. Diﬀerent strategies to choose σ can be found in.[31] If K has to faithfully represent a Markov matrix, then
it has to be normalized by the corresponding degree matrix D :
P = D−1 K.
P now represents a Markov chain. P (xi , xj ) is the probability of transitioning from xi to xj in one a time step.
Similarly the probability of transitioning from xi to xj in t time steps is given by P t (xi , xj ) . Here P t is the matrix
P multiplied to itself t times. Now the Markov matrix P constitutes some notion of local geometry of the data set
X. The major diﬀerence between diﬀusion maps and principal component analysis is that only local features of the
data is considered in diﬀusion maps as opposed to taking correlations of the entire data set.
K deﬁnes a random walk on the data set which means that the kernel captures some local geometry of data set. The
Markov chain deﬁnes fast and slow directions of propagation, based on the values taken by the kernel, and as one
propagates the walk forward in time, the local geometry information aggregates in the same way as local transitions
(deﬁned by diﬀerential equations) of the dynamical system.[30] The concept of diﬀusion arises from the deﬁnition of
a family diﬀusion distance { Dt } t∈N
Dt2 (x, y) = ||pt (x, ·) − pt (y, ·)||2
For a given value of t Dt deﬁnes a distance between any two points of the data set. This means that the value of
Dt (x, y) will be small if there are many paths that connect x to y and vice versa. The quantity Dt (x, y) involves
summing over of all paths of length t, as a result of which Dt is extremely robust to noise in the data as opposed to
geodesic distance. Dt takes into account all the relation between points x and y while calculating the distance and
serves as a better notion of proximity than just Euclidean distance or even geodesic distance.

30.3.15

Hessian Locally-Linear Embedding (Hessian LLE)

Like LLE, Hessian LLE[32] is also based on sparse matrix techniques. It tends to yield results of a much higher quality
than LLE. Unfortunately, it has a very costly computational complexity, so it is not well-suited for heavily-sampled
manifolds. It has no internal model.

30.3.16

Modiﬁed Locally-Linear Embedding (MLLE)

Modiﬁed LLE (MLLE)[33] is another LLE variant which uses multiple weights in each neighborhood to address the
local weight matrix conditioning problem which leads to distortions in LLE maps. MLLE produces robust projections
similar to Hessian LLE, but without the signiﬁcant additional computational cost.

30.3.17

Relational perspective map

Relational perspective map is a multidimensional scaling algorithm. The algorithm ﬁnds a conﬁguration of data points
on a manifold by simulating a multi-particle dynamic system on a closed manifold, where data points are mapped to

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CHAPTER 30. NONLINEAR DIMENSIONALITY REDUCTION

particles and distances (or dissimilarity) between data points represent a repulsive force. As the manifold gradually
grows in size the multi-particle system cools down gradually and converges to a conﬁguration that reﬂects the distance
information of the data points.
Relational perspective map was inspired by a physical model in which positively charged particles move freely on the
surface of a ball. Guided by the Coulomb force between particles, the minimal energy conﬁguration of the particles
will reﬂect the strength of repulsive forces between the particles.
The Relational perspective map was introduced in.[34] The algorithm ﬁrstly used the ﬂat torus as the image manifold,
then it has been extended (in the software VisuMap to use other types of closed manifolds, like the sphere, projective
space, and Klein bottle, as image manifolds.

30.3.18

Local tangent space alignment

Main article: Local tangent space alignment
LTSA[35] is based on the intuition that when a manifold is correctly unfolded, all of the tangent hyperplanes to the
manifold will become aligned. It begins by computing the k-nearest neighbors of every point. It computes the tangent
space at every point by computing the d-ﬁrst principal components in each local neighborhood. It then optimizes to
ﬁnd an embedding that aligns the tangent spaces.

30.3.19

Local multidimensional scaling

Local Multidimensional Scaling[36] performs multidimensional scaling in local regions, and then uses convex optimization to ﬁt all the pieces together.

30.3.20

Maximum variance unfolding

Maximum Variance Unfolding was formerly known as Semideﬁnite Embedding. The intuition for this algorithm is
that when a manifold is properly unfolded, the variance over the points is maximized. This algorithm also begins
by ﬁnding the k-nearest neighbors of every point. It then seeks to solve the problem of maximizing the distance
between all non-neighboring points, constrained such that the distances between neighboring points are preserved.
The primary contribution of this algorithm is a technique for casting this problem as a semideﬁnite programming
problem. Unfortunately, semideﬁnite programming solvers have a high computational cost. The Landmark–MVU
variant of this algorithm uses landmarks to increase speed with some cost to accuracy. It has no model.

30.3.21

Nonlinear PCA

Nonlinear PCA[37] (NLPCA) uses backpropagation to train a multi-layer perceptron to ﬁt to a manifold. Unlike
typical MLP training, which only updates the weights, NLPCA updates both the weights and the inputs. That is, both
the weights and inputs are treated as latent values. After training, the latent inputs are a low-dimensional representation of the observed vectors, and the MLP maps from that low-dimensional representation to the high-dimensional
observation space.

30.3.22

Data-driven high-dimensional scaling

Data-Driven High Dimensional Scaling (DD-HDS)[38] is closely related to Sammon’s mapping and curvilinear component analysis except that (1) it simultaneously penalizes false neighborhoods and tears by focusing on small distances
in both original and output space, and that (2) it accounts for concentration of measure phenomenon by adapting the
weighting function to the distance distribution.

30.3.23

Manifold sculpting

Manifold Sculpting[39] uses graduated optimization to ﬁnd an embedding. Like other algorithms, it computes the
k-nearest neighbors and tries to seek an embedding that preserves relationships in local neighborhoods. It slowly

30.4. METHODS BASED ON PROXIMITY MATRICES

121

scales variance out of higher dimensions, while simultaneously adjusting points in lower dimensions to preserve those
relationships. If the rate of scaling is small, it can ﬁnd very precise embeddings. It boasts higher empirical accuracy
than other algorithms with several problems. It can also be used to reﬁne the results from other manifold learning
algorithms. It struggles to unfold some manifolds, however, unless a very slow scaling rate is used. It has no model.

30.3.24

t-distributed stochastic neighbor embedding

t-distributed stochastic neighbor embedding (t-SNE) [40] is widely used. It is one of a family of stochastic neighbor
embedding methods.

30.3.25

RankVisu

RankVisu[41] is designed to preserve rank of neighborhood rather than distance. RankVisu is especially useful on
diﬃcult tasks (when the preservation of distance cannot be achieved satisfyingly). Indeed, the rank of neighborhood
is less informative than distance (ranks can be deduced from distances but distances cannot be deduced from ranks)
and its preservation is thus easier.

30.3.26

Topologically constrained isometric embedding

Topologically Constrained Isometric Embedding (TCIE)[42] is an algorithm based approximating geodesic distances
after ﬁltering geodesics inconsistent with the Euclidean metric. Aimed at correcting the distortions caused when
Isomap is used to map intrinsically non-convex data, TCIE uses weight least-squares MDS in order to obtain a more
accurate mapping. The TCIE algorithm ﬁrst detects possible boundary points in the data, and during computation of
the geodesic length marks inconsistent geodesics, to be given a small weight in the weighted Stress majorization that
follows.

30.4 Methods based on proximity matrices
A method based on proximity matrices is one where the data is presented to the algorithm in the form of a similarity
matrix or a distance matrix. These methods all fall under the broader class of metric multidimensional scaling.
The variations tend to be diﬀerences in how the proximity data is computed; for example, Isomap, locally linear
embeddings, maximum variance unfolding, and Sammon mapping (which is not in fact a mapping) are examples of
metric multidimensional scaling methods.

30.5 See also
• Discriminant analysis
• Elastic map[43]
• Feature learning
• Growing self-organizing map (GSOM)
• Multilinear subspace learning (MSL)
• Pairwise distance methods
• Self-organizing map (SOM)

30.6 References
[1] John A. Lee, Michel Verleysen, Nonlinear Dimensionality Reduction, Springer, 2007.

122

CHAPTER 30. NONLINEAR DIMENSIONALITY REDUCTION

[2] Gashler, M. and Martinez, T., Temporal Nonlinear Dimensionality Reduction, In Proceedings of the International Joint
Conference on Neural Networks IJCNN'11, pp. 1959–1966, 2011
[3] The illustration is prepared using free software: E.M. Mirkes, Principal Component Analysis and Self-Organizing Maps:
applet. University of Leicester, 2011
[4] Yin, Hujun; Learning Nonlinear Principal Manifolds by Self-Organising Maps, in A.N. Gorban, B. Kégl, D.C. Wunsch,
and A. Zinovyev (Eds.), Principal Manifolds for Data Visualization and Dimension Reduction, Lecture Notes in Computer
Science and Engineering (LNCSE), vol. 58, Berlin, Germany: Springer, 2007, Ch. 3, pp. 68-95. ISBN 978-3-540-737490
[5] A. N. Gorban, A. Zinovyev, Principal manifolds and graphs in practice: from molecular biology to dynamical systems,
International Journal of Neural Systems, Vol. 20, No. 3 (2010) 219–232.
[6] A. Zinovyev, ViDaExpert - Multidimensional Data Visualization Tool (free for non-commercial use). Institut Curie, Paris.
[7] A. Zinovyev, ViDaExpert overview, IHES (Institut des Hautes Études Scientiﬁques), Bures-Sur-Yvette, Île-de-France.
[8] T. Hastie, Principal Curves and Surfaces, Ph.D Dissertation, Stanford Linear Accelerator Center, Stanford University,
Stanford, California, US, November 1984.
[9] A.N. Gorban, B. Kégl, D.C. Wunsch, A. Zinovyev (Eds.), Principal Manifolds for Data Visualisation and Dimension
Reduction, Lecture Notes in Computer Science and Engineering (LNCSE), Vol. 58, Springer, Berlin – Heidelberg – New
York, 2007. ISBN 978-3-540-73749-0
[10] N. Lawrence, Probabilistic Non-linear Principal Component Analysis with Gaussian Process Latent Variable Models, Journal of Machine Learning Research 6(Nov):1783–1816, 2005.
[11] P. Demartines and J. Hérault, Curvilinear Component Analysis: A Self-Organizing Neural Network for Nonlinear Mapping
of Data Sets, IEEE Transactions on Neural Networks, Vol. 8(1), 1997, pp. 148–154
[12] Jigang Sun, Malcolm Crowe, and Colin Fyfe, Curvilinear component analysis and Bregman divergences, In European
Symposium on Artiﬁcial Neural Networks (Esann), pages 81–86. d-side publications, 2010
[13] Christian Walder and Bernhard Schölkopf, Diﬀeomorphic Dimensionality Reduction, Advances in Neural Information
Processing Systems 22, 2009, pp. 1713–1720, MIT Press
[14] B. Schölkopf, A. Smola, K.-R. Müller, Nonlinear Component Analysis as a Kernel Eigenvalue Problem. Neural Computation 10(5):1299-1319, 1998, MIT Press Cambridge, MA, USA, doi:10.1162/089976698300017467
[15] Jihun Ham, Daniel D. Lee, Sebastian Mika, Bernhard Schölkopf. A kernel view of the dimensionality reduction of
manifolds. Proceedings of the 21st International Conference on Machine Learning, Banﬀ, Canada, 2004. doi:10.1145/
1015330.1015417
[16] J. B. Tenenbaum, V. de Silva, J. C. Langford, A Global Geometric Framework for Nonlinear Dimensionality Reduction,
Science 290, (2000), 2319–2323.
[17] S. T. Roweis and L. K. Saul, Nonlinear Dimensionality Reduction by Locally Linear Embedding, Science Vol 290, 22
December 2000, 2323–2326.
[18] Mikhail Belkin and Partha Niyogi, Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering, Advances
in Neural Information Processing Systems 14, 2001, p. 586–691, MIT Press
[19] Mikhail Belkin Problems of Learning on Manifolds, PhD Thesis, Department of Mathematics, The University Of Chicago,
August 2003
[20] Bengio et al. “Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering” in Advances in
Neural Information Processing Systems (2004)
[21] Mikhail Belkin Problems of Learning on Manifolds, PhD Thesis, Department of Mathematics, The University Of Chicago,
August 2003
[22] Ohio-state.edu
[23] Ohio-state.edu
[24] J. Tenenbaum and W. Freeman, Separating style and content with bilinear models, Neural Computation, vol. 12, 2000.
[25] M. Lewandowski, J. Martinez-del Rincon, D. Makris, and J.-C. Nebel, Temporal extension of laplacian eigenmaps for
unsupervised dimensionality reduction of time series, Proceedings of the International Conference on Pattern Recognition
(ICPR), 2010

30.7. EXTERNAL LINKS

123

[26] M. Lewandowski, D. Makris, S.A. Velastin and J.-C. Nebel, Structural Laplacian Eigenmaps for Modeling Sets of Multivariate Sequences, IEEE Transactions on Cybernetics, 44(6): 936-949, 2014
[27] J. Martinez-del-Rincon, M. Lewandowski, J.-C. Nebel and D. Makris, Generalized Laplacian Eigenmaps for Modeling and
Tracking Human Motions, IEEE Transactions on Cybernetics, 44(9), pp 1646-1660, 2014
[28] Wang, Chang; Mahadevan, Sridhar (July 2008). Manifold Alignment using Procrustes Analysis (PDF). The 25th International Conference on Machine Learning. pp. 1120–1127.
[29] Diﬀusion Maps and Geometric Harmonics, Stephane Lafon, PhD Thesis, Yale University, May 2004
[30] Diﬀusion Maps, Ronald R. Coifman and Stephane Lafon,: Science, 19 June 2006
[31] B. Bah, “Diﬀusion Maps: Applications and Analysis”, Masters Thesis, University of Oxford
[32] D. Donoho and C. Grimes, “Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data” Proc
Natl Acad Sci U S A. 2003 May 13; 100(10): 5591–5596
[33] Z. Zhang and J. Wang, “MLLE: Modiﬁed Locally Linear Embedding Using Multiple Weights” http://citeseerx.ist.psu.edu/
viewdoc/summary?doi=10.1.1.70.382
[34] James X. Li, Visualizing high-dimensional data with relational perspective map, Information Visualization (2004) 3, 49–59
[35] Zhang, Zhenyue; Hongyuan Zha (2005). “Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent
Space Alignment”. SIAM Journal on Scientiﬁc Computing 26 (1): 313–338. doi:10.1137/s1064827502419154.
[36] J Venna and S Kaski, Local multidimensional scaling, Neural Networks, 2006
[37] Scholz, M. Kaplan, F. Guy, C. L. Kopka, J. Selbig, J., Non-linear PCA: a missing data approach, In Bioinformatics, Vol.
21, Number 20, pp. 3887–3895, Oxford University Press, 2005
[38] S. Lespinats, M. Verleysen, A. Giron, B. Fertil, DD-HDS: a tool for visualization and exploration of high-dimensional data,
IEEE Transactions on Neural Networks 18 (5) (2007) 1265–1279.
[39] Gashler, M. and Ventura, D. and Martinez, T., Iterative Non-linear Dimensionality Reduction with Manifold Sculpting, In
Platt, J.C. and Koller, D. and Singer, Y. and Roweis, S., editor, Advances in Neural Information Processing Systems 20,
pp. 513–520, MIT Press, Cambridge, MA, 2008
[40] van der Maaten, L.J.P.; Hinton, G.E. (Nov 2008). “Visualizing High-Dimensional Data Using t-SNE” (PDF). Journal of
Machine Learning Research 9: 2579–2605.
[41] Lespinats S., Fertil B., Villemain P. and Herault J., Rankvisu: Mapping from the neighbourhood network, Neurocomputing,
vol. 72 (13–15), pp. 2964–2978, 2009.
[42] Rosman G., Bronstein M. M., Bronstein A. M. and Kimmel R., Nonlinear Dimensionality Reduction by Topologically
Constrained Isometric Embedding, International Journal of Computer Vision, Volume 89, Number 1, 56–68, 2010
[43] ELastic MAPs

30.7 External links
• Isomap
• Generative Topographic Mapping
• Mike Tipping’s Thesis
• Gaussian Process Latent Variable Model
• Locally Linear Embedding
• Relational Perspective Map
• Waﬄes is an open source C++ library containing implementations of LLE, Manifold Sculpting, and some other
manifold learning algorithms.
• Eﬃcient Dimensionality Reduction Toolkit homepage

124

CHAPTER 30. NONLINEAR DIMENSIONALITY REDUCTION

• DD-HDS homepage
• RankVisu homepage
• Short review of Diﬀusion Maps
• Nonlinear PCA by autoencoder neural networks

Chapter 31

One-dimensional space

The number line

In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When
n = 1, the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the
number line, where the position of each point on it can be described by a single number.[1]

31.1 One-dimensional geometry
31.1.1

Polytopes

The only regular polytope in one dimension is the line segment, with the Schläﬂi symbol { }.

31.1.2

Hypersphere

The hypersphere in 1 dimension is a pair of points,[2] sometimes called a 0-sphere as its surface is zero-dimensional.
Its length is

L = 2r
where r is the radius.

31.2 Coordinate systems in one-dimensional space
Main article: Coordinate system
The most popular coordinate systems are the number line and the angle.
• Number line
• Angle
125

126

CHAPTER 31. ONE-DIMENSIONAL SPACE

31.3 References
[1] Гущин, Д. Д. "Пространство как математическое понятие" (in Russian). fmclass.ru. Retrieved 2015-06-06.
[2] Gibilisco, Stan (1983). Understanding Einstein’s Theories of Relativity: Man’s New Perspective on the Cosmos. TAB Books.
p. 89.

Chapter 32

Ordinate

y

(2,3)

3
2

(−3,1)

1

(0,0)
−3

−2

−1

1

x
2

3

−1
−2

(−1.5,−2.5)

−3

Illustration of a Cartesian coordinate plane. The second value in each ordered pair is the ordinate of the corresponding point.

In mathematics, ordinate most often refers to that element of an ordered pair which is plotted on the vertical axis of a
two-dimensional Cartesian coordinate system, as opposed to the abscissa. The term can also refer to the vertical axis
(typically y-axis) of a two-dimensional graph (because that axis is used to deﬁne and measure the vertical coordinates
127

128

CHAPTER 32. ORDINATE

of points in the space). An ordered pair consists of two terms—the abscissa (horizontal, usually x) and the ordinate
(vertical, usually y)—which deﬁne the location of a point in two-dimensional rectangular space.[1]

abscissa ordinate

z}|{ z}|{
( x , y )

32.1 Examples
• For the red point in the above graph (−3, 1), −3 is called the abscissa and 1, the ordinate.

32.2 See also
• Abscissa
• Ordered pair
• Cartesian coordinate conventions
• Dependent and independent variables
• Function (mathematics)
• Relation (mathematics)
• Line chart
• The dictionary deﬁnition of ordinate at Wiktionary

32.3 References
[1] Weisstein, Eric W. “Ordinate”. MathWorld--A Wolfram Web Resource. Retrieved 14 July 2013. External link in |work=
(help)

32.4 External links
• Earliest Known Uses of Some of the Words of Mathematics (O)
This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008
and incorporated under the “relicensing” terms of the GFDL, version 1.3 or later.

Chapter 33

Regular sequence
For regular Cauchy sequence, see Cauchy sequence#In constructive mathematics.
In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent
as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection.

33.1 Deﬁnitions
For a commutative ring R and an R-module M, an element r in R is called a non-zero-divisor on M if r m = 0
implies m = 0 for m in M. An M-regular sequence is a sequence
r1 , ..., rd in R
such that ri is a non-zero-divisor on M/(r1 , ..., ri−₁)M for i = 1, ..., d.[1] Some authors also require that M/(r1 , ...,
rd)M is not zero. Intuitively, to say that r1 , ..., rd is an M-regular sequence means that these elements “cut M down”
as much as possible, when we pass successively from M to M/(r1 )M, to M/(r1 , r2 )M, and so on.
An R-regular sequence is called simply a regular sequence. That is, r1 , ..., rd is a regular sequence if r1 is a nonzero-divisor in R, r2 is a non-zero-divisor in the ring R/(r1 ), and so on. In geometric language, if X is an aﬃne scheme
and r1 , ..., rd is a regular sequence in the ring of regular functions on X, then we say that the closed subscheme {r1 =0,
..., rd=0} ⊂ X is a complete intersection subscheme of X.
For example, x, y(1-x), z(1-x) is a regular sequence in the polynomial ring C[x, y, z], while y(1-x), z(1-x), x is not a
regular sequence. But if R is a Noetherian local ring and the elements ri are in the maximal ideal, or if R is a graded
ring and the ri are homogeneous of positive degree, then any permutation of a regular sequence is a regular sequence.
Let R be a Noetherian ring, I an ideal in R, and M a ﬁnitely generated R-module. The depth of I on M, written
depthR(I, M) or just depth(I, M), is the supremum of the lengths of all M-regular sequences of elements of I. When
R is a Noetherian local ring and M is a ﬁnitely generated R-module, the depth of M, written depthR(M) or just
depth(M), means depthR(m, M); that is, it is the supremum of the lengths of all M-regular sequences in the maximal
ideal m of R. In particular, the depth of a Noetherian local ring R means the depth of R as a R-module. That is, the
depth of R is the maximum length of a regular sequence in the maximal ideal.
For a Noetherian local ring R, the depth of the zero module is ∞,[2] whereas the depth of a nonzero ﬁnitely generated
R-module M is at most the Krull dimension of M (also called the dimension of the support of M).[3]

33.2 Examples
• For a prime number p, the local ring Z₍p₎ is the subring of the rational numbers consisting of fractions whose
denominator is not a multiple of p. The element p is a non-zero-divisor in Z₍p₎, and the quotient ring of Z₍p₎
by the ideal generated by p is the ﬁeld Z/(p). Therefore p cannot be extended to a longer regular sequence in
the maximal ideal (p), and in fact the local ring Z₍p₎ has depth 1.
129

130

CHAPTER 33. REGULAR SEQUENCE

• For any ﬁeld k, the elements x1 , ..., xn in the polynomial ring A = k[x1 , ..., xn] form a regular sequence. It
follows that the localization R of A at the maximal ideal m = (x1 , ..., xn) has depth at least n. In fact, R has
depth equal to n; that is, there is no regular sequence in the maximal ideal of length greater than n.
• More generally, let R be a regular local ring with maximal ideal m. Then any elements r1 , ..., rd of m which
map to a basis for m/m2 as an R/m-vector space form a regular sequence.
An important case is when the depth of a local ring R is equal to its Krull dimension: R is then said to be CohenMacaulay. The three examples shown are all Cohen-Macaulay rings. Similarly, a ﬁnitely generated R-module M is
said to be Cohen-Macaulay if its depth equals its dimension.

33.3 Applications
• If r1 , ..., rd is a regular sequence in a ring R, then the Koszul complex is an explicit free resolution of R/(r1 ,
..., rd) as an R-module, of the form:
d

d

0 → R(d) → · · · → R(1) → R → R/(r1 , . . . , rd ) → 0
In the special case where R is the polynomial ring k[r1 , ..., rd], this gives a resolution of k as an R-module.
• If I is an ideal generated by a regular sequence in a ring R, then the associated graded ring
⊕j≥0 I j /I j+1
is isomorphic to the polynomial ring (R/I)[x1 , ..., xd]. In geometric terms, it follows that a local complete intersection
subscheme Y of any scheme X has a normal bundle which is a vector bundle, even though Y may be singular.

33.4 See also
• Complete intersection ring
• Koszul complex
• Depth (ring theory)
• Cohen-Macaulay ring

33.5 Notes
[1] N. Bourbaki. Algèbre. Chapitre 10. Algèbre Homologique. Springer-Verlag (2006). X.9.6.
[2] A. Grothendieck. EGA IV, Part 1. Publications Mathématiques de l'IHÉS 20 (1964), 259 pp. 0.16.4.5.
[3] N. Bourbaki. Algèbre Commutative. Chapitre 10. Springer-Verlag (2007). Th. X.4.2.

33.6 References
• Bourbaki, Nicolas (2006), Algèbre. Chapitre 10. Algèbre Homologique, Berlin, New York: Springer-Verlag,
doi:10.1007/978-3-540-34493-3, ISBN 978-3-540-34492-6, MR 2327161
• Bourbaki, Nicolas (2007), Algèbre Commutative. Chapitre 10, Berlin, New York: Springer-Verlag, doi:10.1007/9783-540-34395-0, ISBN 978-3-540-34394-3, MR 2333539
• Winfried Bruns; Jürgen Herzog, Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics, 39.
Cambridge University Press, Cambridge, 1993. xii+403 pp. ISBN 0-521-41068-1

33.6. REFERENCES

131

• David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. Springer Graduate Texts in
Mathematics, no. 150. ISBN 0-387-94268-8
• Grothendieck, Alexander (1964), "Éléments de géometrie algébrique IV. Première partie”, Publications Mathématiques de l'Institut des Hautes Études Scientiﬁques 20: 1–259, MR 0173675

Chapter 34

Relative canonical model
In mathematics, the relative canonical model of a singular variety X is a particular canonical variety that maps to
X , which simpliﬁes the structure. The precise deﬁnition is:
If f : Y → X is a resolution deﬁne the adjunction sequence to be the sequence of subsheaves f∗ ωY⊗n ; if ωX is
⊗n
invertible f∗ ωY⊗n = In ωX
where In is the higher adjunction ideal. Problem. Is ⊕n f∗ ωY⊗n ﬁnitely generated? If
this is true then P roj ⊕n f∗ ωY⊗n → X is called the relative canonical model of Y , or the canonical blow-up of X
.[1]
Some basic properties were as follows: The relative canonical model was independent of the choice of resolution.
Some integer multiple r of the canonical divisor of the relative canonical model was Cartier and the number of
exceptional components where this agrees with the same multiple of the canonical divisor of Y is also independent of
the choice of Y. When it equals the number of components of Y it was called crepant.[1] It was not known whether
relative canonical models were Cohen–Macaulay.
Because the relative canonical model is independent of Y , most authors simplify the terminology, referring to it as
the relative canonical model of X rather than either the relative canonical model of Y or the canonical blow-up of
X . The class of varieties that are relative canonical models have canonical singularities. Since that time in the 1970s
other mathematicians solved aﬃrmatively the problem of whether they are Cohen–Macaulay. The minimal model
program started by Shigefumi Mori proved that the sheaf in the deﬁnition always is ﬁnitely generated and therefore
that relative canonical models always exist.

34.1 References
[1] M. Reid, Canonical 3-folds (courtesy copy), proceedings of the Angiers 'Journees de Geometrie Algebrique' 1979

132

Chapter 35

Relative dimension
In mathematics, speciﬁcally linear algebra and geometry, relative dimension is the dual notion to codimension.
In linear algebra, given a quotient map V → Q , the diﬀerence dim V − dim Q is the relative dimension; this equals
the dimension of the kernel.
In ﬁber bundles, the relative dimension of the map is the dimension of the ﬁber.
More abstractly, the codimension of a map is the dimension of the cokernel, while the relative dimension of a map is
the dimension of the kernel.
These are dual in that the inclusion of a subspace V → W of codimension k dualizes to yield a quotient map
W ∗ → V ∗ of relative dimension k, and conversely.
The additivity of codimension under intersection corresponds to the additivity of relative dimension in a ﬁber product.
Just as codimension is mostly used for injective maps, relative dimension is mostly used for surjective maps.

133

Chapter 36

Schauder dimension
In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the
diﬀerence is that Hamel bases use linear combinations that are ﬁnite sums, while for Schauder bases they may be
inﬁnite sums. This makes Schauder bases more suitable for the analysis of inﬁnite-dimensional topological vector
spaces including Banach spaces.
Schauder bases were described by Juliusz Schauder in 1927,[1][2] although such bases were discussed earlier. For
example, the Haar basis was given in 1909, and G. Faber discussed in 1910 a basis for continuous functions on an
interval, sometimes called a Faber–Schauder system.[3]

36.1 Deﬁnitions
Let V denote a Banach space over the ﬁeld F. A Schauder basis is a sequence {bn} of elements of V such that for
every element v ∈ V there exists a unique sequence {αn} of scalars in F so that

v=

∞
∑

αn bn ,

n=0

where the convergence is understood with respect to the norm topology, i.e.,

n

∑

lim
v −
αk bk
= 0.
n→∞

k=0

V

Schauder bases can also be deﬁned analogously in a general topological vector space. As opposed to a Hamel basis,
the elements of the basis must be ordered since the series may not converge unconditionally.
A Schauder basis {bn}n ≥ ₀ is said to be normalized when all the basis vectors have norm 1 in the Banach space V.
A sequence {xn}n ≥ ₀ in V is a basic sequence if it is a Schauder basis of its closed linear span.
Two Schauder bases, {bn} in V and {cn} in W, are said to be equivalent if there exist two constants c > 0 and C
such that for every integer N ≥ 0 and all sequences {αn} of scalars,
N

N

∑

∑

c
αk bk
≤
αk ck

k=0

V

k=0

W

N

∑

≤C
αk bk
.

k=0

V

A family of vectors in V is total if its linear span (the set of ﬁnite linear combinations) is dense in V. If V is a Hilbert
space, an orthogonal basis is a total subset B of V such that elements in B are nonzero and pairwise orthogonal.
Further, when each element in B has norm 1, then B is an orthonormal basis of V.
134

36.2. PROPERTIES

135

36.2 Properties
Let {bn} be a Schauder basis of a Banach space V over F = R or C. It follows from the Banach–Steinhaus theorem
that the linear mappings {Pn} deﬁned by

v=

∞
∑

∑
Pn
αk bk −→
Pn (v) =
αk bk

k=0

n

k=0

are uniformly bounded by some constant C. When C = 1, the basis is called a monotone basis. The maps {Pn} are
the basis projections.
Let {b*n} denote the coordinate functionals, where b*n assigns to every vector v in V the coordinate αn of v in the
above expansion. Each b*n is a bounded linear functional on V. Indeed, for every vector v in V,

|b∗n (v)| ∥bn ∥V = |αn | ∥bn ∥V = ∥αn bn ∥V = ∥Pn (v) − Pn−1 (v)∥V ≤ 2C∥v∥V .
These functionals {b*n} are called biorthogonal functionals associated to the basis {bn}. When the basis {bn} is
normalized, the coordinate functionals {b*n} have norm ≤ 2C in the continuous dual V ′ of V.
A Banach space with a Schauder basis is necessarily separable, but the converse is false, as described below. Since
every vector v in a Banach space V with a Schauder basis is the limit of Pn(v), with Pn of ﬁnite rank and uniformly
bounded, such a space V satisﬁes the bounded approximation property.
A theorem attributed to Mazur[4] asserts that every inﬁnite-dimensional Banach space V contains a basic sequence,
i.e., there is an inﬁnite-dimensional subspace of V that has a Schauder basis. The basis problem is the question
asked by Banach, whether every separable Banach space has a Schauder basis. This was negatively answered by Per
Enﬂo who constructed a separable Banach space failing the approximation property, thus a space without a Schauder
basis.[5]

36.3 Examples
The standard unit vector bases of c0 , and of ℓp for 1 ≤ p < ∞, are monotone Schauder bases. In this unit vector
basis {bn}, the vector bn in V = c0 or in V = ℓp is the scalar sequence {bn, j }j where all coordinates bn, j are 0,
except the nth coordinate:

bn = {bn,j }∞
j=0 ∈ V, bn,j = δn,j ,
where δn, j is the Kronecker delta. The space ℓ∞ is not separable, and therefore has no Schauder basis.
Every orthonormal basis in a separable Hilbert space is a Schauder basis. Every countable orthonormal basis is
equivalent to the standard unit vector basis in ℓ2 .
The Haar system is an example of a basis for Lp ([0, 1]), when 1 ≤ p < ∞.[2] When 1 < p < ∞, another example is the
trigonometric system deﬁned below. The Banach space C([0, 1]) of continuous functions on the interval [0, 1], with
the supremum norm, admits a Schauder basis. The Faber–Schauder system is the most commonly used Schauder
basis for C([0, 1]).[3][6]
Several bases for classical spaces were discovered before Banach’s book appeared (Banach (1932)), but some other
cases remained open for a long time. For example, the question of whether the disk algebra A(D) has a Schauder
basis remained open for more than forty years, until Bočkarev showed in 1974 that a basis constructed from the
Franklin system exists in A(D).[7] One can also prove that the periodic Franklin system[8] is a basis for a Banach
space Ar isomorphic to A(D).[9] This space Ar consists of all complex continuous functions on the unit circle T
whose conjugate function is also continuous. The Franklin system is another Schauder basis for C([0, 1]),[10] and it is
a Schauder basis in Lp ([0, 1]) when 1 ≤ p < ∞.[11] Systems derived from the Franklin system give bases in the space
C 1 ([0, 1]2 ) of diﬀerentiable functions on the unit square.[12] The existence of a Schauder basis in C 1 ([0, 1]2 ) was a
question from Banach’s book.[13]

136

36.3.1

CHAPTER 36. SCHAUDER DIMENSION

Relation to Fourier series

Let {xn} be, in the real case, the sequence of functions
{1, cos(x), sin(x), cos(2x), sin(2x), cos(3x), sin(3x), . . .}
or, in the complex case,
{ ix −ix 2ix −2ix 3ix −3ix
}
1, e , e , e , e
,e ,e
,... .
The sequence {xn} is called the trigonometric system. It is a Schauder basis for the space Lp ([0, 2π]) for any p such
that 1 < p < ∞. For p = 2, this is the content of the Riesz–Fischer theorem, and for p ≠ 2, it is a consequence of the
boundedness on the space Lp ([0, 2π]) of the Hilbert transform on the circle. It follows from this boundedness that
the projections PN deﬁned by
{
f :x→

+∞
∑

}
ck eikx

{
PN

−→

N
∑

PN f : x →

k=−∞

}
ck eikx

k=−N
p

are uniformly bounded on L ([0, 2π]) when 1 < p < ∞. This family of maps {PN} is equicontinuous and tends to
the identity on the dense subset consisting of trigonometric polynomials. It follows that PN f tends to f in Lp -norm
for every f ∈ Lp ([0, 2π]). In other words, {xn} is a Schauder basis of Lp ([0, 2π]).[14]
However, the set {xn} is not a Schauder basis for L1 ([0, 2π]). This means that there are functions in L1 whose
Fourier series does not converge in the L1 norm, or equivalently, that the projections PN are not uniformly bounded
in L1 -norm. Also, the set {xn} is not a Schauder basis for C([0, 2π]).

36.3.2

Bases for spaces of operators

The space K(ℓ2 ) of compact operators on the Hilbert space ℓ2 has a Schauder basis. For every x, y in ℓ2 , let x ⊗ y
denote the rank one operator v ∈ ℓ2 → <v, x> y. If {en }n ≥ ₁ is the standard orthonormal basis of ℓ2 , a basis for
K(ℓ2 ) is given by the sequence[15]
e1 ⊗ e1 , e1 ⊗ e2 , e2 ⊗ e2 , e2 ⊗ e1 , . . . ,
e1 ⊗ en , e2 ⊗ en , . . . , en ⊗ en , en ⊗ en−1 , . . . , en ⊗ e1 , . . .
For every n, the sequence consisting of the n2 ﬁrst vectors in this basis is a suitable ordering of the family {ej ⊗ ek},
for 1 ≤ j, k ≤ n.
The preceding result can be generalized: a Banach space X with a basis has the approximation property, so the space
K(X) of compact operators on X is isometrically isomorphic[16] to the injective tensor product
b ε X ≃ K(X).
X ′⊗
If X is a Banach space with a Schauder basis {en }n ≥ ₁ such that the biorthogonal functionals are a basis of the dual,
that is to say, a Banach space with a shrinking basis, then the space K(X) admits a basis formed by the rank one
operators e *j ⊗ ek : v → e *j (v) ek, with the same ordering as before.[15] This applies in particular to every reﬂexive
Banach space X with a Schauder basis
On the other hand, the space B(ℓ2 ) has no basis, since it is non-separable. Moreover, B(ℓ2 ) does not have the
approximation property.[17]

36.4 Unconditionality
∑
A Schauder basis {bn} is unconditional if whenever the series αn bn converges, it converges unconditionally. For
a Schauder basis {bn}, this is equivalent to the existence of a constant C such that

36.5. SCHAUDER BASES AND DUALITY

137

n
n

∑

∑

αk bk
εk αk bk
≤ C

V

k=0

V

k=0

for all integers n, all scalar coeﬃcients {αk} and all signs εk = ± 1. Unconditionality is an important property since it
allows one to forget about the order of summation. A Schauder basis is symmetric if it is unconditional and uniformly
equivalent to all its permutations: there exists a constant C such that for every integer n, every permutation π of the
integers {0, 1, …, n} , all scalar coeﬃcients {αk} and all signs {εk},
n
n
∑
∑

εk αk bπ(k)
≤ C
αk bk
.

V

k=0

k=0

V

The standard bases of the sequence spaces c0 and ℓp for 1 ≤ p < ∞, as well as every orthonormal basis in a Hilbert
space, are unconditional. These bases are also symmetric.
The trigonometric system is not an unconditional basis in Lp , except for p = 2.
The Haar system is an unconditional basis in Lp for any 1 < p < ∞. The space L1 ([0, 1]) has no unconditional basis.[18]
A natural question is whether every inﬁnite-dimensional Banach space has an inﬁnite-dimensional subspace with an
unconditional basis. This was solved negatively by Timothy Gowers and Bernard Maurey in 1992.[19]

36.5 Schauder bases and duality
A basis {en}n≥₀ of a Banach space X is boundedly complete if for every sequence {an}n≥₀ of scalars such that the
partial sums

Vn =

n
∑

a k ek

k=0

are bounded in X, the sequence {Vn} converges in X. The unit vector basis for ℓp , 1 ≤ p < ∞, is boundedly complete.
However, the unit vector basis is not boundedly complete in c0 . Indeed, if an = 1 for every n, then

∥Vn ∥c0 = max |ak | = 1
0≤k≤n

for every n, but the sequence {Vn} is not convergent in c0 , since ||Vn₊₁ − Vn|| = 1 for every n.
A space X with a boundedly complete basis {en}n≥₀ is isomorphic to a dual space, namely, the space X is isomorphic
to the dual of the closed linear span in the dual X ′ of the biorthogonal functionals associated to the basis {en}.[20]
A basis {en}n≥₀ of X is shrinking if for every bounded linear functional f on X, the sequence of non-negative
numbers

φn = sup{|f (x)| : x ∈ Fn , ∥x∥ ≤ 1}
tends to 0 when n → ∞, where Fn is the linear span of the basis vectors em for m ≥ n. The unit vector basis for ℓp , 1
< p < ∞, or for c0 , is shrinking. It is not shrinking in ℓ1 : if f is the bounded linear functional on ℓ1 given by

f : x = {xn } ∈ ℓ1 →

∞
∑

xn ,

n=0

then φn ≥ f(en) = 1 for every n.

138

CHAPTER 36. SCHAUDER DIMENSION

A basis {en }n ≥ ₀ of X is shrinking if and only if the biorthogonal functionals {e*n }n ≥ ₀ form a basis of the dual
X ′.[21]
Robert C. James characterized reﬂexivity in Banach spaces with basis: the space X with a Schauder basis is reﬂexive if and only if the basis is both shrinking and boundedly complete.[22] James also proved that a space with an
unconditional basis is non-reﬂexive if and only if it contains a subspace isomorphic to c0 or ℓ1 .[23]

36.6 Related concepts
A Hamel basis is a subset B of a vector space V such that every element v ∈ V can uniquely be written as

v=

∑

αb b

b∈B

with αb ∈ F, with the extra condition that the set

{b ∈ B | αb ̸= 0}
is ﬁnite. This property makes the Hamel basis unwieldy for inﬁnite-dimensional Banach spaces; as a Hamel basis
for an inﬁnite-dimensional Banach space has to be uncountable. (Every ﬁnite-dimensional subspace of an inﬁnitedimensional Banach space X has empty interior, and is no-where dense in X. It then follows from the Baire category
theorem that a countable union of these ﬁnite-dimensional subspaces cannot serve as a basis.[24] )

36.7 See also
• Generalized Fourier series
• Orthogonal polynomials
• Haar wavelet
• Banach space

36.8 Notes
[1] see Schauder (1927).
[2] Schauder, Juliusz (1928), “Eine Eigenschaft des Haarschen Orthogonalsystems”, Mathematische Zeitschrift 28: 317–320.
[3] Faber, Georg (1910), "Über die Orthogonalfunktionen des Herrn Haar”, Deutsche Math.-Ver (in German) 19: 104–
112. ISSN 0012-0456; http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X ; http://resolver.sub.
uni-goettingen.de/purl?GDZPPN002122553
[4] for an early published proof, see p. 157, C.3 in Bessaga, C. and Pełczyński, A. (1958), “On bases and unconditional
convergence of series in Banach spaces”, Studia Math. 17: 151–164. In the ﬁrst lines of this article, Bessaga and Pełczyński
write that Mazur’s result appears without proof in Banach’s book —to be precise, on p. 238— but they do not provide a
reference containing a proof.
[5] Enﬂo, Per (July 1973). “A counterexample to the approximation problem in Banach spaces”. Acta Mathematica 130 (1):
309–317. doi:10.1007/BF02392270.
[6] see pp. 48–49 in Schauder (1927). Schauder deﬁnes there a general model for this system, of which the Faber–Schauder
system used today is a special case.
[7] see Bočkarev, S. V. (1974), “Existence of a basis in the space of functions analytic in the disc, and some properties of
Franklin’s system”, (in Russian) Mat. Sb. (N.S.) 95(137): 3–18, 159. Translated in Math. USSR-Sb. 24 (1974), 1–16.
The question is in Banach’s book, Banach (1932) p. 238, §3.

36.9. REFERENCES

139

[8] See p. 161, III.D.20 in Wojtaszczyk (1991).
[9] See p. 192, III.E.17 in Wojtaszczyk (1991).
[10] Franklin, Philip (1928), “A set of continuous orthogonal functions”, Math. Ann. 100: 522–529.
[11] see p. 164, III.D.26 in Wojtaszczyk (1991).
[12] see Ciesielski, Z. (1969), “A construction of basis in C 1 (I 2 )", Studia Math. 33: 243–247, and Schonefeld, Steven (1969),
“Schauder bases in spaces of diﬀerentiable functions”, Bull. Amer. Math. Soc. 75: 586–590.
[13] see p. 238, §3 in Banach (1932).
[14] see p. 40, II.B.11 in Wojtaszczyk (1991).
[15] see Proposition 4.25, p. 88 in Ryan (2002).
[16] see Corollary 4.13, p. 80 in Ryan (2002).
[17] see Szankowski, Andrzej (1981), "B(H) does not have the approximation property”, Acta Math. 147: 89–108.
[18] see p. 24 in Lindenstrauss & Tzafriri (1977).
[19] Gowers, W. Timothy; Maurey, Bernard (6 May 1992). “The unconditional basic sequence problem”. arXiv:math/9205204.
[20] see p. 9 in Lindenstrauss & Tzafriri (1977).
[21] see p. 8 in Lindenstrauss & Tzafriri (1977).
[22] see James, Robert. C. (1950), “Bases and reﬂexivity of Banach spaces”, Ann. of Math. (2) 52: 518–527. See also
Lindenstrauss & Tzafriri (1977) p. 9.
[23] see James, Robert C. (1950), “Bases and reﬂexivity of Banach spaces”, Ann. of Math. (2) 52: 518–527. See also p. 23 in
Lindenstrauss & Tzafriri (1977).
[24] Carothers, N. L. (2005), A short course on Banach space theory, Cambridge University Press ISBN 0-521-60372-2

This article incorporates material from Countable basis on PlanetMath, which is licensed under the Creative Commons
Attribution/Share-Alike License.

36.9 References
• Schauder, Juliusz (1927), “Zur Theorie stetiger Abbildungen in Funktionalraumen”, Mathematische Zeitschrift
(in German) 26: 47–65, doi:10.1007/BF01475440.
• Banach, Stefan (1932), Théorie des opérations linéaires, Monograﬁe Matematyczne 1, Warszawa: Subwencji
Funduszu Kultury Narodowej, Zbl 0005.20901.
• Lindenstrauss, Joram; Tzafriri, Lior (1977), Classical Banach Spaces I, Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Berlin: Springer-Verlag, ISBN 3-540-08072-4.
• Ryan, Raymond A. (2002), Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics, London: Springer-Verlag, pp. xiv+225, ISBN 1-85233-437-1.
• Schaefer, Helmut H. (1971), Topological vector spaces, Graduate Texts in Mathematics 3, New York: SpringerVerlag, pp. xi+294, ISBN 0-387-98726-6.
• Wojtaszczyk, Przemysław (1991), Banach spaces for analysts, Cambridge Studies in Advanced Mathematics
25, Cambridge: Cambridge University Press, pp. xiv+382, ISBN 0-521-35618-0.
• Golubov, B.I. (2001), “Faber–Schauder system”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,
ISBN 978-1-55608-010-4 .
• Heil, Christopher E. (1997). “A basis theory primer” (PDF)..
• Franklin system. B.I. Golubov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.
org/index.php?title=Franklin_system&oldid=16655

140

CHAPTER 36. SCHAUDER DIMENSION

36.10 Further reading
• Kufner, Alois (2013), Function spaces, De Gruyter Series in Nonlinear analysis and applicatioons 14, Prague:
Academia Publishing House of the Czechoslovak Academy of Sciences, de Gruyter

Chapter 37

Seven-dimensional space
In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 7,
the set of all such locations is called 7-dimensional space. Often such a space are studied as a vector space, without
any notion of distance. Seven-dimensional Euclidean space is seven-dimensional space equipped with a Euclidean
metric, which is deﬁned by the dot product.
More generally, the term may refer to a seven-dimensional vector space over any ﬁeld, such as a seven-dimensional
complex vector space, which has 14 real dimensions. It may also refer to a seven-dimensional manifold such as a
7-sphere, or a variety of other geometric constructions.
Seven-dimensional spaces have a number of special properties, many of them related to the octonions. An especially
distinctive property is that a cross product can be deﬁned only in three or seven dimensions. This is related to Hurwitz’s
theorem, which prohibits the existence of algebraic structures like the quaternions and octonions in dimensions other
than 2, 4, and 8. The ﬁrst exotic spheres ever discovered were seven-dimensional.

37.1 Geometry
37.1.1

7-polytope

Main article: Uniform 7-polytope
A polytope in seven dimensions is called a 7-polytope. The most studied are the regular polytopes, of which there
are only three in seven dimensions: the 7-simplex, 7-cube, and 7-orthoplex. A wider family are the uniform 7polytopes, constructed from fundamental symmetry domains of reﬂection, each domain deﬁned by a Coxeter group.
Each uniform polytope is deﬁned by a ringed Coxeter-Dynkin diagram. The 7-demicube is a unique polytope from
the D7 family, and 321 , 231 , and 132 polytopes from the E7 family.

37.1.2

6-sphere

The 6-sphere or hypersphere in seven-dimensional Euclidean space is the six-dimensional surface equidistant from a
point, e.g. the origin. It has symbol S6 , with formal deﬁnition for the 6-sphere with radius r of
{
}
S 6 = x ∈ R7 : ∥x∥ = r .
The volume of the space bounded by this 6-sphere is

V7 =

16π 3 7
r
105

which is 4.72477 × r7 , or 0.0369 of the 7-cube that contains the 6-sphere.
141

142

CHAPTER 37. SEVEN-DIMENSIONAL SPACE

37.2 Applications
37.2.1

Cross product

Main article: Seven-dimensional cross product
As mentioned above, a cross product in seven dimensions analogous to the usual three can be deﬁned, and in fact a
cross product can only be deﬁned in three and seven dimensions.

37.2.2

Exotic spheres

Main article: Exotic sphere
In 1956, John Milnor constructed an exotic sphere in 7 dimensions and showed that there are at least 7 diﬀerentiable
structures on the 7-sphere. In 1963 he showed that the exact number of such structures is 28.

37.3 See also
• Euclidean geometry
• List of geometry topics
• List of regular polytopes

37.4 References
• H.S.M. Coxeter: Regular Polytopes. Dover, 1973
• J.W. Milnor: On manifolds homeomorphic to the 7-sphere. Annals of Mathematics 64, 1956

37.5 External links
• Hazewinkel, Michiel, ed. (2001), “Euclidean geometry”, Encyclopedia of Mathematics, Springer, ISBN 9781-55608-010-4

Chapter 38

Six-dimensional space
Six-dimensional space is any space that has six dimensions, that is, six degrees of freedom, and that needs six pieces
of data, or coordinates, to specify a location in this space. There are an inﬁnite number of these, but those of most interest are simpler ones that model some aspect of the environment. Of particular interest is six-dimensional Euclidean
space, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional elliptical space and hyperbolic spaces
are also studied, with constant positive and negative curvature.
Formally, six-dimensional Euclidean space, ℝ6 , is generated by considering all real 6-tuples as 6-vectors in this space.
As such it has the properties of all Euclidean spaces, so it is linear, has a metric and a full set of vector operations.
In particular the dot product between two 6-vectors is readily deﬁned, and can be used to calculate the metric. 6 × 6
matrices can be used to describe transformations such as rotations that keep the origin ﬁxed.
More generally, any space that can be described locally with six coordinates, not necessarily Euclidean ones, is sixdimensional. One example is the surface of the 6-sphere, S6 . This is the set of all points in seven-dimensional
Euclidean space ℝ7 that are equidistant from the origin. This constraint reduces the number of coordinates needed to
describe a point on the 6-sphere by one, so it has six dimensions. Such non-Euclidean spaces are far more common
than Euclidean spaces, and in six dimensions they have far more applications.

38.1 Geometry
38.1.1

6-polytope

Main article: 6-polytope
A polytope in six dimensions is called a 6-polytope. The most studied are the regular polytopes, of which there are
only three in six dimensions: the 6-simplex, 6-cube, and 6-orthoplex. A wider family are the uniform 6-polytopes,
constructed from fundamental symmetry domains of reﬂection, each domain deﬁned by a Coxeter group. Each
uniform polytope is deﬁned by a ringed Coxeter-Dynkin diagram. The 6-demicube is a unique polytope from the D6
family, and 221 and 122 polytopes from the E6 family.

38.1.2

5-sphere

The 5-sphere, or hypersphere in six dimensions, is the ﬁve-dimensional surface equidistant from a point. It has symbol
S5 , and the equation for the 5-sphere, radius r, centre the origin is
{
}
S 5 = x ∈ R6 : ∥x∥ = r .
The volume of six-dimensional space bounded by this 5-sphere is

V6 =

π 3 r6
6
143

144

CHAPTER 38. SIX-DIMENSIONAL SPACE

which is 5.16771 × r6 , or 0.0807 of the smallest 6-cube that contains the 5-sphere.

38.1.3

6-sphere

The 6-sphere, or hypersphere in seven dimensions, is the six-dimensional surface equidistant from a point. It has
symbol S6 , and the equation for the 6-sphere, radius r, centre the origin is
{
}
S 6 = x ∈ R7 : ∥x∥ = r .
The volume of the space bounded by this 6-sphere is

V7 =

16π 3 r7
105

which is 4.72477 × r7 , or 0.0369 of the smallest 7-cube that contains the 6-sphere.

38.2 Applications
38.2.1

Transformations in three dimensions

In three-dimensional space a generalised transformation has six degrees of freedom, three translations along the three
coordinate axes and three from the rotation group SO(3). Often these transformations are handled separately as
they have very diﬀerent geometrical structures, but there are ways of dealing with them that treat them as a single
six-dimensional object.

Homogeneous coordinates
Main article: Homogeneous coordinates
Using four-dimensional Homogeneous coordinates it is possible to describe a general transformation using a single 4
× 4 matrix. This matrix has six degrees of freedom, which can identiﬁed with the six elements of the matrix above
the main diagonal, as all others are determined by these.

Screw theory
Main article: Screw theory
In screw theory angular and linear velocity are combined into one six-dimensional object, called a twist. A similar
object called a wrench combines forces and torques in six dimensions. These can be treated as six-dimensional vectors
that transform linearly when changing frame of reference. Translations and rotations cannot be done this way, but
are related to a twist by exponentiation.

Phase space
Main article: Phase space
Phase space is a space made up of the position and momentum of a particle, which can be plotted together in a phase
diagram to highlight the relationship between the quantities. A general particle moving in three dimensions has a
phase space with six dimensions, too many to plot but they can be analysed mathematically.[1]

38.2. APPLICATIONS

145

Phase portrait of the Van der Pol oscillator

38.2.2

Rotations in four dimensions

Main article: Rotations in 4-dimensional Euclidean space
The rotation group in four dimensions, SO(4), has six degrees of freedom. This can be seen by considering the 4 ×
4 matrix that represents a rotation: as it is an orthogonal matrix the matrix is determined, up to a change in sign, by
e.g. the six elements above the main diagonal. But this group is not linear, and it has a more complex structure than
other applications seen so far.
Another way of looking at this group is with quaternion multiplication. Every rotation in four dimensions can be
achieved by multiplying by a pair of unit quaternions, one before and one after the vector. These quaternion are
unique, up to a change in sign for both of them, and generate all rotations when used this way, so the product of their
groups, S3 × S3 , is a double cover of SO(4), which must have six dimensions.
Although the space we live in is considered three-dimensional, there are practical applications for four-dimensional
space. Quaternions, one of the ways to describe rotations in three dimensions, consist of a four-dimensional space.
Rotations between quaternions, for interpolation for example, take place in four dimensions. Spacetime, which has
three space dimensions and one time dimension is also four-dimensional, though with a diﬀerent structure to Euclidean
space.

38.2.3

Plücker coordinates

Main article: Plücker coordinates
Plücker coordinates are a way of representing lines in three dimensions using six homogeneous coordinates. As
homogeneous coordinates they have only ﬁve degrees of freedom, corresponding to the ﬁve degrees of freedom of
a general line, but they are treated as 6-vectors for some purposes. For example the check for the intersection of

146

CHAPTER 38. SIX-DIMENSIONAL SPACE

two lines is a 6-dimensional dot product between two sets of Plücker coordinates, one of which has exchanged its
displacement and moment parts.

38.2.4

Electromagnetism

In electromagnetism, the electromagnetic ﬁeld is generally thought of as being made of two things, the electric ﬁeld
and magnetic ﬁeld. They are both three-dimensional vector ﬁelds, related to each other by Maxwell’s equations.
A second approach is to combine them in a single object, the six-dimensional electromagnetic tensor, a tensor or
bivector valued representation of the electromagnetic ﬁeld. Using this Maxwell’s equations can be condensed from
four equations into a particularly compact single equation:

∂F = J
where F is the bivector form of the electromagnetic tensor, J is the four-current and ∂ is a suitable diﬀerential
operator.[2]

38.2.5

String theory

In physics string theory is an attempt to describe general relativity and quantum mechanics with a single mathematical
model. Although it is an attempt to model our universe it takes place in a space with more dimensions than the four of
space-time that we are familiar with. In particular a number of string theories take place in a ten-dimensional space,
adding an extra six dimensions. These extra dimensions are required by the theory, but as they cannot be observed
are thought to be quite diﬀerent, perhaps compactiﬁed to form a six-dimensional space with a particular geometry
too small to be observable.
Since 1997, another string theory has come to light that works in six dimensions. Little string theories are nongravitational string theories in ﬁve and six dimensions that arise when considering limits of ten-dimensional string
theory.[3]

38.3 Theoretical background
38.3.1

Bivectors in four dimensions

A number of the above applications can be related to each other algebraically by considering the real, six-dimensional
bivectors in four dimensions. These can be written Λ2 ℝ4 for the set of bivectors in Euclidean space or Λ2 ℝ3,1 for the
set of bivectors in spacetime. The Plücker coordinates are bivectors in ℝ4 while the electromagnetic tensor discussed
in the previous section is a bivector in ℝ3,1 . Bivectors can be used to generate rotations in either ℝ4 or ℝ3,1 through
the exponential map (e.g. applying the exponential map of all bivectors in Λ2 ℝ4 generates all rotations in ℝ4 ). They
can also be related to general transformations in three dimensions through homogeneous coordinates, which can be
thought of as modiﬁed rotations in ℝ4 .
The bivectors arise from sums of all possible wedge products between pairs of 4-vectors. They therefore have C4
2 = 6 components, and can be written most generally as

B = B12 e12 + B13 e13 + B14 e14 + B23 e23 + B24 e24 + B34 e34
They are the ﬁrst bivectors that cannot all be generated by products of pairs of vectors. Those that can are simple
bivectors and the rotations they generate are simple rotations. Other rotations in four dimensions are double and
isoclinic rotations and correspond to non-simple bivectors that cannot be generated by single wedge product.[4]

38.3.2

6-vectors

6-vectors are simply the vectors of six-dimensional Euclidean space. Like other such vectors they are linear, can be
added subtracted and scaled like in other dimensions. Rather than using letters of the alphabet, higher dimensions

38.3. THEORETICAL BACKGROUND

147

Calabi–Yau manifold (3D projection)

usually use suﬃxes to designate dimensions, so a general six-dimensional vector can be written a = (a1 , a2 , a3 , a4 , a5 ,
a6 ). Written like this the six basis vectors are (1, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0,
0, 0, 0, 1, 0) and (0, 0, 0, 0, 0, 1).
Of the vector operators the cross product cannot be used in six dimensions; instead the wedge product of two 6-vectors
results in a bivector with 15 dimensions. The dot product of two vectors is

a · b = a1 b1 + a2 b2 + a3 b3 + a4 b4 + a5 b5 + a6 b6 .
It can be used to ﬁnd the angle between two vectors and the norm,

|a| =

√
√
a · a = a1 2 + a2 2 + a3 2 + a4 2 + a5 2 + a6 2 .

This can be used for example to calculate the diagonal of a 6-cube; with one corner at the origin, edges aligned to the
axes and side length 1 the opposite corner could be at (1, 1, 1, 1, 1, 1), the norm of which is
√
√
1 + 1 + 1 + 1 + 1 + 1 = 6 = 2.4495,

148

CHAPTER 38. SIX-DIMENSIONAL SPACE

which is the length of the vector and so of the diagonal of the 6-cube.

38.3.3

Complex 3-space

The complex plane C has two real dimensions, so C3 is a six-dimensional space. William Rowan Hamilton identiﬁed this space in 1853[5] as the bivectors of is biquaternions. He had introduced vectors as 3-dimensional parts of
quaternions, so when the tensor product C ⊗H became biquaternions, the complex 3-dimensional part was bivectors.
The exponential map takes bivectors to the unit sphere of the biquaternion algebra, which is isomorphic to the Lorentz
group. Hence, as Ronald Shaw and Graham Bowtell[6] have noted, bivectors are logarithms of Lorentz transformations. Generally vector analysis is conﬁned to three dimensions, but in Vector Analysis (1901) the six-dimensional
space of bivectors was used by J. W. Gibbs and E. B. Wilson.[7]
In the diﬀerential geometry of complex manifolds some six-dimensional spaces arise as algebraic manifolds. Examples include the quintic threefold and the Barth-Nieto quintic. According to Piergiorgio Odifreddi, the classiﬁcation
of complex three-dimensional manifolds “was one of the spectacular results obtained by the Japanese school of geometry of Heisuke Hironaka, Shing Tung Yau, and Shigefumi Mori. For this work they were awarded the Fields
Medal in 1970, 1983, and 1990, respectively.”[8]

38.4 Footnotes
[1] Arthur Besier (1969). Perspectives of Modern Physics. McGraw-Hill.
[2] Lounesto (2001), pp. 109–110
[3] Aharony (2000)
[4] Lounesto (2001), pp. 86-89
[5] William Rowan Hamilton (1853) Lectures on Quaternions, p 665, Royal Irish Academy, link from Cornell University
Historical Mathematics Collection
[6] Ronald Shaw and Graham Bowtell (1969) “The Bivector Logarithm of a Lorentz Transformation”, Quarterly Journal of
Mathematics 20:497–503
[7] Edwin Bidwell Wilson (1901) Vector Analysis, pages 426 to 436, “Harmonic Vibrations and Bivectors”
[8] Piergiorgio Odifreddi (2004) The Mathematical Century, page 82, Princeton University Press ISBN 0-691-09294-X

38.5 References
• Lounesto, Pertti (2001). Cliﬀord Algebras and Spinors. Cambridge: Cambridge University Press. ISBN 9780-521-00551-7.
• Aharony, Ofer (2000). “A Brief Review of “Little String Theories"". Quantum Grav. 17 (5). arXiv:hepth/9911147. Bibcode:2000CQGra..17..929A. doi:10.1088/0264-9381/17/5/302.

Chapter 39

String theory
For the study of strings of characters, see Concatenation theory.
For a more accessible and less technical introduction to this topic, see Introduction to M-theory.
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced
by one-dimensional objects called strings. String theory describes how these strings propagate through space and
interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle,
with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one
of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries
gravitational force. Thus string theory is a theory of quantum gravity.
String theory is a broad and varied subject that attempts to address a number of deep questions of fundamental physics.
String theory has been applied to a variety of problems in black hole physics, early universe cosmology, nuclear
physics, and condensed matter physics, and it has stimulated a number of major developments in pure mathematics.
Because string theory potentially provides a uniﬁed description of gravity and particle physics, it is a candidate for a
theory of everything, a self-contained mathematical model that describes all fundamental forces and forms of matter.
Despite much work on these problems, it is not known to what extent string theory describes the real world or how
much freedom the theory allows to choose the details.
String theory was ﬁrst studied in the late 1960s as a theory of the strong nuclear force, before being abandoned in
favor of quantum chromodynamics. Subsequently, it was realized that the very properties that made string theory
unsuitable as a theory of nuclear physics made it a promising candidate for a quantum theory of gravity. The earliest
version of string theory, bosonic string theory, incorporated only the class of particles known as bosons. It later
developed into superstring theory, which posits a connection called supersymmetry between bosons and the class of
particles called fermions. Five consistent versions of superstring theory were developed before it was conjectured in
the mid-1990s that they were all diﬀerent limiting cases of a single theory in eleven dimensions known as M-theory.
In late 1997, theorists discovered an important relationship called the AdS/CFT correspondence, which relates string
theory to another type of physical theory called a quantum ﬁeld theory.
One of the challenges of string theory is that the full theory does not yet have a satisfactory deﬁnition in all circumstances. Another issue is that the theory is thought to describe an enormous landscape of possible universes, and this
has complicated eﬀorts to develop theories of particle physics based on string theory. These issues have led some in
the community to criticize these approaches to physics and question the value of continued research on string theory
uniﬁcation.

39.1 Fundamentals
In the twentieth century, two theoretical frameworks emerged for formulating the laws of physics. One of these
frameworks was Albert Einstein's general theory of relativity, a theory that explains the force of gravity and the
structure of space and time. The other was quantum mechanics, a radically diﬀerent formalism for describing physical
phenomena using probability. By the late 1970s, these two frameworks had proven to be suﬃcient to explain most
of the observed features of the universe, from elementary particles to atoms to the evolution of stars and the universe
as a whole.[1]
149

150

CHAPTER 39. STRING THEORY

The fundamental objects of string theory are open and closed strings.

In spite of these successes, there are still many problems that remain to be solved. One of the deepest problems
in modern physics is the problem of quantum gravity.[1] The general theory of relativity is formulated within the
framework of classical physics, whereas the other fundamental forces are described within the framework of quantum
mechanics. A quantum theory of gravity is needed in order to reconcile general relativity with the principles of
quantum mechanics, but diﬃculties arise when one attempts to apply the usual prescriptions of quantum theory to the
force of gravity.[2] In addition to the problem of developing a consistent theory of quantum gravity, there are many
other fundamental problems in the physics of atomic nuclei, black holes, and the early universe.[lower-alpha 1]
String theory is a theoretical framework that attempts to address these questions and many others. The starting point
for string theory is the idea that the point-like particles of particle physics can also be modeled as one-dimensional
objects called strings. String theory describes how strings propagate through space and interact with each other. In
a given version of string theory, there is only one kind of string, which may look like a small loop or segment of
ordinary string, and it can vibrate in diﬀerent ways. On distance scales larger than the string scale, a string will look
just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the
string. In this way, all of the diﬀerent elementary particles may be viewed as vibrating strings. In string theory, one of
the vibrational states of the string gives rise to the graviton, a quantum mechanical particle that carries gravitational
force. Thus string theory is a theory of quantum gravity.[3]
One of the main developments of the past several decades in string theory was the discovery of certain “dualities”,
mathematical transformations that identify one physical theory with another. Physicists studying string theory have
discovered a number of these dualities between diﬀerent versions of string theory, and this has led to the conjecture
that all consistent versions of string theory are subsumed in a single framework known as M-theory.[4]
Studies of string theory have also yielded a number of results on the nature of black holes and the gravitational
interaction. There are certain paradoxes that arise when one attempts to understand the quantum aspects of black
holes, and work on string theory has attempted to clarify these issues. In late 1997 this line of work culminated
in the discovery of the anti-de Sitter/conformal ﬁeld theory correspondence or AdS/CFT.[5] This is a theoretical
result which relates string theory to other physical theories which are better understood theoretically. The AdS/CFT
correspondence has implications for the study of black holes and quantum gravity, and it has been applied to other
subjects, including nuclear[6] and condensed matter physics.[7][8]
Since string theory incorporates all of the fundamental interactions, including gravity, many physicists hope that
it fully describes our universe, making it a theory of everything. One of the goals of current research in string
theory is to ﬁnd a solution of the theory that reproduces the observed spectrum of elementary particles, with a small
cosmological constant, containing dark matter and a plausible mechanism for cosmic inﬂation. While there has been

39.1. FUNDAMENTALS

151

progress toward these goals, it is not known to what extent string theory describes the real world or how much freedom
the theory allows to choose the details.[9]
One of the challenges of string theory is that the full theory does not yet have a satisfactory deﬁnition in all circumstances. The scattering of strings is most straightforwardly deﬁned using the techniques of perturbation theory,
but it is not known in general how to deﬁne string theory nonperturbatively.[10] It is also not clear whether there
is any principle by which string theory selects its vacuum state, the physical state that determines the properties of
our universe.[11] These problems have led some in the community to criticize these approaches to the uniﬁcation of
physics and question the value of continued research on these problems.[12]

39.1.1

Strings

Main article: String (physics)
The application of quantum mechanics to physical objects such as the electromagnetic ﬁeld, which are extended in

Interaction in the quantum world: worldlines of point-like particles or a worldsheet swept up by closed strings in string theory.

space and time, is known as quantum ﬁeld theory. In particle physics, quantum ﬁeld theories form the basis for our
understanding of elementary particles, which are modeled as excitations in the fundamental ﬁelds.[13]
In quantum ﬁeld theory, one typically computes the probabilities of various physical events using the techniques of
perturbation theory. Developed by Richard Feynman and others in the ﬁrst half of the twentieth century, perturbative
quantum ﬁeld theory uses special diagrams called Feynman diagrams to organize computations. One imagines that
these diagrams depict the paths of point-like particles and their interactions.[13]
The starting point for string theory is the idea that the point-like particles of quantum ﬁeld theory can also be modeled as one-dimensional objects called strings.[14] The interaction of strings is most straightforwardly deﬁned by
generalizing the perturbation theory used in ordinary quantum ﬁeld theory. At the level of Feynman diagrams, this
means replacing the one-dimensional diagram representing the path of a point particle by a two-dimensional surface representing the motion of a string.[15] Unlike in quantum ﬁeld theory, string theory does not yet have a full
non-perturbative deﬁnition, so many of the theoretical questions that physicists would like to answer remain out of
reach.[16]
In theories of particle physics based on string theory, the characteristic length scale of strings is assumed to be on the
order of the Planck length, or 10−35 meters, the scale at which the eﬀects of quantum gravity are believed to become
signiﬁcant.[15] On much larger length scales, such as the scales visible in physics laboratories, such objects would be
indistinguishable from zero-dimensional point particles, and the vibrational state of the string would determine the

152

CHAPTER 39. STRING THEORY

type of particle. One of the vibrational states of a string corresponds to the graviton, a quantum mechanical particle
that carries the gravitational force.[3]
The original version of string theory was bosonic string theory, but this version described only bosons, a class of particles which transmit forces between the matter particles, or fermions. Bosonic string theory was eventually superseded
by theories called superstring theories. These theories describe both bosons and fermions, and they incorporate a theoretical idea called supersymmetry. This is a mathematical relation that exists in certain physical theories between
the bosons and fermions. In theories with supersymmetry, each boson has a counterpart which is a fermion, and vice
versa.[17]
There are several versions of superstring theory: type I, type IIA, type IIB, and two ﬂavors of heterotic string theory
(SO(32) and E 8 ×E 8 ). The diﬀerent theories allow diﬀerent types of strings, and the particles that arise at low energies
exhibit diﬀerent symmetries. For example, the type I theory includes both open strings (which are segments with
endpoints) and closed strings (which form closed loops), while types IIA and IIB include only closed strings.[18]

39.1.2

Extra dimensions

An example of compactiﬁcation: At large distances, a two dimensional surface with one circular dimension looks one-dimensional.

In everyday life, there are three familiar dimensions of space: height, width and length. Einstein’s general theory of
relativity treats time as a dimension on par with the three spatial dimensions; in general relativity, space and time
are not modeled as separate entities but are instead uniﬁed to a four-dimensional spacetime. In this framework, the
phenomenon of gravity is viewed as a consequence of the geometry of spacetime.[19]
In spite of the fact that the universe is well described by four-dimensional spacetime, there are several reasons why
physicists consider theories in other dimensions. In some cases, by modeling spacetime in a diﬀerent number of
dimensions, a theory becomes more mathematically tractable, and one can perform calculations and gain general
insights more easily.[lower-alpha 2] There are also situations where theories in two or three spacetime dimensions are
useful for describing phenomena in condensed matter physics.[20] Finally, there exist scenarios in which there could
actually be more than four dimensions of spacetime which have nonetheless managed to escape detection.[21]
One notable feature of string theories is that these theories require extra dimensions of spacetime for their mathematical consistency. In bosonic string theory, spacetime is 26-dimensional, while in superstring theory it is ten-

39.1. FUNDAMENTALS

153

dimensional. In order to describe real physical phenomena using string theory, one must therefore imagine scenarios
in which these extra dimensions would not be observed in experiments.[22]

A cross section of a quintic Calabi–Yau manifold

Compactiﬁcation is one way of modifying the number of dimensions in a physical theory. In compactiﬁcation, some
of the extra dimensions are assumed to “close up” on themselves to form circles.[23] In the limit where these curled up
dimensions become very small, one obtains a theory in which spacetime has eﬀectively a lower number of dimensions.
A standard analogy for this is to consider a multidimensional object such as a garden hose. If the hose is viewed from
a suﬃcient distance, it appears to have only one dimension, its length. However, as one approaches the hose, one
discovers that it contains a second dimension, its circumference. Thus, an ant crawling on the surface of the hose
would move in two dimensions.[24]
Compactiﬁcation can be used to construct models in which spacetime is eﬀectively four-dimensional. However, not
every way of compactifying the extra dimensions produces a model with the right properties to describe nature. In
a viable model of particle physics, the compact extra dimensions must be shaped like a Calabi–Yau manifold.[23] A
Calabi–Yau manifold is a special space which is typically taken to be six-dimensional in applications to string theory.
It is named after mathematicians Eugenio Calabi and Shing-Tung Yau.[25]
Another approach to reducing the number of dimensions is the so-called brane-world scenario. In this approach,
physicists assume that the observable universe is a four-dimensional subspace of a higher dimensional space. In such

154

CHAPTER 39. STRING THEORY

models, the force-carrying bosons of particle physics arise from open strings with endpoints attached to the fourdimensional subspace, while gravity arises from closed strings propagating through the larger ambient space. This
idea plays an important role in attempts to develop models of real world physics based on string theory, and it provides
a natural explanation for the weakness of gravity compared to the other fundamental forces.[26]

39.1.3

Dualities

Main articles: S-duality and T-duality
One notable fact about string theory is that the diﬀerent versions of the theory all turn out to be related in highly
nontrivial ways. One of the relationships that can exist between diﬀerent string theories is called S-duality. This is a
relationship which says that a collection of strongly interacting particles in one theory can, in some cases, be viewed as
a collection of weakly interacting particles in a completely diﬀerent theory. Roughly speaking, a collection of particles
is said to be strongly interacting if they combine and decay often and weakly interacting if they do so infrequently.
Type I string theory turns out to be equivalent by S-duality to the SO(32) heterotic string theory. Similarly, type IIB
string theory is related to itself in a nontrivial way by S-duality.[27]
Another relationship between diﬀerent string theories is T-duality. Here one considers strings propagating around a
circular extra dimension. T-duality states that a string propagating around a circle of radius R is equivalent to a string
propagating around a circle of radius 1/R in the sense that all observable quantities in one description are identiﬁed
with quantities in the dual description. For example, a string has momentum as it propagates around a circle, and it
can also wind around the circle one or more times. The number of times the string winds around a circle is called
the winding number. If a string has momentum p and winding number n in one description, it will have momentum
n and winding number p in the dual description. For example, type IIA string theory is equivalent to type IIB string
theory via T-duality, and the two versions of heterotic string theory are also related by T-duality.[27]
In general, the term duality refers to a situation where two seemingly diﬀerent physical systems turn out to be equivalent in a nontrivial way. Two theories related by a duality need not be string theories. For example, Montonen–Olive
duality is example of an S-duality relationship between quantum ﬁeld theories. The AdS/CFT correspondence is
example of a duality which relates string theory to a quantum ﬁeld theory. If two theories are related by a duality,
it means that one theory can be transformed in some way so that it ends up looking just like the other theory. The
two theories are then said to be dual to one another under the transformation. Put diﬀerently, the two theories are
mathematically diﬀerent descriptions of the same phenomena.[28]

39.1.4

Branes

Main article: Brane
In string theory and related theories, a brane is a physical object that generalizes the notion of a point particle to
higher dimensions. For example, a point particle can be viewed as a brane of dimension zero, while a string can be
viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension p, these
are called p-branes. The word brane comes from the word “membrane” which refers to a two-dimensional brane.[29]
Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics.
They have mass and can have other attributes such as charge. A p-brane sweeps out a (p+1)-dimensional volume in
spacetime called its worldvolume. Physicists often study ﬁelds analogous to the electromagnetic ﬁeld which live on
the worldvolume of a brane.[29]
In string theory, D-branes are an important class of branes that arise when one considers open strings. As an open
string propagates through spacetime, its endpoints are required to lie on a D-brane. The letter “D” in D-brane refers
to a certain mathematical condition on the system known as the Dirichlet boundary condition. The study of D-branes
in string theory has led to important results such as the AdS/CFT correspondence, which has shed light on many
problems in quantum ﬁeld theory.[30]
Branes are also frequently studied from a purely mathematical point of view. Mathematically, branes can be described
as objects of certain categories, such as the derived category of coherent sheaves on a complex algebraic variety, or
the Fukaya category of a symplectic manifold.[31] The connection between the physical notion of a brane and the
mathematical notion of a category has led to important mathematical insights in the ﬁelds of algebraic and symplectic
geometry[32] and representation theory.[33]

39.1. FUNDAMENTALS

155

Type I

SO(32) heterotic

E8xE8 heterotic

M-theory
Type IIA

Type IIB

A diagram of string theory dualities. Yellow arrows indicate S-duality. Blue arrows indicate T-duality.

156

CHAPTER 39. STRING THEORY

Open strings attached to a pair of D-branes

39.2 M-theory
Main article: M-theory
Prior to 1995, theorists believed that there were ﬁve consistent versions of superstring theory (type I, type IIA, type
IIB, and two versions of heterotic string theory). This understanding changed in 1995 when Edward Witten suggested
that the ﬁve theories were just special limiting cases of an eleven-dimensional theory called M-theory. Witten’s conjecture was based on the work of a number of other physicists, including Ashoke Sen, Chris Hull, Paul Townsend,
and Michael Duﬀ. His announcement led to a ﬂurry of research activity now known as the second superstring revolution.[34]

39.2.1

Uniﬁcation of superstring theories

In the 1970s, many physicists became interested in supergravity theories, which combine general relativity with supersymmetry. Whereas general relativity makes sense in any number of dimensions, supergravity places an upper limit
on the number of dimensions.[35] In 1978, work by Werner Nahm showed that the maximum spacetime dimension
in which one can formulate a consistent supersymmetric theory is eleven.[36] In the same year, Eugene Cremmer,
Bernard Julia, and Joel Scherk of the École Normale Supérieure showed that supergravity not only permits up to
eleven dimensions but is in fact most elegant in this maximal number of dimensions.[37][38]
Initially, many physicists hoped that by compactifying eleven-dimensional supergravity, it might be possible to construct realistic models of our four-dimensional world. The hope was that such models would provide a uniﬁed description of the four fundamental forces of nature: electromagnetism, the strong and weak nuclear forces, and gravity.
Interest in eleven-dimensional supergravity soon waned as various ﬂaws in this scheme were discovered. One of the
problems was that the laws of physics appear to distinguish between clockwise and counterclockwise, a phenomenon
known as chirality. Edward Witten and others observed this chirality property cannot be readily derived by compactifying from eleven dimensions.[38]
In the ﬁrst superstring revolution in 1984, many physicists turned to string theory as a uniﬁed theory of particle

39.2. M-THEORY

157

11D supergravity

E8×E8 heterotic
Type IIA

M-theory
Type IIB
SO(32) heterotic

Type I
A schematic illustration of the relationship between M-theory, the ﬁve superstring theories, and eleven-dimensional supergravity. The
shaded region represents a family of diﬀerent physical scenarios that are possible in M-theory. In certain limiting cases corresponding
to the cusps, it is natural to describe the physics using one of the six theories labeled there.

physics and quantum gravity. Unlike supergravity theory, string theory was able to accommodate the chirality of
the standard model, and it provided a theory of gravity consistent with quantum eﬀects.[38] Another feature of string
theory that many physicists were drawn to in the 1980s and 1990s was its high degree of uniqueness. In ordinary
particle theories, one can consider any collection of elementary particles whose classical behavior is described by
an arbitrary Lagrangian. In string theory, the possibilities are much more constrained: by the 1990s, physicists had
argued that there were only ﬁve consistent supersymmetric versions of the theory.[38]
Although there were only a handful of consistent superstring theories, it remained a mystery why there was not just
one consistent formulation.[38] However, as physicists began to examine string theory more closely, they realized that
these theories are related in intricate and nontrivial ways. They found that a system of strongly interacting strings
can, in some cases, be viewed as a system of weakly interacting strings. This phenomenon is known as S-duality.
It was studied by Ashoke Sen in the context of heterotic strings in four dimensions[39][40] and by Chris Hull and
Paul Townsend in the context of the type IIB theory.[41] Theorists also found that diﬀerent string theories may be
related by T-duality. This duality implies that strings propagating on completely diﬀerent spacetime geometries may
be physically equivalent.[42]
At around the same time, as many physicists were studying the properties of strings, a small group of physicists
was examining the possible applications of higher dimensional objects. In 1987, Eric Bergshoeﬀ, Ergin Sezgin,
and Paul Townsend showed that eleven-dimensional supergravity includes two-dimensional branes.[43] Intuitively,
these objects look like sheets or membranes propagating through the eleven-dimensional spacetime. Shortly after
this discovery, Michael Duﬀ, Paul Howe, Takeo Inami, and Kellogg Stelle considered a particular compactiﬁcation
of eleven-dimensional supergravity with one of the dimensions curled up into a circle.[44] In this setting, one can
imagine the membrane wrapping around the circular dimension. If the radius of the circle is suﬃciently small, then
this membrane looks just like a string in ten-dimensional spacetime. In fact, Duﬀ and his collaborators showed that
this construction reproduces exactly the strings appearing in type IIA superstring theory.[45]
Speaking at a string theory conference in 1995, Edward Witten made the surprising suggestion that all ﬁve superstring theories were in fact just diﬀerent limiting cases of a single theory in eleven spacetime dimensions. Witten’s
announcement drew together all of the previous results on S- and T-duality and the appearance of higher dimensional

158

CHAPTER 39. STRING THEORY

branes in string theory.[46] In the months following Witten’s announcement, hundreds of new papers appeared on the
Internet conﬁrming diﬀerent parts of his proposal.[47] Today this ﬂurry of work is known as the second superstring
revolution.[48]
Initially, some physicists suggested that the new theory was a fundamental theory of membranes, but Witten was
skeptical of the role of membranes in the theory. In a paper from 1996, Hořava and Witten wrote “As it has been
proposed that the eleven-dimensional theory is a supermembrane theory but there are some reasons to doubt that
interpretation, we will non-committally call it the M-theory, leaving to the future the relation of M to membranes.”[49]
In the absence of an understanding of the true meaning and structure of M-theory, Witten has suggested that the M
should stand for “magic”, “mystery”, or “membrane” according to taste, and the true meaning of the title should be
decided when a more fundamental formulation of the theory is known.[50]

39.2.2

Matrix theory

Main article: Matrix theory (physics)
In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular
kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A
matrix model describes the behavior of a set of matrices within the framework of quantum mechanics.[51]
One important example of a matrix model is the BFSS matrix model proposed by Tom Banks, Willy Fischler, Stephen
Shenker, and Leonard Susskind in 1997. This theory describes the behavior of a set of nine large matrices. In their
original paper, these authors showed, among other things, that the low energy limit of this matrix model is described
by eleven-dimensional supergravity. These calculations led them to propose that the BFSS matrix model is exactly
equivalent to M-theory. The BFSS matrix model can therefore be used as a prototype for a correct formulation of
M-theory and a tool for investigating the properties of M-theory in a relatively simple setting.[51]
The development of the matrix model formulation of M-theory has led physicists to consider various connections
between string theory and a branch of mathematics called noncommutative geometry. This subject is a generalization of ordinary geometry in which mathematicians deﬁne new geometric notions using tools from noncommutative
algebra.[52] In a paper from 1998, Alain Connes, Michael R. Douglas, and Albert Schwarz showed that some aspects
of matrix models and M-theory are described by a noncommutative quantum ﬁeld theory, a special kind of physical
theory in which spacetime is described mathematically using noncommutative geometry.[53] This established a link
between matrix models and M-theory on the one hand, and noncommutative geometry on the other hand. It quickly
led to the discovery of other important links between noncommutative geometry and various physical theories.[54][55]

39.3 Black holes
In general relativity, a black hole is deﬁned as a region of spacetime in which the gravitational ﬁeld is so strong that
no particle or radiation can escape. In the currently accepted models of stellar evolution, black holes are thought to
arise when massive stars undergo gravitational collapse, and many galaxies are thought to contain supermassive black
holes at their centers. Black holes are also important for theoretical reasons, as they present profound challenges for
theorists attempting to understand the quantum aspects of gravity. String theory has proved to be an important tool
for investigating the theoretical properties of black holes because it provides a framework in which theorists can study
their thermodynamics.[56]

39.3.1

Bekenstein–Hawking formula

In the branch of physics called statistical mechanics, entropy is a measure of the randomness or disorder of a physical
system. This concept was studied in the 1870s by the Austrian physicist Ludwig Boltzmann, who showed that the
thermodynamic properties of a gas could be derived from the combined properties of its many constituent molecules.
Boltzmann argued that by averaging the behaviors of all the diﬀerent molecules in a gas, one can understand macroscopic properties such as volume, temperature, and pressure. In addition, this perspective led him to give a precise
deﬁnition of entropy as the natural logarithm of the number of diﬀerent states of the molecules (also called microstates) that give rise to the same macroscopic features.[57]
In the twentieth century, physicists began to apply the same concepts to black holes. In most systems such as gases,

39.4. ADS/CFT CORRESPONDENCE

159

the entropy scales with the volume. In the 1970s, the physicist Jacob Bekenstein suggested that the entropy of a black
hole is instead proportional to the surface area of its event horizon, the boundary beyond which matter and radiation
is lost to its gravitational attraction.[58] When combined with ideas of the physicist Stephen Hawking,[59] Bekenstein’s
work yielded a precise formula for the entropy of a black hole. The formula expresses the entropy S as

S=

c3 kA
4ℏG

where c is the speed of light, k is Boltzmann’s constant, ħ is the reduced Planck constant, G is Newton’s constant, and
A is the surface area of the event horizon.[60]
Like any physical system, a black hole has an entropy deﬁned in terms of the number of diﬀerent microstates that
lead to the same macroscopic features. The Bekenstein–Hawking entropy formula gives the expected value of the
entropy of a black hole, but by the 1990s, physicists still lacked a derivation of this formula by counting microstates
in a theory of quantum gravity. Finding such a derivation of this formula was considered an important test of the
viability of any theory of quantum gravity such as string theory.[61]

39.3.2

Derivation within string theory

In a paper from 1996, Andrew Strominger and Cumrun Vafa showed how to derive the Beckenstein–Hawking formula
for certain black holes in string theory.[62] Their calculation was based on the observation that D-branes—which look
like ﬂuctuating membranes when they are weakly interacting—become dense, massive objects with event horizons
when the interactions are strong. In other words, a system of strongly interacting D-branes in string theory is indistinguishable from a black hole. Strominger and Vafa analyzed such D-brane systems and calculated the number of
diﬀerent ways of placing D-branes in spacetime so that their combined mass and charge is equal to a given mass and
charge for the resulting black hole. Their calculation reproduced the Bekenstein–Hawking formula exactly, including
the factor of 1/4.[63] Subsequent work by Strominger, Vafa, and others reﬁned the original calculations and gave the
precise values of the “quantum corrections” needed to describe very small black holes.[64][65]
The black holes that Strominger and Vafa considered in their original work were quite diﬀerent from real astrophysical
black holes. One diﬀerence was that Strominger and Vafa considered only extremal black holes in order to make
the calculation tractable. These are deﬁned as black holes with the lowest possible mass compatible with a given
charge.[66] Strominger and Vafa also restricted attention to black holes in ﬁve-dimensional spacetime with unphysical
supersymmetry.[67]
Although it was originally developed in this very particular and physically unrealistic context in string theory, the
entropy calculation of Strominger and Vafa has led to a qualitative understanding of how black hole entropy can be
accounted for in any theory of quantum gravity. Indeed, in 1998, Strominger argued that the original result could be
generalized to an arbitrary consistent theory of quantum gravity without relying on strings or supersymmetry.[68] In
collaboration with several other authors in 2010, he showed that some results on black hole entropy could be extended
to non-extremal astrophysical black holes.[69][70]

39.4 AdS/CFT correspondence
Main article: AdS/CFT correspondence
One approach to formulating string theory and studying its properties is provided by the anti-de Sitter/conformal
ﬁeld theory (AdS/CFT) correspondence. This is a theoretical result which implies that string theory is in some cases
equivalent to a quantum ﬁeld theory. In addition to providing insights into the mathematical structure of string theory,
the AdS/CFT correspondence has shed light on many aspects of quantum ﬁeld theory in regimes where traditional
calculational techniques are ineﬀective.[6] The AdS/CFT correspondence was ﬁrst proposed by Juan Maldacena in late
1997.[71] Important aspects of the correspondence were elaborated in articles by Steven Gubser, Igor Klebanov, and
Alexander Markovich Polyakov,[72] and by Edward Witten.[73] By 2010, Maldacena’s article had over 7000 citations,
becoming the most highly cited article in the ﬁeld of high energy physics.[lower-alpha 3]

160

39.4.1

CHAPTER 39. STRING THEORY

Overview of the correspondence

In the AdS/CFT correspondence, the geometry of spacetime is described in terms of a certain vacuum solution
of Einstein’s equation called anti-de Sitter space.[74] In very elementary terms, anti-de Sitter space is a mathematical
model of spacetime in which the notion of distance between points (the metric) is diﬀerent from the notion of distance
in ordinary Euclidean geometry. It is closely related to hyperbolic space, which can be viewed as a disk as illustrated
on the left.[75] This image shows a tessellation of a disk by triangles and squares. One can deﬁne the distance between
points of this disk in such a way that all the triangles and squares are the same size and the circular outer boundary is
inﬁnitely far from any point in the interior.[76]
One can imagine a stack of hyperbolic disks where each disk represents the state of the universe at a given time.
The resulting geometric object is three-dimensional anti-de Sitter space.[75] It looks like a solid cylinder in which any
cross section is a copy of the hyperbolic disk. Time runs along the vertical direction in this picture. The surface of
this cylinder plays an important role in the AdS/CFT correspondence. As with the hyperbolic plane, anti-de Sitter
space is curved in such a way that any point in the interior is actually inﬁnitely far from this boundary surface.[76]

time

anti-de Sitter space
conformal
boundary

Three-dimensional anti-de Sitter space is like a stack of hyperbolic disks, each one representing the state of the universe at a given
time. The resulting spacetime looks like a solid cylinder.

This construction describes a hypothetical universe with only two space dimensions and one time dimension, but it
can be generalized to any number of dimensions. Indeed, hyperbolic space can have more than two dimensions and
one can “stack up” copies of hyperbolic space to get higher-dimensional models of anti-de Sitter space.[75]

39.4. ADS/CFT CORRESPONDENCE

161

An important feature of anti-de Sitter space is its boundary (which looks like a cylinder in the case of threedimensional anti-de Sitter space). One property of this boundary is that, within a small region on the surface around
any given point, it looks just like Minkowski space, the model of spacetime used in nongravitational physics.[77] One
can therefore consider an auxiliary theory in which “spacetime” is given by the boundary of anti-de Sitter space.
This observation is the starting point for AdS/CFT correspondence, which states that the boundary of anti-de Sitter
space can be regarded as the “spacetime” for a quantum ﬁeld theory. The claim is that this quantum ﬁeld theory is
equivalent to a gravitational theory, such as string theory, in the bulk anti-de Sitter space in the sense that there is
a “dictionary” for translating entities and calculations in one theory into their counterparts in the other theory. For
example, a single particle in the gravitational theory might correspond to some collection of particles in the boundary
theory. In addition, the predictions in the two theories are quantitatively identical so that if two particles have a 40
percent chance of colliding in the gravitational theory, then the corresponding collections in the boundary theory
would also have a 40 percent chance of colliding.[78]

39.4.2

Applications to quantum gravity

The discovery of the AdS/CFT correspondence was a major advance in physicists’ understanding of string theory and
quantum gravity. One reason for this is that the correspondence provides a formulation of string theory in terms of
quantum ﬁeld theory, which is well understood by comparison. Another reason is that it provides a general framework
in which physicists can study and attempt to resolve the paradoxes of black holes.[56]
In 1975, Stephen Hawking published a calculation which suggested that black holes are not completely black but
emit a dim radiation due to quantum eﬀects near the event horizon.[59] At ﬁrst, Hawking’s result posed a problem for
theorists because it suggested that black holes destroy information. More precisely, Hawking’s calculation seemed
to conﬂict with one of the basic postulates of quantum mechanics, which states that physical systems evolve in time
according to the Schrödinger equation. This property is usually referred to as unitarity of time evolution. The apparent
contradiction between Hawking’s calculation and the unitarity postulate of quantum mechanics came to be known as
the black hole information paradox.[79]
The AdS/CFT correspondence resolves the black hole information paradox, at least to some extent, because it shows
how a black hole can evolve in a manner consistent with quantum mechanics in some contexts. Indeed, one can
consider black holes in the context of the AdS/CFT correspondence, and any such black hole corresponds to a conﬁguration of particles on the boundary of anti-de Sitter space.[80] These particles obey the usual rules of quantum
mechanics and in particular evolve in a unitary fashion, so the black hole must also evolve in a unitary fashion, respecting the principles of quantum mechanics.[81] In 2005, Hawking announced that the paradox had been settled
in favor of information conservation by the AdS/CFT correspondence, and he suggested a concrete mechanism by
which black holes might preserve information.[82]

39.4.3

Applications to quantum ﬁeld theory

Main articles: AdS/QCD correspondence and AdS/CMT correspondence
In addition to its applications to theoretical problems in quantum gravity, the AdS/CFT correspondence has been
applied to a variety of problems in quantum ﬁeld theory. One physical system that has been studied using the AdS/CFT
correspondence is the quark–gluon plasma, an exotic state of matter produced in particle accelerators. This state
of matter arises for brief instants when heavy ions such as gold or lead nuclei are collided at high energies. Such
collisions cause the quarks that make up atomic nuclei to deconﬁne at temperatures of approximately two trillion
kelvins, conditions similar to those present at around 10−11 seconds after the Big Bang.[83]
The physics of the quark–gluon plasma is governed by a theory called quantum chromodynamics, but this theory is
mathematically intractable in problems involving the quark–gluon plasma.[lower-alpha 4] In an article appearing in 2005,
Đàm Thanh Sơn and his collaborators showed that the AdS/CFT correspondence could be used to understand some
aspects of the quark–gluon plasma by describing it in the language of string theory.[84] By applying the AdS/CFT
correspondence, Sơn and his collaborators were able to describe the quark gluon plasma in terms of black holes
in ﬁve-dimensional spacetime. The calculation showed that the ratio of two quantities associated with the quark–
gluon plasma, the shear viscosity and volume density of entropy, should be approximately equal to a certain universal
constant. In 2008, the predicted value of this ratio for the quark–gluon plasma was conﬁrmed at the Relativistic
Heavy Ion Collider at Brookhaven National Laboratory.[85][86]
The AdS/CFT correspondence has also been used to study aspects of condensed matter physics. Over the decades,
experimental condensed matter physicists have discovered a number of exotic states of matter, including superconductors

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CHAPTER 39. STRING THEORY

A magnet levitating above a high-temperature superconductor. Today some physicists are working to understand high-temperature
superconductivity using the AdS/CFT correspondence.[7]

and superﬂuids. These states are described using the formalism of quantum ﬁeld theory, but some phenomena are difﬁcult to explain using standard ﬁeld theoretic techniques. Some condensed matter theorists including Subir Sachdev
hope that the AdS/CFT correspondence will make it possible to describe these systems in the language of string
theory and learn more about their behavior.[85]
So far some success has been achieved in using string theory methods to describe the transition of a superﬂuid to an
insulator. A superﬂuid is a system of electrically neutral atoms that ﬂows without any friction. Such systems are often
produced in the laboratory using liquid helium, but recently experimentalists have developed new ways of producing
artiﬁcial superﬂuids by pouring trillions of cold atoms into a lattice of criss-crossing lasers. These atoms initially
behave as a superﬂuid, but as experimentalists increase the intensity of the lasers, they become less mobile and then
suddenly transition to an insulating state. During the transition, the atoms behave in an unusual way. For example, the
atoms slow to a halt at a rate that depends on the temperature and on Planck’s constant, the fundamental parameter
of quantum mechanics, which does not enter into the description of the other phases. This behavior has recently
been understood by considering a dual description where properties of the ﬂuid are described in terms of a higher
dimensional black hole.[87]

39.5 Phenomenology
Main article: String phenomenology
In addition to being an idea of considerable theoretical interest, string theory provides a framework for constructing
models of real world physics that combine general relativity and particle physics. Phenomenology is the branch of
theoretical physics in which physicists construct realistic models of nature from more abstract theoretical ideas. String
phenomenology is the part of string theory that attempts to construct realistic models based on string theory.
Partly because of theoretical and mathematical diﬃculties and partly because of the extremely high energies needed
to test these theories experimentally, there is so far no experimental evidence that would unambiguously point to any

39.5. PHENOMENOLOGY

163

of these models being a correct fundamental description of nature. This has led some in the community to criticize
these approaches to uniﬁcation and question the value of continued research on these problems.[12]

39.5.1

Particle physics

The currently accepted theory describing elementary particles and their interactions is known as the standard model
of particle physics. This theory provides a uniﬁed description of three of the fundamental forces of nature: electromagnetism and the strong and weak nuclear forces. Despite its remarkable success in explaining a wide range of
physical phenomena, the standard model cannot be a complete description of reality. This is because the standard
model fails to incorporate the force of gravity and because of problems such as the hierarchy problem and the inability
to explain the structure of fermion masses or dark matter.
String theory has been used to construct a variety of models of particle physics going beyond the standard model. Typically, such models are based on the idea of compactiﬁcation. Starting with the ten- or eleven-dimensional spacetime
of string or M-theory, physicists postulate a shape for the extra dimensions. By choosing this shape appropriately, they
can construct models roughly similar to the standard model of particle physics, together with additional undiscovered
particles.[88] One popular way of deriving realistic physics from string theory is to start with the heterotic theory in
ten dimensions and assume that the six extra dimensions of spacetime are shaped like a six-dimensional Calabi–Yau
manifold. Such compactiﬁcations oﬀer many ways of extracting realistic physics from string theory. Other similar
methods can be used to construct realistic models of our four-dimensional world based on M-theory.[89]

39.5.2

Cosmology

Main article: String cosmology
The Big Bang theory is the prevailing cosmological model for the universe from the earliest known periods through its

A map of the cosmic microwave background produced by the Wilkinson Microwave Anisotropy Probe

subsequent large-scale evolution. Despite its success in explaining many observed features of the universe including
galactic redshifts, the relative abundance of light elements such as hydrogen and helium, and the existence of a
cosmic microwave background, there are several questions that remain unanswered. For example, the standard Big
Bang model does not explain why the universe appears to be same in all directions, why it appears ﬂat on very large
distance scales, or why certain hypothesized particles such as magnetic monopoles are not observed in experiments.[90]
Currently, the leading candidate for a theory going beyond the Big Bang is the theory of cosmic inﬂation. Developed by Alan Guth and others in the 1980s, inﬂation postulates a period of extremely rapid accelerated expansion of
the universe prior to the expansion described by the standard Big Bang theory. The theory of cosmic inﬂation preserves the successes of the Big Bang while providing a natural explanation for some of the mysterious features of the

164

CHAPTER 39. STRING THEORY

universe.[91] The theory has also received striking support from observations of the cosmic microwave background,
the radiation that has ﬁlled the sky since around 380,000 years after the Big Bang.[92]
In the theory of inﬂation, the rapid initial expansion of the universe is caused by a hypothetical particle called the
inﬂaton. The exact properties of this particle are not ﬁxed by the theory but should ultimately be derived from a
more fundamental theory such as string theory.[93] Indeed, there have been a number of attempts to identify an
inﬂation within the spectrum of particles described by string theory and to study inﬂation using string theory. While
these approaches might eventually ﬁnd support in observational data such as measurements of the cosmic microwave
background, the application of string theory to cosmology is still in its early stages.[94]

39.6 Connections to mathematics
In addition to inﬂuencing research in theoretical physics, string theory has stimulated a number of major developments
in pure mathematics. Like many developing ideas in theoretical physics, string theory does not at present have a
mathematically rigorous formulation in which all of its concepts can be deﬁned precisely. As a result, physicists who
study string theory are often guided by physical intuition to conjecture relationships between the seemingly diﬀerent
mathematical structures that are used to formalize diﬀerent parts of the theory. These conjectures are later proved
by mathematicians, and in this way, string theory serves as a source of new ideas in pure mathematics.[95]

39.6.1

Mirror symmetry

Main article: Mirror symmetry (string theory)
After Calabi–Yau manifolds had entered physics as a way to compactify extra dimensions in string theory, many
physicists began studying these manifolds. In the late 1980s, several physicists noticed that given such a compactiﬁcation of string theory, it is not possible to reconstruct uniquely a corresponding Calabi–Yau manifold.[96] Instead,
two diﬀerent versions of string theory, type IIA and type IIB, can be compactiﬁed on completely diﬀerent Calabi–
Yau manifolds giving rise to the same physics. In this situation, the manifolds are called mirror manifolds, and the
relationship between the two physical theories is called mirror symmetry.[97]
Regardless of whether Calabi–Yau compactiﬁcations of string theory provide a correct description of nature, the
existence of the mirror duality between diﬀerent string theories has signiﬁcant mathematical consequences. The
Calabi–Yau manifolds used in string theory are of interest in pure mathematics, and mirror symmetry allows mathematicians to solve problems in enumerative geometry, a branch of mathematics concerned with counting the numbers
of solutions to geometric questions.[31][98]
Enumerative geometry studies a class of geometric objects called algebraic varieties which are deﬁned by the vanishing
of polynomials. For example, the Clebsch cubic illustrated on the right is an algebraic variety deﬁned using a certain
polynomial of degree three in four variables. A celebrated result of nineteenth-century mathematicians Arthur Cayley
and George Salmon states that there are exactly 27 straight lines that lie entirely on such a surface.[99]
Generalizing this problem, one can ask how many lines can be drawn on a quintic Calabi–Yau manifold, such as the
one illustrated above, which is deﬁned by a polynomial of degree ﬁve. This problem was solved by the nineteenthcentury German mathematician Hermann Schubert, who found that there are exactly 2,875 such lines. In 1986,
geometer Sheldon Katz proved that the number of curves, such as circles, that are deﬁned by polynomials of degree
two and lie entirely in the quintic is 609,250.[100]
By the year 1991, most of the classical problems of enumerative geometry had been solved and interest in enumerative
geometry had begun to diminish.[101] The ﬁeld was reinvigorated in May 1991 when physicists Philip Candelas, Xenia
de la Ossa, Paul Green, and Linda Parks showed that mirror symmetry could be used to translate diﬃcult mathematical
questions about one Calabi–Yau manifold into easier questions about its mirror.[102] In particular, they used mirror
symmetry to show that a six-dimensional Calabi–Yau manifold can contain exactly 317,206,375 curves of degree
three.[101] In addition to counting degree-three curves, Candelas and his collaborators obtained a number of more
general results for counting rational curves which went far beyond the results obtained by mathematicians.[103]
Originally, these results of Candelas were justiﬁed on physical grounds. However, mathematicians generally prefer
rigorous proofs that do not require an appeal to physical intuition. Inspired by physicists’ work on mirror symmetry, mathematicians have therefore constructed their own arguments proving the enumerative predictions of mirror symmetry.[lower-alpha 5] Today mirror symmetry is an active area of research in mathematics, and mathematicians are working to develop a more complete mathematical understanding of mirror symmetry based on physicists’

39.6. CONNECTIONS TO MATHEMATICS

165

The Clebsch cubic is an example of a kind of geometric object called an algebraic variety. A classical result of enumerative geometry
states that there are exactly 27 straight lines that lie entirely on this surface.

intuition.[104] Major approaches to mirror symmetry include the homological mirror symmetry program of Maxim
Kontsevich[32] and the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, and Eric Zaslow.[105]

39.6.2

Monstrous moonshine

Main article: Monstrous moonshine
Group theory is the branch of mathematics that studies the concept of symmetry. For example, one can consider a
geometric shape such as an equilateral triangle. There are various operations that one can perform on this triangle
without changing its shape. One can rotate it through 120°, 240°, or 360°, or one can reﬂect in any of the lines labeled
S 0 , S 1 , or S 2 in the picture. Each of these operations is called a symmetry, and the collection of these symmetries
satisﬁes certain technical properties making it into what mathematicians call a group. In this particular example, the
group is known as the dihedral group of order 6 because it has six elements. A general group may describe ﬁnitely
many or inﬁnitely many symmetries; if there are only ﬁnitely many symmetries, it is called a ﬁnite group.[106]
Mathematicians often strive for a classiﬁcation (or list) of all mathematical objects of a given type. It is generally
believed that ﬁnite groups are too diverse to admit a useful classiﬁcation. A more modest but still challenging problem
is to classify all ﬁnite simple groups. These are ﬁnite groups which may be used as building blocks for constructing
arbitrary ﬁnite groups in the same way that prime numbers can be used to construct arbitrary whole numbers by taking
products.[lower-alpha 6] One of the major achievements of contemporary group theory is the classiﬁcation of ﬁnite simple
groups, a mathematical theorem which provides a list of all possible ﬁnite simple groups.[107]

166

CHAPTER 39. STRING THEORY

An equilateral triangle can be rotated through 120°, 240°, or 360°, or reﬂected in any of the three lines pictured without changing
its shape.

This classiﬁcation theorem identiﬁes several inﬁnite families of groups as well as 26 additional groups which do not
ﬁt into any family. The latter groups are called the “sporadic” groups, and each one owes its existence to a remarkable
combination of circumstances. The largest sporadic group, the so-called monster group, has over 1053 elements,
more than a thousand times the number of atoms in the Earth.[108]
A seemingly unrelated construction is the j-function of number theory. This object belongs to a special class of
functions called modular functions, whose graphs form a certain kind of repeating pattern.[109] Although this function
appears in a branch of mathematics which seems very diﬀerent from the theory of ﬁnite groups, the two subjects turn
out to be intimately related. In the late 1970s, mathematicians John McKay and John Thompson noticed that certain
numbers arising in the analysis of the monster group (namely, the dimensions of its irreducible representations) are
related to numbers that appear in a formula for the j-function (namely, the coeﬃcients of its Fourier series).[110] This
relationship was further developed by John Horton Conway and Simon Norton[111] who called it monstrous moonshine
because it seemed so far fetched.[112]
In 1992, Richard Borcherds constructed a bridge between the theory of modular functions and ﬁnite groups and, in
the process, explained the observations of McKay and Thompson.[113][114] Borcherds’ work used ideas from string
theory in an essential way, extending earlier results of Igor Frenkel, James Lepowsky, and Arne Meurman, who had
realized the monster group as the symmetries of a particular version of string theory.[115] In 1998, Borcherds was
awarded the Fields medal for his work.[116]
Since the 1990s, the connection between string theory and moonshine has led to further results in mathematics and
physics.[108] In 2010, physicists Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa discovered connections between
a diﬀerent sporadic group, the Mathieu group M 24 , and a certain version of string theory.[117] Miranda Cheng,
John Duncan, and Jeﬀrey A. Harvey proposed a generalization of this moonshine phenomenon called umbral moon-

39.7. HISTORY

167

A graph of the j-function in the complex plane

shine,[118] and their conjecture was proved mathematically by Duncan, Michael Griﬃn, and Ken Ono.[119] Witten
has also speculated that the version of string theory appearing in monstrous moonshine might be related to a certain
simpliﬁed model of gravity in three spacetime dimensions.[120]

39.7 History
Main article: History of string theory

39.7.1

Early results

Some of the structures reintroduced by string theory arose for the ﬁrst time much earlier as part of the program of
classical uniﬁcation started by Albert Einstein. The ﬁrst person to add a ﬁfth dimension to a theory of gravity was
Gunnar Nordström in 1914, who noted that gravity in ﬁve dimensions describes both gravity and electromagnetism in
four. Nordström attempted to unify electromagnetism with his theory of gravitation, which was however superseded
by Einstein’s general relativity in 1919. Thereafter, German mathematician Theodor Kaluza combined the ﬁfth
dimension with general relativity, and only Kaluza is usually credited with the idea. In 1926, the Swedish physicist
Oskar Klein gave a physical interpretation of the unobservable extra dimension—it is wrapped into a small circle.
Einstein introduced a non-symmetric metric tensor, while much later Brans and Dicke added a scalar component to
gravity. These ideas would be revived within string theory, where they are demanded by consistency conditions.
String theory was originally developed during the late 1960s and early 1970s as a never completely successful theory
of hadrons, the subatomic particles like the proton and neutron that feel the strong interaction. In the 1960s, Geoﬀrey
Chew and Steven Frautschi discovered that the mesons make families called Regge trajectories with masses related
to spins in a way that was later understood by Yoichiro Nambu, Holger Bech Nielsen and Leonard Susskind to be the
relationship expected from rotating strings. Chew advocated making a theory for the interactions of these trajectories

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CHAPTER 39. STRING THEORY

Leonard Susskind

that did not presume that they were composed of any fundamental particles, but would construct their interactions
from self-consistency conditions on the S-matrix. The S-matrix approach was started by Werner Heisenberg in the

39.7. HISTORY

169

1940s as a way of constructing a theory that did not rely on the local notions of space and time, which Heisenberg
believed break down at the nuclear scale. While the scale was oﬀ by many orders of magnitude, the approach he
advocated was ideally suited for a theory of quantum gravity.
Working with experimental data, R. Dolen, D. Horn and C. Schmid developed some sum rules for hadron exchange.
When a particle and antiparticle scatter, virtual particles can be exchanged in two qualitatively diﬀerent ways. In
the s-channel, the two particles annihilate to make temporary intermediate states that fall apart into the ﬁnal state
particles. In the t-channel, the particles exchange intermediate states by emission and absorption. In ﬁeld theory, the
two contributions add together, one giving a continuous background contribution, the other giving peaks at certain
energies. In the data, it was clear that the peaks were stealing from the background—the authors interpreted this as
saying that the t-channel contribution was dual to the s-channel one, meaning both described the whole amplitude
and included the other.

Gabriele Veneziano

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CHAPTER 39. STRING THEORY

The result was widely advertised by Murray Gell-Mann, leading Gabriele Veneziano to construct a scattering amplitude that had the property of Dolen-Horn-Schmid duality, later renamed world-sheet duality. The amplitude needed
poles where the particles appear, on straight line trajectories, and there is a special mathematical function whose
poles are evenly spaced on half the real line— the Gamma function— which was widely used in Regge theory. By
manipulating combinations of Gamma functions, Veneziano was able to ﬁnd a consistent scattering amplitude with
poles on straight lines, with mostly positive residues, which obeyed duality and had the appropriate Regge scaling at
high energy. The amplitude could ﬁt near-beam scattering data as well as other Regge type ﬁts, and had a suggestive
integral representation that could be used for generalization.
Over the next years, hundreds of physicists worked to complete the bootstrap program for this model, with many
surprises. Veneziano himself discovered that for the scattering amplitude to describe the scattering of a particle
that appears in the theory, an obvious self-consistency condition, the lightest particle must be a tachyon. Miguel
Virasoro and Joel Shapiro found a diﬀerent amplitude now understood to be that of closed strings, while Ziro Koba
and Holger Nielsen generalized Veneziano’s integral representation to multiparticle scattering. Veneziano and Sergio
Fubini introduced an operator formalism for computing the scattering amplitudes that was a forerunner of world-sheet
conformal theory, while Virasoro understood how to remove the poles with wrong-sign residues using a constraint on
the states. Claud Lovelace calculated a loop amplitude, and noted that there is an inconsistency unless the dimension
of the theory is 26. Charles Thorn, Peter Goddard and Richard Brower went on to prove that there are no wrong-sign
propagating states in dimensions less than or equal to 26.
In 1969, Yoichiro Nambu, Holger Bech Nielsen, and Leonard Susskind recognized that the theory could be given
a description in space and time in terms of strings. The scattering amplitudes were derived systematically from
the action principle by Peter Goddard, Jeﬀrey Goldstone, Claudio Rebbi, and Charles Thorn, giving a space-time
picture to the vertex operators introduced by Veneziano and Fubini and a geometrical interpretation to the Virasoro
conditions.
In 1970, Pierre Ramond added fermions to the model, which led him to formulate a two-dimensional supersymmetry
to cancel the wrong-sign states. John Schwarz and André Neveu added another sector to the fermi theory a short
time later. In the fermion theories, the critical dimension was 10. Stanley Mandelstam formulated a world sheet
conformal theory for both the bose and fermi case, giving a two-dimensional ﬁeld theoretic path-integral to generate
the operator formalism. Michio Kaku and Keiji Kikkawa gave a diﬀerent formulation of the bosonic string, as a string
ﬁeld theory, with inﬁnitely many particle types and with ﬁelds taking values not on points, but on loops and curves.
In 1974, Tamiaki Yoneya discovered that all the known string theories included a massless spin-two particle that
obeyed the correct Ward identities to be a graviton. John Schwarz and Joel Scherk came to the same conclusion and
made the bold leap to suggest that string theory was a theory of gravity, not a theory of hadrons. They reintroduced
Kaluza–Klein theory as a way of making sense of the extra dimensions. At the same time, quantum chromodynamics was recognized as the correct theory of hadrons, shifting the attention of physicists and apparently leaving the
bootstrap program in the dustbin of history.
String theory eventually made it out of the dustbin, but for the following decade all work on the theory was completely
ignored. Still, the theory continued to develop at a steady pace thanks to the work of a handful of devotees. Ferdinando
Gliozzi, Joel Scherk, and David Olive realized in 1976 that the original Ramond and Neveu Schwarz-strings were
separately inconsistent and needed to be combined. The resulting theory did not have a tachyon, and was proven to
have space-time supersymmetry by John Schwarz and Michael Green in 1981. The same year, Alexander Polyakov
gave the theory a modern path integral formulation, and went on to develop conformal ﬁeld theory extensively. In
1979, Daniel Friedan showed that the equations of motions of string theory, which are generalizations of the Einstein
equations of General Relativity, emerge from the Renormalization group equations for the two-dimensional ﬁeld
theory. Schwarz and Green discovered T-duality, and constructed two superstring theories—IIA and IIB related by
T-duality, and type I theories with open strings. The consistency conditions had been so strong, that the entire theory
was nearly uniquely determined, with only a few discrete choices.

39.7.2

First superstring revolution

In the early 1980s, Edward Witten discovered that most theories of quantum gravity could not accommodate chiral
fermions like the neutrino. This led him, in collaboration with Luis Alvarez-Gaumé to study violations of the conservation laws in gravity theories with anomalies, concluding that type I string theories were inconsistent. Green
and Schwarz discovered a contribution to the anomaly that Witten and Alvarez-Gaumé had missed, which restricted
the gauge group of the type I string theory to be SO(32). In coming to understand this calculation, Edward Witten
became convinced that string theory was truly a consistent theory of gravity, and he became a high-proﬁle advocate.

39.7. HISTORY

171

John Schwarz

Following Witten’s lead, between 1984 and 1986, hundreds of physicists started to work in this ﬁeld, and this is
sometimes called the ﬁrst superstring revolution.
During this period, David Gross, Jeﬀrey Harvey, Emil Martinec, and Ryan Rohm discovered heterotic strings. The
gauge group of these closed strings was two copies of E8, and either copy could easily and naturally include the
standard model. Philip Candelas, Gary Horowitz, Andrew Strominger and Edward Witten found that the Calabi–Yau
manifolds are the compactiﬁcations that preserve a realistic amount of supersymmetry, while Lance Dixon and others
worked out the physical properties of orbifolds, distinctive geometrical singularities allowed in string theory. Cumrun
Vafa generalized T-duality from circles to arbitrary manifolds, creating the mathematical ﬁeld of mirror symmetry.
Daniel Friedan, Emil Martinec and Stephen Shenker further developed the covariant quantization of the superstring

172

CHAPTER 39. STRING THEORY

Edward Witten

using conformal ﬁeld theory techniques. David Gross and Vipul Periwal discovered that string perturbation theory was
divergent. Stephen Shenker showed it diverged much faster than in ﬁeld theory suggesting that new non-perturbative
objects were missing.
In the 1990s, Joseph Polchinski discovered that the theory requires higher-dimensional objects, called D-branes and
identiﬁed these with the black-hole solutions of supergravity. These were understood to be the new objects suggested
by the perturbative divergences, and they opened up a new ﬁeld with rich mathematical structure. It quickly became
clear that D-branes and other p-branes, not just strings, formed the matter content of the string theories, and the
physical interpretation of the strings and branes was revealed—they are a type of black hole. Leonard Susskind had
incorporated the holographic principle of Gerardus 't Hooft into string theory, identifying the long highly excited
string states with ordinary thermal black hole states. As suggested by 't Hooft, the ﬂuctuations of the black hole
horizon, the world-sheet or world-volume theory, describes not only the degrees of freedom of the black hole, but all
nearby objects too.

39.7.3

Second superstring revolution

In 1995, at the annual conference of string theorists at the University of Southern California (USC), Edward Witten
gave a speech on string theory that in essence united the ﬁve string theories that existed at the time, and giving birth
to a new 11-dimensional theory called M-theory. M-theory was also foreshadowed in the work of Paul Townsend at

39.7. HISTORY

173

Joseph Polchinski

approximately the same time. The ﬂurry of activity that began at this time is sometimes called the second superstring
revolution.[34]
During this period, Tom Banks, Willy Fischler, Stephen Shenker and Leonard Susskind formulated matrix theory, a
full holographic description of M-theory using IIA D0 branes.[51] This was the ﬁrst deﬁnition of string theory that
was fully non-perturbative and a concrete mathematical realization of the holographic principle. It is an example
of a gauge-gravity duality and is now understood to be a special case of the AdS/CFT correspondence. Andrew

174

CHAPTER 39. STRING THEORY

Juan Maldacena

Strominger and Cumrun Vafa calculated the entropy of certain conﬁgurations of D-branes and found agreement with
the semi-classical answer for extreme charged black holes.[62] Petr Hořava and Witten found the eleven-dimensional
formulation of the heterotic string theories, showing that orbifolds solve the chirality problem. Witten noted that the
eﬀective description of the physics of D-branes at low energies is by a supersymmetric gauge theory, and found geometrical interpretations of mathematical structures in gauge theory that he and Nathan Seiberg had earlier discovered
in terms of the location of the branes.
In 1997, Juan Maldacena noted that the low energy excitations of a theory near a black hole consist of objects close to
the horizon, which for extreme charged black holes looks like an anti-de Sitter space.[71] He noted that in this limit the
gauge theory describes the string excitations near the branes. So he hypothesized that string theory on a near-horizon

39.8. CRITICISM

175

extreme-charged black-hole geometry, an anti-deSitter space times a sphere with ﬂux, is equally well described by the
low-energy limiting gauge theory, the N = 4 supersymmetric Yang–Mills theory. This hypothesis, which is called the
AdS/CFT correspondence, was further developed by Steven Gubser, Igor Klebanov and Alexander Polyakov,[72] and
by Edward Witten,[73] and it is now well-accepted. It is a concrete realization of the holographic principle, which has
far-reaching implications for black holes, locality and information in physics, as well as the nature of the gravitational
interaction.[56] Through this relationship, string theory has been shown to be related to gauge theories like quantum
chromodynamics and this has led to more quantitative understanding of the behavior of hadrons, bringing string
theory back to its roots.[84]

39.8 Criticism
39.8.1

Number of solutions

To construct models of particle physics based on string theory, physicists typically begin by specifying a shape for
the extra dimensions of spacetime. Each of these diﬀerent shapes corresponds to a diﬀerent possible universe, or
“vacuum state”, with a diﬀerent collection of particles and forces. String theory as it is currently understood has an
enormous number of vacuum states, typically estimated to be around 10500 , and these might be suﬃciently diverse
to accommodate almost any phenomena that might be observed at low energies.[121]
Many critics of string theory have expressed concerns about the large number of possible universes described by string
theory. In his book Not Even Wrong, Peter Woit, a lecturer in the mathematics department at Columbia University,
has argued that the large number of diﬀerent physical scenarios renders string theory vacuous as a framework for
constructing models of particle physics. According to Woit,
The possible existence of, say, 10500 consistent diﬀerent vacuum states for superstring theory probably destroys the hope of using the theory to predict anything. If one picks among this large set just
those states whose properties agree with present experimental observations, it is likely there still will be
such a large number of these that one can get just about whatever value one wants for the results of any
new observation.[122]
Some physicists believe this large number of solutions is actually a virtue because it may allow a natural anthropic explanation of the observed values of physical constants, in particular the small value of the cosmological constant.[122]
The anthropic principle is the idea that some of the numbers appearing in the laws of physics are not ﬁxed by any
fundamental principle but must be compatible with the evolution of intelligent life. In 1987, Steven Weinberg published an article in which he argued that the cosmological constant could not have been too large, or else galaxies and
intelligent life would not have been able to develop.[123] Weinberg suggested that there might be a huge number of
possible consistent universes, each with a diﬀerent value of the cosmological constant, and observations indicate a
small value of the cosmological constant only because humans happen to live in a universe that has allowed intelligent
life, and hence observers, to exist.[124]
String theorist Leonard Susskind has argued that string theory provides a natural anthropic explanation of the small
value of the cosmological constant.[125] According to Susskind, the diﬀerent vacuum states of string theory might be
realized as diﬀerent universes within a larger multiverse. The fact that the observed universe has a small cosmological
constant is just a tautological consequence of the fact that a small value is required for life to exist.[126] Many prominent
theorists and critics have disagreed with Susskind’s conclusions.[127] According to Woit, “in this case [anthropic
reasoning] is nothing more than an excuse for failure. Speculative scientiﬁc ideas fail not just when they make
incorrect predictions, but also when they turn out to be vacuous and incapable of predicting anything.”[128]

39.8.2

Background independence

Main article: Background independence
One of the fundamental principles of Einstein’s general theory of relativity is the idea that the laws of physics should
be background independent. This means that the geometry of spacetime is not speciﬁed from the outset but is instead
determined dynamically by the theory. In general relativity, the geometry of spacetime can evolve in time, responding
to whatever matter is present.[129]

176

CHAPTER 39. STRING THEORY

One of the older criticisms of string theory is that it is not manifestly background independent. In string theory, one
must typically specify a ﬁxed reference geometry for spacetime, and all other possible geometries are described as
perturbations of this ﬁxed one. In his book The Trouble With Physics, physicist Lee Smolin of the Perimeter Institute
for Theoretical Physics claims that this is the principal weakness of string theory as a theory of quantum gravity,
saying that string theory has failed to incorporate this important insight from general relativity.[130]
Others have disagreed with Smolin’s characterization of string theory. In a review of Smolin’s book, string theorist
Joseph Polchinski writes
[Smolin] is mistaking an aspect of the mathematical language being used for one of the physics being
described. New physical theories are often discovered using a mathematical language that is not the most
suitable for them… In string theory it has always been clear that the physics is background-independent
even if the language being used is not, and the search for more suitable language continues. Indeed,
as Smolin belatedly notes, [AdS/CFT] provides a solution to this problem, one that is unexpected and
powerful.[131]
Polchinski notes that an important open problem in quantum gravity is to develop holographic descriptions of gravity
which do not require the gravitational ﬁeld to be asymptotically anti-de Sitter.[131]

39.8.3

Sociological issues

Since the superstring revolutions of the 1980s and 1990s, string theory has become the dominant paradigm of high
energy theoretical physics.[132] Some string theorists have expressed the view that there does not exist an equally
successful alternative theory addressing the deep questions of fundamental physics. In an interview from 1987, Nobel
laureate David Gross made the following controversial comments about the reasons for the popularity of string theory:
The most important [reason] is that there are no other good ideas around. That’s what gets most
people into it. When people started to get interested in string theory they didn't know anything about
it. In fact, the ﬁrst reaction of most people is that the theory is extremely ugly and unpleasant, at least
that was the case a few years ago when the understanding of string theory was much less developed. It
was diﬃcult for people to learn about it and to be turned on. So I think the real reason why people have
got attracted by it is because there is no other game in town. All other approaches of constructing grand
uniﬁed theories, which were more conservative to begin with, and only gradually became more and more
radical, have failed, and this game hasn't failed yet.[133]
Several other high proﬁle theorists and commentators have expressed similar views, suggesting that there are no viable
alternatives to string theory.[134]
Many critics of string theory have commented on this state of aﬀairs. In his book criticizing string theory, Peter
Woit views the status of string theory research as unhealthy and detrimental to the future of fundamental physics.
He argues that the extreme popularity of string theory among theoretical physicists is partly a consequence of the
ﬁnancial structure of academia and the ﬁerce competition for scarce resources.[135] In his book The Road to Reality,
mathematical physicist Roger Penrose expresses similar views, stating “The often frantic competitiveness that this
ease of communication engenders leads to 'bandwagon' eﬀects, where researchers fear to be left behind if they do
not join in.”[136] Penrose also claims that the technical diﬃculty of modern physics forces young scientists to rely
on the preferences of established researchers, rather than forging new paths of their own.[137] Lee Smolin expresses
a slightly diﬀerent position in his critique, claiming that string theory grew out of a tradition of particle physics
which discourages speculation about the foundations of physics, while his preferred approach, loop quantum gravity,
encourages more radical thinking. According to Smolin,
String theory is a powerful, well-motivated idea and deserves much of the work that has been devoted
to it. If it has so far failed, the principal reason is that its intrinsic ﬂaws are closely tied to its strengths—
and, of course, the story is unﬁnished, since string theory may well turn out to be part of the truth. The
real question is not why we have expended so much energy on string theory but why we haven't expended
nearly enough on alternative approaches.[138]
Smolin goes on to oﬀer a number of prescriptions for how scientists might encourage a greater diversity of approaches
to quantum gravity research.[139]

39.9. REFERENCES

177

39.9 References
39.9.1

Notes

[1] For example, physicists are still working to understand the phenomenon of quark conﬁnement, the paradoxes of black
holes, and the origin of dark energy.
[2] For example, in the context of the AdS/CFT correspondence, theorists often formulate and study theories of gravity in
unphysical numbers of spacetime dimensions.
[3] “Top Cited Articles during 2010 in hep-th”. Retrieved 25 July 2013.
[4] More precisely, one cannot apply the methods of perturbative quantum ﬁeld theory.
[5] Two independent mathematical proofs of mirror symmetry were given by Givental 1996, 1998 and Lian, Liu, Yau 1997,
1999, 2000.
[6] More precisely, a nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.
The Jordan–Hölder theorem exhibits ﬁnite simple groups as the building blocks for all ﬁnite groups.

39.9.2

Citations

[1] Becker, Becker, and Schwarz 2007, p. 1
[2] Zwiebach 2009, p. 6
[3] Becker, Becker, and Schwarz 2007, pp. 2–3
[4] Becker, Becker, and Schwarz 2007, pp. 9–12
[5] Becker, Becker, and Schwarz 2007, pp. 14–15
[6] Klebanov and Maldacena 2009
[7] Merali 2011
[8] Sachdev 2013
[9] Becker, Becker, and Schwarz 2007, pp. 3, 15–16
[10] Becker, Becker, and Schwarz 2007, p. 8
[11] Becker, Becker, and Schwarz 13–14
[12] Woit 2006
[13] Zee 2010
[14] Becker, Becker, and Schwarz 2007, p. 2
[15] Becker, Becker, and Schwarz 2007, p. 6
[16] Zwiebach 2009, p. 12
[17] Becker, Becker, and Schwarz 2007, p. 4
[18] Zwiebach 2009, p. 324
[19] Wald 1984, p. 4
[20] Zee 2010, Parts V and VI
[21] Zwiebach 2009, p. 9
[22] Zwiebach 2009, p. 8
[23] Yau and Nadis 2010, Ch. 6
[24] Greene 2000, p. 186

178

[25] Yau and Nadis 2010, p. ix
[26] Randall and Sundrum 1999
[27] Becker, Becker, and Schwarz 2007
[28] Zwiebach 2009, p. 376
[29] Moore 2005, p. 214
[30] Moore 2005, p. 215
[31] Aspinwall et al. 2009
[32] Kontsevich 1995
[33] Kapustin and Witten 2007
[34] Duﬀ 1998
[35] Duﬀ 1998, p. 64
[36] Nahm 1978
[37] Cremmer, Julia, and Scherk 1978
[38] Duﬀ 1998, p. 65
[39] Sen 1994a
[40] Sen 1994b
[41] Hull and Townsend 1995
[42] Duﬀ 1998, p. 67
[43] Bergshoeﬀ, Sezgin, and Townsend 1987
[44] Duﬀ et al. 1987
[45] Duﬀ 1998, p. 66
[46] Witten 1995
[47] Duﬀ 1998, pp. 67–68
[48] Becker, Becker, and Schwarz 2007, p. 296
[49] Hořava and Witten 1996
[50] Duﬀ 1996, sec. 1
[51] Banks et al. 1997
[52] Connes 1994
[53] Connes, Douglas, and Schwarz 1998
[54] Nekrasov and Schwarz 1998
[55] Seiberg and Witten 1999
[56] de Haro et al. 2013, p. 2
[57] Yau and Nadis 2010, p. 187–188
[58] Bekenstein 1973
[59] Hawking 1975
[60] Wald 1984, p. 417
[61] Yau and Nadis 2010, p. 189
[62] Strominger and Vafa 1996

CHAPTER 39. STRING THEORY

39.9. REFERENCES

[63] Yau and Nadis 2010, pp. 190–192
[64] Maldacena, Strominger, and Witten 1997
[65] Ooguri, Strominger, and Vafa 2004
[66] Yau and Nadis 2010, pp. 192–193
[67] Yau and Nadis 2010, pp. 194–195
[68] Strominger 1998
[69] Guica et al. 2009
[70] Castro, Maloney, and Strominger 2010
[71] Maldacena 1998
[72] Gubser, Klebanov, and Polyakov 1998
[73] Witten 1998
[74] Klebanov and Maldacena 2009, p. 28
[75] Maldacena 2005, p. 60
[76] Maldacena 2005, p. 61
[77] Zwiebach 2009, p. 552
[78] Maldacena 2005, pp. 61–62
[79] Susskind 2008
[80] Zwiebach 2009, p. 554
[81] Maldacena 2005, p. 63
[82] Hawking 2005
[83] Zwiebach 2009, p. 559
[84] Kovtun, Son, and Starinets 2001
[85] Merali 2011, p. 303
[86] Luzum and Romatschke 2008
[87] Sachdev 2013, p. 51
[88] Candelas et al. 1985
[89] Yau and Nadis 2010, pp. 147–150
[90] Becker, Becker, and Schwarz 2007, pp. 530–531
[91] Becker, Becker, and Schwarz 2007, p. 531
[92] Becker, Becker, and Schwarz 2007, p. 538
[93] Becker, Becker, and Schwarz 2007, p. 533
[94] Becker, Becker, and Schwarz 2007, pp. 539–543
[95] Deligne et al. 1999, p. 1
[96] Hori et al. 2003, p. xvii
[97] Aspinwall et al. 2009, p. 13
[98] Hori et al. 2003
[99] Yau and Nadis 2010, p. 167
[100] Yau and Nadis 2010, p. 166

179

180

[101] Yau and Nadis 2010, p. 169
[102] Candelas et al. 1991
[103] Yau and Nadis 2010, p. 171
[104] Hori et al. 2003, p. xix
[105] Strominger, Yau, and Zaslow 1996
[106] Dummit and Foote 2004
[107] Dummit and Foote 2004, pp. 102–103
[108] Klarreich 2015
[109] Gannon 2006, p. 2
[110] Gannon 2006, p. 4
[111] Conway and Norton 1979
[112] Gannon 2006, p. 5
[113] Gannon 2006, p. 8
[114] Borcherds 1992
[115] Frenkel, Lepowsky, and Meurman 1988
[116] Gannon 2006, p. 11
[117] Eguchi, Ooguri, and Tachikawa 2010
[118] Cheng, Duncan, and Harvey 2013
[119] Duncan, Griﬃn, and Ono 2015
[120] Witten 2007
[121] Woit 2006, pp. 240–242
[122] Woit 2006, p. 242
[123] Weinberg 1987
[124] Woit 2006, p. 243
[125] Susskind 2005
[126] Woit 2006, pp. 242–243
[127] Woit 2006, p. 240
[128] Woit 2006, p. 249
[129] Smolin 2006, p. 81
[130] Smolin 2006, p. 184
[131] Polchinski 2007
[132] Penrose 2004, p. 1017
[133] Woit 2006, pp. 224–225
[134] Woit 2006, Ch. 16
[135] Woit 2006, p. 239
[136] Penrose 2004, p. 1018
[137] Penrose 2004, pp. 1019–1020
[138] Smolin 2006, p. 349
[139] Smolin 2006, Ch. 20

CHAPTER 39. STRING THEORY

39.9. REFERENCES

39.9.3

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Congress of Mathematicians: 120–139. arXiv:alg-geom/9411018. Bibcode:1994alg.geom.11018K.
• Kovtun, P. K.; Son, Dam T.; Starinets, A. O. (2001). “Viscosity in strongly interacting quantum ﬁeld theories
from black hole physics”. Physical review letters 94 (11): 111601. arXiv:hep-th/0405231. Bibcode:2005PhRvL..94k1601K.
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• Lian, Bong; Liu, Kefeng; Yau, Shing-Tung (1997). “Mirror principle, I”. Asian Journal of Math 1: 729–763.
arXiv:alg-geom/9712011. Bibcode:1997alg.geom.12011L.
• Lian, Bong; Liu, Kefeng; Yau, Shing-Tung (1999a). “Mirror principle, II”. Asian Journal of Math 3: 109–146.
arXiv:math/9905006. Bibcode:1999math......5006L.
• Lian, Bong; Liu, Kefeng; Yau, Shing-Tung (1999b). “Mirror principle, III”. Asian Journal of Math 3: 771–
800. arXiv:math/9912038. Bibcode:1999math.....12038L.

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• Lian, Bong; Liu, Kefeng; Yau, Shing-Tung (2000). “Mirror principle, IV”. Surveys in Diﬀerential Geometry:
475–496. arXiv:math/0007104. Bibcode:2000math......7104L.
• Luzum, Matthew; Romatschke, Paul (2008). “Conformal relativistic viscous hydrodynamics: Applications to
RHIC results at √sNN=200 GeV”. Physical Review C 78 (3). arXiv:0804.4015. doi:10.1103/PhysRevC.78.034915.
• Maldacena, Juan (1998). “The Large N limit of superconformal ﬁeld theories and supergravity”. Advances in
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• Maldacena, Juan (2005). “The Illusion of Gravity” (PDF). Scientiﬁc American 293 (5): 56–63. Bibcode:2005SciAm.293e..56M.
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• Sen, Ashoke (1994a). “Strong-weak coupling duality in four-dimensional string theory”. International Journal
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Comes Next. New York: Houghton Miﬄin Co. ISBN 0-618-55105-0.
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184

CHAPTER 39. STRING THEORY

• Susskind, Leonard (2005). The Cosmic Landscape: String Theory and the Illusion of Intelligent Design. Back
Bay Books. ISBN 978-0316013338.
• Susskind, Leonard (2008). The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for
Quantum Mechanics. Little, Brown and Company. ISBN 978-0-316-01641-4.
• Wald, Robert (1984). General Relativity. University of Chicago Press. ISBN 978-0-226-87033-5.
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2607.
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• Witten, Edward (2007). “Three-dimensional gravity revisited”. arXiv:0706.3359 [hep-th].
• Woit, Peter (2006). Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law.
Basic Books. p. 105. ISBN 0-465-09275-6.
• Yau, Shing-Tung; Nadis, Steve (2010). The Shape of Inner Space: String Theory and the Geometry of the
Universe’s Hidden Dimensions. Basic Books. ISBN 978-0-465-02023-2.
• Zee, Anthony (2010). Quantum Field Theory in a Nutshell (2nd ed.). Princeton University Press. ISBN
978-0-691-14034-6.
• Zwiebach, Barton (2009). A First Course in String Theory. Cambridge University Press. ISBN 978-0-52188032-9.

39.10 Further reading
39.10.1

Popularizations

General
• Greene, Brian (2003). The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate
Theory. New York: W.W. Norton & Company. ISBN 0-393-05858-1.
• Greene, Brian (2004). The Fabric of the Cosmos: Space, Time, and the Texture of Reality. New York: Alfred
A. Knopf. ISBN 0-375-41288-3.
Critical
• Penrose, Roger (2005). The Road to Reality: A Complete Guide to the Laws of the Universe. Knopf. ISBN
0-679-45443-8.
• Smolin, Lee (2006). The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What
Comes Next. New York: Houghton Miﬄin Co. ISBN 0-618-55105-0.
• Woit, Peter (2006). Not Even Wrong: The Failure of String Theory And the Search for Unity in Physical Law.
London: Jonathan Cape &: New York: Basic Books. ISBN 978-0-465-09275-8.

39.10.2

Textbooks

For physicists
• Becker, Katrin; Becker, Melanie; Schwarz, John (2007). String Theory and M-theory: A Modern Introduction.
Cambridge University Press. ISBN 978-0-521-86069-7.

39.11. EXTERNAL LINKS

185

• Green, Michael; Schwarz, John; Witten, Edward (2012). Superstring theory. Vol. 1: Introduction. Cambridge
University Press. ISBN 978-1107029118.
• Green, Michael; Schwarz, John; Witten, Edward (2012). Superstring theory. Vol. 2: Loop amplitudes, anomalies and phenomenology. Cambridge University Press. ISBN 978-1107029132.
• Polchinski, Joseph (1998). String Theory Vol. 1: An Introduction to the Bosonic String. Cambridge University
Press. ISBN 0-521-63303-6.
• Polchinski, Joseph (1998). String Theory Vol. 2: Superstring Theory and Beyond. Cambridge University Press.
ISBN 0-521-63304-4.
• Zwiebach, Barton (2009). A First Course in String Theory. Cambridge University Press. ISBN 978-0-52188032-9.
For mathematicians
• Deligne, Pierre; Etingof, Pavel; Freed, Daniel; Jeﬀery, Lisa; Kazhdan, David; Morgan, John; Morrison, David;
Witten, Edward, eds. (1999). Quantum Fields and Strings: A Course for Mathematicians, Vol. 2. American
Mathematical Society. ISBN 978-0821819883.

39.11 External links
• The Elegant Universe—A three-hour miniseries with Brian Greene by NOVA (original PBS Broadcast Dates:
October 28, 8–10 p.m. and November 4, 8–9 p.m., 2003). Various images, texts, videos and animations
explaining string theory.
• Not Even Wrong—A blog critical of string theory
• The Oﬃcial String Theory Web Site
• Why String Theory—An introduction to string theory.

Chapter 40

Two-dimensional space

y

(2,3)

3
2

(−3,1)

1

(0,0)
−3

−2

−1

1

x
2

3

−1
−2

(−1.5,−2.5)

−3

Bi-dimensional Cartesian coordinate system

This page is about 2-dimensional Euclidean For the general theory of 2D objects, see Surface.
In physics and mathematics, two-dimensional space or bi-dimensional space is a geometric model of the planar
186

40.1. HISTORY

187

projection of the physical universe. The two dimensions are commonly called length and width. Both directions lie
in the same plane.
A sequence of n real numbers can be understood as a location in n-dimensional space. When n = 2, the set of all such
locations is called two-dimensional space or bi-dimensional space, and usually is thought of as a Euclidean space.

40.1 History
Books I through IV and VI of Euclid’s Elements dealt with two-dimensional geometry, developing such notions as
similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of
the angles in a triangle, and the three cases in which triangles are “equal” (have the same area), among many other
topics.
Later, the plane was described in a so-called Cartesian coordinate system, a coordinate system that speciﬁes each point
uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two ﬁxed
perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or
just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can
also be deﬁned as the positions of the perpendicular projections of the point onto the two axes, expressed as signed
distances from the origin.
The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat,
although Fermat also worked in three dimensions, and did not publish the discovery.[1] Both authors used a single
axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of
axes was introduced later, after Descartes’ La Géométrie was translated into Latin in 1649 by Frans van Schooten and
his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes’
work.[2]
Later, the plane was thought of as a ﬁeld, where any two points could be multiplied and, except for 0, divided. This
was known as the complex plane. The complex plane is sometimes called the Argand plane because it is used in
Argand diagrams. These are named after Jean-Robert Argand (1768–1822), although they were ﬁrst described by
Norwegian-Danish land surveyor and mathematician Caspar Wessel (1745–1818).[3] Argand diagrams are frequently
used to plot the positions of the poles and zeroes of a function in the complex plane.

40.2 In geometry
See also: Euclidean geometry

40.2.1

Coordinate systems

Main article: Coordinate system
In mathematics, analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space
by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin. They
are usually labeled x and y. Relative to these axes, the position of any point in two-dimensional space is given by an
ordered pair of real numbers, each number giving the distance of that point from the origin measured along the given
axis, which is equal to the distance of that point from the other axis.
Another widely used coordinate system is the polar coordinate system, which speciﬁes a point in terms of its distance
from the origin and its angle relative to a rightward reference ray.
• Cartesian coordinate system
• Polar coordinate system

188

40.2.2

CHAPTER 40. TWO-DIMENSIONAL SPACE

Polytopes

Main article: Polygon
In two dimensions, there are inﬁnitely many polytopes: the polygons. The ﬁrst few regular ones are shown below:
Convex
The Schläﬂi symbol {p} represents a regular p-gon.
Degenerate (spherical)
The regular henagon {1} and regular digon {2} can be considered degenerate regular polygons. They can exist
nondegenerately in non-Euclidean spaces like on a 2-sphere or a 2-torus.
Non-convex
There exist inﬁnitely many non-convex regular polytopes in two dimensions, whose Schläﬂi symbols consist of rational
numbers {n/m}. They are called star polygons and share the same vertex arrangements of the convex regular polygons.
In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläﬂi symbols
{n/m} for all m such that m < n/2 (strictly speaking {n/m} = {n/(n − m)}) and m and n are coprime.

40.2.3

Circle

Main article: Circle
The hypersphere in 2 dimensions is a circle, sometimes called a 1-sphere (S 1 ) because it is an one-dimensional
manifold. In a Euclidean plane, it has the length 2πr and the area of its interior is

A = πr2
where r is the radius.

40.2.4

Other shapes

Main article: List of two-dimensional geometric shapes
There are an inﬁnitude of other curved shapes in two dimensions, notably including the conic sections: the ellipse,
the parabola, and the hyperbola.

40.3 In linear algebra
Another mathematical way of viewing two-dimensional space is found in linear algebra, where the idea of independence is crucial. The plane has two dimensions because the length of a rectangle is independent of its width. In the
technical language of linear algebra, the plane is two-dimensional because every point in the plane can be described
by a linear combination of two independent vectors.

40.3.1

Dot product, angle, and length

Main article: Dot product

40.3. IN LINEAR ALGEBRA

189

The dot product of two vectors A = [A1 , A2 ] and B = [B1 , B2 ] is deﬁned as:[4]

A · B = A1 B1 + A2 B2
A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points.
The magnitude of a vector A is denoted by ∥A∥ . In this viewpoint, the dot product of two Euclidean vectors A and
B is deﬁned by[5]

A · B = ∥A∥ ∥B∥ cos θ,
where θ is the angle between A and B.
The dot product of a vector A by itself is

A · A = ∥A∥2 ,
which gives

190

CHAPTER 40. TWO-DIMENSIONAL SPACE

√
A · A,

∥A∥ =

the formula for the Euclidean length of the vector.

40.4 In calculus
40.4.1

Gradient

In a rectangular coordinate system, the gradient is given by

∇f =

∂f
∂f
i+
j
∂x
∂y

40.4.2

Line integrals and double integrals

For some scalar ﬁeld f : U ⊆ R2 → R, the line integral along a piecewise smooth curve C ⊂ U is deﬁned as
∫

∫

b

f ds =

f (r(t))|r′ (t)| dt.

a

C

where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b) give the endpoints
of C and a < b .
For a vector ﬁeld F : U ⊆ R2 → R2 , the line integral along a piecewise smooth curve C ⊂ U, in the direction of r, is
deﬁned as
∫

∫

b

F(r) · dr =

F(r(t)) · r′ (t) dt.

a

C

where · is the dot product and r: [a, b] → C is a bijective parametrization of the curve C such that r(a) and r(b) give
the endpoints of C.
A double integral refers to an integral within a region D in R2 of a function f (x, y), and is usually written as:
∫∫
f (x, y) dx dy.
D

40.4.3

Fundamental theorem of line integrals

Main article: Fundamental theorem of line integrals
The fundamental theorem of line integrals, says that a line integral through a gradient ﬁeld can be evaluated by
evaluating the original scalar ﬁeld at the endpoints of the curve.
Let φ : U ⊆ R2 → R . Then
∫
φ (q) − φ (p) =

∇φ(r) · dr.
γ[p, q]

40.5. IN TOPOLOGY

40.4.4

191

Green’s theorem

Main article: Green’s theorem
Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by
C. If L and M are functions of (x, y) deﬁned on an open region containing D and have continuous partial derivatives
there, then[6][7]
∫∫ (

I
(L dx + M dy) =
C

D

∂M
∂L
−
∂x
∂y

)
dx dy

where the path of integration along C is counterclockwise.

40.5 In topology
In topology, the plane is characterized as being the unique contractible 2-manifold.
Its dimension is characterized by the fact that removing a point from the plane leaves a space that is connected, but
not simply connected.

40.6 In graph theory
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in
such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges
cross each other.[8] Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can
be deﬁned as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane
curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all
curves are disjoint except on their extreme points.

40.7 References
[1] “Analytic geometry”. Encyclopædia Britannica (Encyclopædia Britannica Online ed.). 2008.
[2] Burton 2011, p. 374
[3] Wessel’s memoir was presented to the Danish Academy in 1797; Argand’s paper was published in 1806. (Whittaker &
Watson, 1927, p. 9)
[4] S. Lipschutz, M. Lipson (2009). Linear Algebra (Schaum’s Outlines) (4th ed.). McGraw Hill. ISBN 978-0-07-154352-1.
[5] M.R. Spiegel, S. Lipschutz, D. Spellman (2009). Vector Analysis (Schaum’s Outlines) (2nd ed.). McGraw Hill. ISBN
978-0-07-161545-7.
[6] Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press,
2010, ISBN 978-0-521-86153-3
[7] Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipschutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009,
ISBN 978-0-07-161545-7
[8] Trudeau, Richard J. (1993). Introduction to Graph Theory (Corrected, enlarged republication. ed.). New York: Dover
Pub. p. 64. ISBN 978-0-486-67870-2. Retrieved 8 August 2012. Thus a planar graph, when drawn on a ﬂat surface,
either has no edge-crossings or can be redrawn without them.

192

40.8 See also
• Three-dimensional space
• Two-dimensional graph

CHAPTER 40. TWO-DIMENSIONAL SPACE

Chapter 41

VC dimension
In statistical learning theory, or sometimes computational learning theory, the VC dimension (for Vapnik–Chervonenkis
dimension) is a measure of the capacity (complexity, expressive power, richness, or ﬂexibility) of a statistical classiﬁcation algorithm, deﬁned as the cardinality of the largest set of points that the algorithm can shatter. It is a core
concept in Vapnik–Chervonenkis theory, and was originally deﬁned by Vladimir Vapnik and Alexey Chervonenkis.
Informally, the capacity of a classiﬁcation model is related to how complicated it can be. For example, consider the
thresholding of a high-degree polynomial: if the polynomial evaluates above zero, that point is classiﬁed as positive,
otherwise as negative. A high-degree polynomial can be wiggly, so it can ﬁt a given set of training points well. But
one can expect that the classiﬁer will make errors on other points, because it is too wiggly. Such a polynomial has a
high capacity. A much simpler alternative is to threshold a linear function. This function may not ﬁt the training set
well, because it has a low capacity. This notion of capacity is made rigorous below.

41.1 Shattering
A classiﬁcation model f with some parameter vector θ is said to shatter a set of data points (x1 , x2 , . . . , xn ) if, for
all assignments of labels to those points, there exists a θ such that the model f makes no errors when evaluating that
set of data points.
The VC dimension of a model f is the maximum number of points that can be arranged so that f shatters them.
More formally, it is h′ where h′ is the maximum h such that some data point set of cardinality h can be shattered by
f.
For example, consider a straight line as the classiﬁcation model: the model used by a perceptron. The line should
separate positive data points from negative data points. There exist sets of 3 points that can indeed be shattered
using this model (any 3 points that are not collinear can be shattered). However, no set of 4 points can be shattered:
by Radon’s theorem, any four points can be partitioned into two subsets with intersecting convex hulls, so it is not
possible to separate one of these two subsets from the other. Thus, the VC dimension of this particular classiﬁer
is 3. It is important to remember that while one can choose any arrangement of points, the arrangement of those
points cannot change when attempting to shatter for some label assignment. Note, only 3 of the 23 = 8 possible label
assignments are shown for the three points.

41.2 Uses
The VC dimension has utility in statistical learning theory, because it can predict a probabilistic upper bound on the
test error of a classiﬁcation model.
Vapnik [1] proved that the probability of the test error distancing from an upper bound (on data that is drawn i.i.d.
from the same distribution as the training set) is given by
)
(
√
=1−η
P error test ≤ error training + h(log(2N /h)+1)−log(η/4)
N
where h is the VC dimension of the classiﬁcation model, 0 ≤ η ≤ 1 , and N is the size of the training set (restriction:
193

194

CHAPTER 41. VC DIMENSION

this formula is valid when h ≪ N ). Similar complexity bounds can be derived using Rademacher complexity, but
Rademacher complexity can sometimes provide more insight than VC dimension calculations into such statistical
methods such as those using kernels. The generalization of the VC dimension for multi-valued functions is the
Natarajan dimension.
In computational geometry, VC dimension is one of the critical parameters in the size of ε-nets, which determines
the complexity of approximation algorithms based on them; range sets without ﬁnite VC dimension may not have
ﬁnite ε-nets at all.

41.3 See also
• Sauer–Shelah lemma, a bound on the number of sets in a set system in terms of the VC dimension
• Karpinski-Macintyre theorem, a bound on the VC dimension of general Pfaﬃan formulas

41.4 References
[1] Vapnik, Vladimir. The nature of statistical learning theory. springer, 2000.

• Andrew Moore’s VC dimension tutorial
• Vapnik, Vladimir. “The nature of statistical learning theory”. springer, 2000.
• V. Vapnik and A. Chervonenkis. “On the uniform convergence of relative frequencies of events to their probabilities.” Theory of Probability and its Applications, 16(2):264–280, 1971.
• A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth. “Learnability and the Vapnik–Chervonenkis
dimension.” Journal of the ACM, 36(4):929–865, 1989.
• Christopher Burges Tutorial on SVMs for Pattern Recognition (containing information also for VC dimension)
• Bernard Chazelle. “The Discrepancy Method.”
• B.K. Natarajan. “On Learning sets and functions.” Machine Learning, 4, 67-97, 1989.

Chapter 42

Zero-dimensional space
This article is about zero dimension in topology. For several kinds of zero space in algebra, see zero object (algebra).
In mathematics, a zero-dimensional topological space (or nildimensional) is a topological space that has dimension
zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space.[1][2] An
illustration of a nildimensional space is a point.[3]

42.1 Deﬁnition
Speciﬁcally:
• A topological space is zero-dimensional with respect to the Lebesgue covering dimension if every ﬁnite open
cover of the space has a ﬁnite reﬁnement which is a cover of the space by open sets such that any point in the
space is contained in exactly one open set of this reﬁnement.
• A topological space is zero-dimensional with respect to the small inductive dimension if it has a base consisting
of clopen sets.
The two notions above agree for separable, metrisable spaces.

42.2 Properties of spaces with covering dimension zero
• A zero-dimensional Hausdorﬀ space is necessarily totally disconnected, but the converse fails. However, a
locally compact Hausdorﬀ space is zero-dimensional if and only if it is totally disconnected. (See (Arhangel’skii
2008, Proposition 3.1.7, p.136) for the non-trivial direction.)
• Zero-dimensional Polish spaces are a particularly convenient setting for descriptive set theory. Examples of
such spaces include the Cantor space and Baire space.
• Hausdorﬀ zero-dimensional spaces are precisely the subspaces of topological powers 2I where 2 = {0, 1} is
given the discrete topology. Such a space is sometimes called a Cantor cube. If I is countably inﬁnite, 2I is
the Cantor space.

42.3 Notes
• Arhangel’skii, Alexander; Tkachenko, Mikhail (2008), Topological Groups and Related Structures, Atlantis
Studies in Mathematics, Vol. 1, Atlantis Press, ISBN 90-78677-06-6
• Engelking, Ryszard (1977). General Topology. PWN, Warsaw.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.
195

196

CHAPTER 42. ZERO-DIMENSIONAL SPACE

42.4 References
[1] “zero dimensional”. planetmath.org. Retrieved 2015-06-06.
[2] Hazewinkel, Michiel (1989). Encyclopaedia of Mathematics, Volume 3. Kluwer Academic Publishers. p. 190.
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42.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

197

42.5 Text and image sources, contributors, and licenses
42.5.1

Text

• 2 1/2D Source: https://en.wikipedia.org/wiki/2_1/2D?oldid=535427127 Contributors: Zundark, Michael Hardy, Malcolma, EoGuy,
Yobot, Hajatvrc and Microextruders
• Abscissa Source: https://en.wikipedia.org/wiki/Abscissa?oldid=687619746 Contributors: Zundark, Andre Engels, Matusz, Ellmist, Grape~enwiki,
Stevenj, Schneelocke, Emperorbma, Dysprosia, Hyacinth, Topbanana, Robbot, Hadal, Dufekin, Bender235, BillCook, Algocu, Oleg
Alexandrov, Woohookitty, Linas, TAKASUGI Shinji, Nihiltres, Maxal, Thecurran, EamonnPKeane, Bhny, Stephenb, Eb Oesch, SmackBot, Octahedron80, Tomhubbard, Wizard191, Tophtucker, Rt3368, Deﬂective, David Eppstein, R'n'B, Belovedfreak, VolkovBot, Someguy1221,
Tomaxer, Flyer22 Reborn, Sean.hoyland, Siskus, Käptn Weltall, Avoided, Addbot, CarsracBot, Theraven502, Erutuon, OlEnglish, KamikazeBot, Reindra, Fighter 10, Twri, Xqbot, Amaury, Smallman12q, FrescoBot, RedBot, Double sharp, Autoreplay, Clarice Reis, EmausBot,
Nappolita, Minimac’s Clone, Solomonfromﬁnland, Stringybark, Sjjamsa, Aupif, ClueBot NG, TheAwesomeMathGenie, Mdb23b, ReconditeRodent, Quenhitran, Bryanrutherford0, Imafagetty, Malabsky and Anonymous: 44
• Cartesian coordinate system Source: https://en.wikipedia.org/wiki/Cartesian_coordinate_system?oldid=688896779 Contributors: Damian
Yerrick, Chuck Smith, Bryan Derksen, Zundark, Tarquin, Mark Ryan, Andre Engels, Heron, Montrealais, Patrick, Michael Hardy, Dcljr,
JWSchmidt, Александър, AugPi, Skyfaller, Smack, Pizza Puzzle, Nikola Smolenski, Drz~enwiki, Emperorbma, Charles Matthews,
Timwi, Dysprosia, Colipon, The Anomebot, Maximus Rex, Bevo, Chuunen Baka, Robbot, Romanm, Modulatum, Sverdrup, Henrygb,
Prara, Michael Snow, Jleedev, Enochlau, Snobot, Giftlite, DocWatson42, Jason Quinn, Jorge Stolﬁ, Manuel Anastácio, SoWhy, MrMambo, Joseph Myers, C4~enwiki, Adashiel, Discospinster, Rich Farmbrough, Guanabot, Paul August, Bender235, Elwikipedista~enwiki,
BenjBot, Edwinstearns, Rgdboer, Mavhc, Che090572, Kjkolb, Larry V, HasharBot~enwiki, OGoncho, Jumbuck, Alansohn, SnowFire, PAR, Yossiea~enwiki, Fordan, Dionoea, Kbolino, Falcorian, Oleg Alexandrov, Mel Etitis, Woohookitty, Linas, LOL, Fireﬁshy,
Mpatel, Waldir, Ruziklan, Palica, Tony1849, Graham87, Jobnikon, BD2412, Rjwilmsi, MJSkia1, Ygrek, OneWeirdDude, MarSch,
Jmcc150, Salix alba, Brighterorange, Titoxd, FlaBot, Mathbot, Nihiltres, Maxal, Cmbrannon, Slant, Fresheneesz, Srleﬄer, Chobot,
Helios, DVdm, YurikBot, Wavelength, RobotE, Charles Gaudette, Pip2andahalf, RussBot, Hede2000, Piet Delport, Gustavb, Byj2000,
RabidDeity, Dhollm, Bucketsofg, Dbﬁrs, Cheeser1, DeadEyeArrow, Dast, MathsIsFun, HereToHelp, Tony Liao~enwiki, JLaTondre,
Gesslein, Archer7, Allens, Katieh5584, Meegs, Cmglee, Nekura, DVD R W, Sardanaphalus, SmackBot, RDBury, Adam majewski,
Incnis Mrsi, Delldot, Canthusus, ParlorGames, Gilliam, Skizzik, Mirokado, Kurykh, Miquonranger03, Silly rabbit, Basalisk, Octahedron80, Nbarth, DHN-bot~enwiki, Hongooi, Antonrojo, Can't sleep, clown will eat me, RedHillian, SundarBot, Cameron Nedland,
Memming, Downtown dan seattle, Doodle77, Andeggs, Sadi Carnot, Andrei Stroe, Lambiam, Dbtfz, Kusarbo, Cronholm144, Bjankuloski06en~enwiki, Aleenf1, IronGargoyle, 041744, JHunterJ, Stwalkerster, Mets501, Wizard191, Maelor, JForget, CmdrObot, Jackzhp,
Dgw, NickW557, 345Kai, WeggeBot, Yaris678, WillowW, LouisBB, ST47, Edgerck, Juansempere, DumbBOT, TJ09, Vanished User jdksfajlasd, Zalgo, Lanepierce, Thijs!bot, Epbr123, Headbomb, Federhalter, Escarbot, Cyclonenim, AntiVandalBot, Gioto, Mhaitham.shammaa,
Modernist, JAnDbot, Chausx, PhilKnight, DeclinedShadow, Magioladitis, Connormah, VoABot II, Catslash, Mrfence, CattleGirl, Fabrictramp, Srice13, Maniwar, David Eppstein, DerHexer, MartinBot, Rettetast, Snozzer, J.delanoy, Captain panda, Sasajid, Trusilver,
Numbo3, Century0, Cpiral, Lantonov, Stan J Klimas, Samtheboy, NewEnglandYankee, Davecrosby uk, CardinalDan, RJASE1, Idiomabot, KillerOfThem, VolkovBot, JohnBlackburne, Mun206, TXiKiBoT, Anonymous Dissident, Steven J. Anderson, Martin451, Uncaringgunner, Domitius, Topherjasmin09, Andy Dingley, Dirkbb, Blindman.rms, AlleborgoBot, Logan, Katzmik, SieBot, Euryalus, Yintan,
Joaosampaio, Flyer22 Reborn, Man It’s So Loud In Here, Paolo.dL, Oxymoron83, Atmamatma, Jurlinga, Hello71, Hobartimus, Svick,
Anchor Link Bot, Randomblue, Escape Orbit, Martarius, ClueBot, The Thing That Should Not Be, Mild Bill Hiccup, DragonBot, Excirial, Abrech, SockPuppetForTomruen, Thingg, Mattreedywiki, Johnuniq, Darkicebot, CaptainVideo890, Skunkboy74, Vanostran, Hyperweb79, Addbot, Binary TSO, DougsTech, Fgnievinski, Blethering Scot, Jncraton, Fieldday-sunday, Leszek Jańczuk, MrOllie, Allliam,
TheFreeloader, Tide rolls, Alanfeynman, Lightbot, Nobono9, Zorrobot, Վազգեն, Wmplayer, Wwannsda, Luckas-bot, Yobot, Fraggle81,
Anypodetos, Nallimbot, Vltava 68, Tempodivalse, AnomieBOT, 1exec1, Rajmathi mehta, Piano non troppo, Ulric1313, Materialscientist, Citation bot, Oftopladb, Xqbot, Waﬄeman12, Jeﬀwang, NorbDigiBeaver, Almabot, Frosted14, Red van man, A.amitkumar, FrescoBot, Appropo, Majopius, Masterknighted, Rhino bucket, Wireless Keyboard, Þjóðólfr, Pmokeefe, Jsjunkie, Rogiemac, Shanmugamp7,
Rausch, ‫کاشف عقیل‬, Lotje, Colin Cochrane, Dasteve, Suﬀusion of Yellow, Shanker Pur, NameIsRon, Timh3221, Slon02, EmausBot,
Acather96, WikitanvirBot, GoingBatty, RA0808, Wikipelli, Slawekb, CanonLawJunkie, Knight1993, Junelvillejo, Quondum, MonoAV,
Chewings72, WMC, ClueBot NG, Gareth Griﬃth-Jones, KlappCK, Wcherowi, MelbourneStar, Satellizer, Movses-bot, Widr, Helpful
Pixie Bot, DBigXray, Popsh, Papadim.G, Questioneﬁsica, Mark Arsten, Blue Mist 1, Phl.jns, Nbrothers, Pbierre, Pratyya Ghosh, SergeantHippyZombie, Radio15dude, ChrisGualtieri, Bigloser12345loser, EuroCarGT, Ramesepirate, Ducknish, Kelvinsong, Kingbowen, Webclient101, Indiana State, RazrRekr201, Jc86035, Acetotyce, ProtossPylon, Liekturtles, SamX, Wamiq, Sicaeﬀect, Ginsuloft, Primalshell,
UY Scuti, Stamptrader, Andreatristan, Akhilburle, Wilson Widyadhana, BlueFenixReborn, Roshmita, Elsa1098, Hdkeudhdjisjedu and
Anonymous: 477
• Codimension Source: https://en.wikipedia.org/wiki/Codimension?oldid=657214672 Contributors: Zundark, Patrick, Charles Matthews,
MathMartin, Tosha, Giftlite, Fropuﬀ, HasharBot~enwiki, Joriki, Grammarbot, FlaBot, Mathbot, Wavelength, Crasshopper, KasugaHuang, User24, SmackBot, RDBury, David Farris, TimBentley, Nbarth, Valoem, Andri Egilsson, Nick Number, EagleFan, Akulo,
R'n'B, LokiClock, Anonymous Dissident, Philmac, Mild Bill Hiccup, Addbot, Luckas-bot, Yobot, Zandr4, TechBot, Erik9bot, Nunc aut
numquam, HUnTeR4subs, Brad7777 and Anonymous: 9
• Complex dimension Source: https://en.wikipedia.org/wiki/Complex_dimension?oldid=614212764 Contributors: Michael Hardy, Charles
Matthews, SmackBot, CBM, David Eppstein, VolkovBot, CohesionBot, Addbot, Erik9bot and D.Lazard
• Complex network zeta function Source: https://en.wikipedia.org/wiki/Complex_network_zeta_function?oldid=563206803 Contributors: Michael Hardy, Charles Matthews, Andreas Kaufmann, Rjwilmsi, Crystallina, Coppertwig, Oshanker, DOI bot, Yobot, Citation bot
1, Spectral sequence and Anonymous: 1
• Concentration dimension Source: https://en.wikipedia.org/wiki/Concentration_dimension?oldid=656502748 Contributors: Michael
Hardy, Sullivan.t.j, David Eppstein, Melcombe, Yobot and Helpful Pixie Bot
• Degrees of freedom Source: https://en.wikipedia.org/wiki/Degrees_of_freedom?oldid=685055239 Contributors: The Anome, Peterlin~enwiki, Michael Hardy, 168..., Looxix~enwiki, EasilyAmused, CatherineMunro, Jﬁtzg, Charles Matthews, Fibonacci, Rogper~enwiki,
Bearcat, Robbot, Seglea, Wikibot, MathKnight, Amp, Zhen Lin, RyanKnoll, Icairns, Warﬁeldian, Brianjd, Mormegil, CALR, BrokenSegue, Duk, John Fader, Vizcarra, MIT Trekkie, Oleg Alexandrov, FlaBot, Margosbot~enwiki, Chobot, YurikBot, Russell C. Sibley,

198

CHAPTER 42. ZERO-DIMENSIONAL SPACE

Alex Bakharev, Rhythm, KocjoBot~enwiki, Dingar, Pissant, Pardy, Thijs!bot, Grimlock, David Eppstein, Jiuguang Wang, STBotD,
VolkovBot, OKBot, Addbot, Fgnievinski, Zorrobot, Yobot, Amirobot, Nallimbot, Xqbot, Fortdj33, Pinethicket, TobeBot, EmausBot,
Hhhippo, ZéroBot, D.Lazard, Qurrat ul an, ClueBot NG, Rezabot, Illia Connell, Mark viking, Hamoudafg, JeremiahY and Anonymous:
28
• Degrees of freedom (physics and chemistry) Source: https://en.wikipedia.org/wiki/Degrees_of_freedom_(physics_and_chemistry)
?oldid=688859292 Contributors: Kku, Charles Matthews, Gandalf61, Giftlite, Tromer, Finn-Zoltan, Karol Langner, Icairns, Dmr2,
John Vandenberg, Vizcarra, Monk127, Jheald, Count Iblis, Oleg Alexandrov, Pol098, Adiel, TeaDrinker, Siddhant, Conscious, Grubber, Archelon, NawlinWiki, Josteinaj, Ignitus, Tom Morris, SmackBot, Rex the ﬁrst, Incnis Mrsi, Dingar, ThorinMuglindir, Nbarth,
Archimerged, Cydebot, Boardhead, Thijs!bot, Lagesag~enwiki, Kilva, Al Lemos, AntiVandalBot, Jj137, Gökhan, JAnDbot, Grimlock,
Mythealias, R'n'B, Adavidb, VolkovBot, TXiKiBoT, Marc Vuﬀray, Antoni Barau, Kbrose, SieBot, Damorbel, Bernard Marx, Flyer22
Reborn, Oxymoron83, Taggard, Myrikhan, Dabomb87, Sfan00 IMG, Vql, The Founders Intent, XLinkBot, Leia, Addbot, Meisam,
Luckas-bot, Nallimbot, Sander roy, AnomieBOT, Rubinbot, Xqbot, Nfr-Maat, J04n, GrouchoBot, Nickmn, RibotBOT, Mnmngb, Thehelpfulbot, FrescoBot, Victor of Gaugamela, Yitping, Adlerbot, Jordgette, Heurisko, EmausBot, WikitanvirBot, Hhhippo, Johnnycracka,
Quondum, Hpubliclibrary, Maschen, ClueBot NG, BG19bot, Gnosygnu, 1994bhaskar, Supra256, Klilidiplomus, Aisteco, Hurricaneerica,
Haider ali siddiqui, JeAlLi and Anonymous: 64
• Dimension Source: https://en.wikipedia.org/wiki/Dimension?oldid=682028999 Contributors: AxelBoldt, BF, Bryan Derksen, Zundark,
The Anome, Josh Grosse, Vignaux, XJaM, Stevertigo, Frecklefoot, Patrick, Boud, Michael Hardy, Isomorphic, Menchi, Ixfd64, Kalki,
Delirium, Looxix~enwiki, William M. Connolley, Angela, Julesd, Glenn, AugPi, Poor Yorick, Jiang, Raven in Orbit, Vargenau, Pizza
Puzzle, Schneelocke, Charles Matthews, Dysprosia, Furrykef, Jgm, Omegatron, Kizor, Lowellian, Gandalf61, Blainster, Bkell, Tobias
Bergemann, Centrx, Giftlite, Beolach, Lupin, Unconcerned, Falcon Kirtaran, Alvestrand, Utcursch, CryptoDerk, Antandrus, Joseph Myers, Lesgles, Jokestress, Mike Storm, Tomruen, Bodnotbod, Marcos, Mare-Silverus, Brianjd, D6, Discospinster, Guanabot, Gadykozma,
Pjacobi, Mani1, Paul August, Corvun, Andrejj, Ground, RJHall, Mr. Billion, Lycurgus, Rgdboer, Zenohockey, Bobo192, W8TVI,
Rbj, Dungodung, Shlomital, Hesperian, Friviere, Quaoar, Msh210, Mo0, Arthena, Seans Potato Business, InShaneee, Malo, Metron4,
Wtmitchell, Velella, TheRealFennShysa, Culix, Lerdsuwa, Ringbang, Oleg Alexandrov, Bobrayner, Boothy443, Linas, ScottDavis, Arcann, Peter Hitchmough, OdedSchramm, Waldir, Christopher Thomas, Mandarax, Graham87, Marskell, Dpv, Rjwilmsi, Jmcc150, THE
KING, RobertG, Mathbot, Margosbot~enwiki, Ewlyahoocom, Gurch, Karelj, Saswann, Chobot, Sharkface217, Volunteer Marek, Bgwhite, The Rambling Man, Wavelength, Reverendgraham, Hairy Dude, Rt66lt, Jimp, RussBot, Michael Slone, Rsrikanth05, Bisqwit, CyclopsX, Rhythm, NickBush24, BaldMonkey, TVilkesalo~enwiki, Welsh, Trovatore, Eighty~enwiki, Dureo, Vb, Number 57, Elizabeyth,
Eurosong, Arthur Rubin, JoanneB, Vicarious, Geoﬀrey.landis, JLaTondre, ThunderBird, Profero, GrinBot~enwiki, Segv11, DVD R
W, Sardanaphalus, SmackBot, RDBury, Haza-w, Lestrade, VigilancePrime, Cavenba, Lord Snoeckx, Rojomoke, Frymaster, Spireguy,
Gilliam, Brianski, Rpmorrow, Chris the speller, Ayavaron, Kurykh, TimBentley, Persian Poet Gal, Nbarth, Christobal54, William Allen
Simpson, Foxjwill, Swilk, Tamfang, Moonsword, HoodedMan, Gurps npc, Yidisheryid, Kittybrewster, Addshore, Amazins490, Stevenmitchell, COMPFUNK2, PiMaster3, PiPhD, Dreadstar, Richard001, RandomP, Danielkwalsh, JakGd1, Snakeyes (usurped), Andeggs,
SashatoBot, Lambiam, Richard L. Peterson, MagnaMopus, SteveG23, Footballrocks41237, Pthag, Thomas Howey, Jim.belk, Loadmaster,
Waggers, Mets501, Michael Greiner, Daviddaniel37, Autonova, Fredil Yupigo, Iridescent, Kiwi8, Mulder416sBot, Courcelles, Rccarman,
Tanthalas39, KyraVixen, Randalllin, 345Kai, Requestion, MarsRover, WeggeBot, Myasuda, FilipeS, Equendil, Vectro, Cydebot, Hanju,
Tawkerbot4, Xantharius, Dragonﬂare82, Arb, JamesAM, Thijs!bot, Epbr123, Mbell, Mojo Hand, SomeStranger, The Hybrid, Nick
Number, Heroeswithmetaphors, Escarbot, Stannered, AntiVandalBot, Mrbip, Salgueiro~enwiki, Kent Witham, The man stephen, The
Transhumanist, Matthew Fennell, Vanished user s4irtj34tivkj12erhskj46thgdg, Beaumont, Cynwolfe, Acroterion, Bongwarrior, VoABot
II, Alvatros~enwiki, Rafuki 33, Wikidudeman, JamesBWatson, TheChard, Stijn Vermeeren, Indon, Illspirit, David Eppstein, Fang 23,
Spellmaster, JoergenB, DerHexer, Patstuart, DukeTwicep, Greenguy1090, Torrr, Gasheadsteve, Leyo, J.delanoy, Captain panda, Enchepilon, Elizabethrhodes, EscapingLife, Inimino, Maurice Carbonaro, DR TORMEY, Good-afternun!, It Is Me Here, Thomﬁlm, TomasBat,
Han Solar de Harmonics, SemblaceII, CardinalDan, Hamzabahaa, VolkovBot, Seldon1, JohnBlackburne, Orthologist, LokiClock, Jacroe,
Philip Trueman, Mdmkolbe, TXiKiBoT, Gaara144, Vipinhari, Technopat, Aodessey, Anonymous Dissident, Immorality, Berchin, Ferengi, Jackfork, LariAnn, Gbaor, Andyo2000, Jmath666, Wenli, HX Aeternus, Wolfrock, Lamro, Enviroboy, Mike4ty4, Symane, Rybu,
BriEnBest, YonaBot, Racer x124, Pelyukhno.erik, Oysterguitarist, Harry~enwiki, SuperLightningKick, Techman224, OKBot, Anchor
Link Bot, JL-Bot, Mr. Granger, Martarius, ClueBot, WurmWoode, The Thing That Should Not Be, Gamehero, Roal AT, DragonBot,
Kitsunegami, Excirial, D1a4l2s3y5, Grb 1991, Cute lolicon, Puceron, Jennonpress, SchreiberBike, ChrisHodgesUK, Qwfp, Alousybum,
TimothyRias, Rror, Egyptianboy15223, NellieBly, Me, Myself, and I, Addbot, Leszek Jańczuk, NjardarBot, Joonojob, Sillyfolkboy,
NerdBoy1392, Ranjitvohra, Renatokeshet, Numbo3-bot, Tide rolls, Lightbot, Yobot, Tester999, Tohd8BohaithuGh1, Fraggle81, Sarrus,
Zenquin, AnomieBOT, Andrewrp, AdjustShift, Materialscientist, Hunnjazal, 90 Auto, Citation bot, Maxis ftw, Nagoshi1, Staysfresh,
Danny955, MeerkatNerd1, Anna Frodesiak, Mlpearc, AbigailAbernathy, Srich32977, Charvest, Natural Cut, Sonoluminesence, Currydump, FrescoBot, Васин Юрий, PhysicsExplorer, Sae1962, Rtycoon, Drew R. Smith, RandomDSdevel, Pinethicket, I dream of
horses, Edderso, Jonesey95, MastiBot, Σ, December21st2012Freak, AGiorgio08, SkyMachine, Double sharp, Sheogorath, Lotje, Chrisjameshull, 4, Diannaa, Jesse V., Xnn, Distortiondude, Mean as custard, Alison22, Pokdhjdj, J36miles, Envirodan, Ovizelu, RA0808,
Scleria, Slightsmile, Wikipelli, Hhhippo, Traxs7, Medeis, StudyLakshan, Sarapaxton, D.Lazard, SpikeTD, Markshutter, Ewa5050, JaySebastos, L Kensington, Bomazi, BioPupil, RockMagnetist, 28bot, Isocliﬀ, Khestwol, ClueBot NG, Wcherowi, Thekk2007, Lanthanum138, O.Koslowski, MerlIwBot, Bibcode Bot, Love’s Labour Lost, Snaevar-bot, Nospildoh, Bereziny, Jxuan, Rahul.quara, Naeem21,
Ownedroad9, Balance of paradox, Brat162, ChrisGualtieri, Kelvinsong, Uevboweburvkuwbekl, RobertAnderson1432, Hillbillyholiday,
Theeverst87, Titz69, Awesome2013, Wamiq, Penitence, Dez Moines, Dodi 8238, I3roly, Anonymous-232, Brandon Ernst, Kylejaylee,
Chuluojun, Prachi.apomr, KH-1, Loraof, BakedLikaBiscuit, Inkanyamba, Knight victor, Srednuas Lenoroc, TopDym and Anonymous:
485
• Dimension (vector space) Source: https://en.wikipedia.org/wiki/Dimension_(vector_space)?oldid=685831772 Contributors: AxelBoldt,
Zundark, Tarquin, Michael Hardy, Wshun, TakuyaMurata, AugPi, Charles Matthews, Gandalf61, MathMartin, Giftlite, Lethe, Bob.v.R,
Krakhan, Army1987, Obradovic Goran, Jumbuck, Oleg Alexandrov, Mathbot, YurikBot, Tetracube, JahJah, Banus, SmackBot, Chris the
speller, Bluebot, Oli Filth, Pierrecurie, Nbarth, Rludlow, Dreadstar, Tilin, SashatoBot, Jim.belk, Jbolden1517, Cydebot, Kanags, Epbr123,
Faigl.ladislav, JAnDbot, Quentar~enwiki, David Eppstein, Geometry guy, Jmath666, Neparis, Rybu, Ivan Štambuk, Niceguyedc, Marc
van Leeuwen, MystBot, Addbot, Kein Einstein, Legobot, Luckas-bot, Yobot, Star Flyer, Xqbot, Flavio Guitian, Edderso, EmausBot,
Quondum, Zynwyx, Mesoderm, Helpful Pixie Bot, AvocatoBot, Grv87, Impsswoon and Anonymous: 23
• Dimension of an algebraic variety Source: https://en.wikipedia.org/wiki/Dimension_of_an_algebraic_variety?oldid=620029417 Contributors: Zundark, TakuyaMurata, Charles Matthews, Giftlite, D6, SmackBot, RobHar, Dthomsen8, Addbot, AnomieBOT, D.Lazard,
ClueBot NG and Anonymous: 8

42.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

199

• Dimension theory (algebra) Source: https://en.wikipedia.org/wiki/Dimension_theory_(algebra)?oldid=673460478 Contributors: TakuyaMurata, Sligocki, CBM, Headbomb, RobHar, Addbot, AkhtaBot, AnomieBOT, John of Reading and Anonymous: 2
• Dimensional metrology Source: https://en.wikipedia.org/wiki/Dimensional_metrology?oldid=628135739 Contributors: Michael Hardy,
ALoopingIcon, Mikeblas, SmackBot, JMiall, Whpq, Mattisse, MarshBot, Ron.swonger, Flyer22 Reborn, Trojancowboy, Tarheel95,
MystBot, Addbot, Hackipedia, Ibraheem alex, EmausBot, John of Reading, Roberticus, RADHA HARIHARAN, Brad7777, Sabrinaspringer, ChrisGualtieri and Anonymous: 13
• Eight-dimensional space Source: https://en.wikipedia.org/wiki/Eight-dimensional_space?oldid=653940584 Contributors: Mporter, Edcolins, Tomruen, Rgdboer, Woohookitty, Arthur Rubin, SmackBot, Colonies Chris, Sammy1339, CmdrObot, ShelfSkewed, JaGa, JohnBlackburne, Raymondwinn, JL-Bot, Martarius, Niceguyedc, Muhandes, SockPuppetForTomruen, AnomieBOT, Materialscientist, Xqbot,
MGA73bot, Double sharp, 4, BertSeghers, EmausBot, John of Reading, Traxs7, Alborzagros, Crown Prince, ClueBot NG, Brat162, Duplij, ChrisGualtieri, Mogism, Mark viking and Anonymous: 5
• Exterior dimension Source: https://en.wikipedia.org/wiki/Exterior_dimension?oldid=609910971 Contributors: Michael Hardy, Kku,
Rjwilmsi, David Eppstein, Yobot, Materialscientist, Wayne Slam, ClueBot NG, Jonny Calvete, Brad7777 and Anonymous: 4
• Five-dimensional space Source: https://en.wikipedia.org/wiki/Five-dimensional_space?oldid=687238670 Contributors: Zundark, Charles
Matthews, Jeﬀq, Adam78, Mporter, Gene Ward Smith, Tomruen, Agonist, Mike Rosoft, Kehamrick, Bobo192, Army1987, Keenan Pepper, BDD, Sandover, Linas, WadeSimMiser, Mpatel, Ashmoo, Kbdank71, Hiberniantears, Nneonneo, CannotResolveSymbol, Mathbot,
Dorsk188, Neitherday, Rt66lt, Geologician, Hellbus, Trovatore, Brian Crawford, Tony1, Tetracube, Closedmouth, Arthur Rubin, Hurricanehink, Alasdair, JLaTondre, Katieh5584, Trickstar, Blake Dayton, SmackBot, AKismet, Mr. Mediocre, Folajimi, Tamfang, Chlewbot,
OrphanBot, Decltype, Monotonehell, Treyt021, Cheesy Yeast, Starkiller, Igoldste, Nadyes, ShelfSkewed, INVERTED, Cydebot, Cahk,
Reywas92, A Softer Answer, Pascal.Tesson, Robertinventor, Jon-13, Marek69, Big Bird, AntiVandalBot, Steveprutz, VoABot II, Drewcifer3000, Wdﬂake, Urco, Shentino, Pekaje, PrestonH, Jr285714, J.delanoy, STBotD, Deor, JohnBlackburne, LokiClock, HENRYtastic,
Berchin, BarbaraBananas, Meters, Tiddly Tom, Caltas, BlueAzure, DavidDW, Cpl Giroro, Denisarona, Pwrong, ClueBot, ArdClose, Ender of Games, Maniac18, Onward2home, Radaustin, Jusdafax, Aidan oz, N3me$i$, Jeﬀrey Wordsmith, Vanished user uih38riiw4hjlsd,
Cminard, HexaChord, Addbot, Soundout, Medisaid, Kyle1278, Lightbot, Luckas-bot, Pcap, AnomieBOT, Turul2, Arbustoardiente,
Jim1138, AdjustShift, Xqbot, FrescoBot, Nicolas Perrault III, Akasanof, I dream of horses, Dazedbythebell, BigDwiki, Gazeboland, Double sharp, Panel Guy, LawBot, 4, Skakkle, Tbhotch, Nevin.williams, Onel5969, GoingBatty, K6ka, Amlentjes, AvicAWB, Alborzagros,
Petrb, ClueBot NG, Wcherowi, Gilderien, Widr, Helpful Pixie Bot, Jeraphine Gryphon, Carnac32, Wikiecam, QuarkyPi, Physiomath,
OCCullens, Louis993546, Dobie80, JYBot, Raumaank, Lugia2453, Stephen A Field, Cweedmeesy, Vanamonde93, Eyesnore, AnthonyJ
Lock, Myth of jesus, Plumpy Humperdinkle, Mohammed.Adil1989 and Anonymous: 161
• Flatland Source: https://en.wikipedia.org/wiki/Flatland?oldid=689158699 Contributors: Brion VIBBER, Ortolan88, Edward, Pmmenneg, Oliver Pereira, Dcljr, DavidWBrooks, Rossami, Palfrey, Patrick28657, Charles Matthews, Tpbradbury, Maximus Rex, Itai, Val42, Fibonacci, BenRG, Treutwein, P0lyglut, Merovingian, Blainster, Hippietrail, Adam78, Kevin Saﬀ, Giftlite, DocWatson42, Tspoon~enwiki,
Bﬁnn, Tagishsimon, Neilc, Keith Edkins, LucasVB, Ruzulo, Tomruen, Hellisp, MakeRocketGoNow, Jfpierce, Abelson, RJHall, Ylee,
Crunchy Frog, El C, Bobo192, NetBot, Barno, Cje~enwiki, Michael614, Ziggurat, Rockhopper10r, PWilkinson, Alansohn, Ricardo
monteiro, Jordan117, CyberSkull, Lectonar, Velella, HenkvD, Philthecow, Mathmo, Pol098, Dodiad, Macaddct1984, Stefanomione,
Graham87, ConradKilroy, Sjö, Koavf, Koos Jol, Salix alba, Bhaak, Mike Peel, Mathbot, ZoneSeek, Twipley, Strangnet, Cmadler, Wavelength, Hairy Dude, Ppinheiro, Dannycas, Xihr, Shawn81, Gaius Cornelius, RandallJones, Nick, Jpbowen, Zwobot, DeadEyeArrow,
Silverhill, Ms2ger, Tetracube, Nikkimaria, Plankhead, KenoSarawa, Rredwell, JDspeeder1, SmackBot, InverseHypercube, Rovenhot,
Gilliam, Hmains, Kevinalewis, Gorman, Mysticaloctopus, Thumperward, OrangeDog, Coﬃn, James Fryer, Sadads, Delta Tango, Emurphy42, Egsan Bacon, Tamfang, JudahH, Cybercobra, MasterSheep~enwiki, Kimpire, Thor Dockweiler, TenPoundHammer, John, Subanark, James.S, Ocatecir, Programmar, Eebleﬁsh, Ryulong, AEMoreira042281, Turtleheart, Shoeofdeath, Twas Now, Zero sharp, Dclayh,
Esn, The Letter J, CmdrObot, CBM, Equendil, Gogo Dodo, Tkynerd, Briantw, Danindenver, Casliber, Thijs!bot, Epbr123, PHaze,
Ash274, Aericanwizard, WikiSlasher, Widefox, Kbthompson, Donwithnoname, Sonicsuns, Planetary, Robina Fox, Albany NY, Slothman32, Belg4mit, Siddharth Mehrotra, Douglas Jardine, Charlyz, Tremilux, Froid, Catgut, MartinBot, Anaxial, Garkbit, Huzzlet the bot,
Uncle Dick, SlowJog, Quizzle, Literacola, Warlordbcm1, Lambdove, Docued, BrettAllen, Remember the dot, GrahamHardy, Dikke poes,
JohnBlackburne, Rtrace, DoorsAjar, TXiKiBoT, Gdorner, Utgard Loki, Laddehlingerjr, Broadbot, Room429, Cogburnd02, Kurowoofwoof111, Synthebot, Turgan, Wikineer, Yngvarr, Dogah, TCO, France3470, Goustien, Ipsherman, Fratrep, MrsKrishan, Blacklemon67,
Timeastor, Martarius, ClueBot, LAX, The Thing That Should Not Be, LukeShu, Niceguyedc, Shaydie, Atdenney, DumZiBoT, XLinkBot,
Micmachete, DaL33T, Marchije, Shoemaker’s Holiday, Addbot, Oﬀenbach, BONKEROO, Sudirclu, AkhtaBot, Sussmanbern, Lightbot,
Zorrobot, Artichoke-Boy, Luckas-bot, Yobot, TaBOT-zerem, Theornamentalist, Anand011892, AnomieBOT, Citation bot, Seblopedia, GB fan, Matrixmania, Grey ghost, Xqbot, Jayarathina, Jmundo, Armydelay99, GrouchoBot, Kylelovesyou, RibotBOT, FrescoBot,
Fortdj33, Paine Ellsworth, Sqwirrel, Theglasslipper, Akinari42, Pinethicket, RedBot, TRBP, Lotje, Duoduoduo, Jesse V., Siduofan,
Hbooger45, Jsawka33, Danojohnson, ZéroBot, Informative3, OnePt618, OpenlibraryBot, SBaker43, Odysseus1479, Zmax15, EdoBot,
Manytexts, ClueBot NG, Wcherowi, Chester Markel, Jkta97, Mpaa, Cntras, Weirdnessy0912, Helpful Pixie Bot, BG19bot, Nospildoh,
Robert the Devil, SD5bot, Cobalt174, Hillbillyholiday, Fzegdhzhee, Flat Out, Ugog Nizdast, Paul2520, Balljust, OccultZone, Fixuture,
Hax123, XBoner, Shinyanga, ♥Golf, Rob at Houghton, Deunanknute, Mvaldivieso and Anonymous: 251
• Four-dimensional space Source: https://en.wikipedia.org/wiki/Four-dimensional_space?oldid=687769927 Contributors: Zundark, The
Anome, SJK, William Avery, Patrick, Michael Hardy, Dcljr, Delirium, Ahoerstemeier, PeterBrooks, Charles Matthews, Reddi, Dysprosia, Robbot, Fredrik, Gandalf61, AceMyth, Dina, Tobias Bergemann, Adam78, Giftlite, Mporter, Cobaltbluetony, Lethe, Fropuﬀ,
Bovlb, Daniel Brockman, Jossi, Bumm13, Tomruen, Bodnotbod, Icairns, Sam Hocevar, Mare-Silverus, Discospinster, Pak21, FT2,
JoeSmack, Ben Standeven, Aecis, Rgdboer, Lankiveil, Bobo192, Army1987, Shlomital, Physicistjedi, Ultra megatron, Alansohn, SemperBlotto, Cjthellama, Hu, Wtmitchell, Jheald, Mikeo, Axeman89, Gamiar, Oleg Alexandrov, Feezo, OwenX, Linas, LOL, Yansa,
Kmg90, Maartenvdbent, Jon Harald Søby, Marvelvsdc, KyuuA4, Jclemens, Zoz, Phoenix-forgotten, Chipuni, Attitude2000, MarSch,
Salix alba, NeonMerlin, DoubleBlue, Algebra, FlaBot, TiagoTiago, SiriusB, Nihiltres, RexNL, Gurch, DannyDaWriter, TeaDrinker,
LeCire~enwiki, Cpcheung, TheSun, King of Hearts, DaGizza, Jared Preston, DVdm, VolatileChemical, Amaurea, Algebraist, YurikBot,
Wavelength, Maelin, Splintercellguy, Sceptre, Retodon8, Michael Slone, KamuiShirou, SpuriousQ, Hellbus, Shawn81, Yamara, MightyGiant, Ihope127, Havok, NawlinWiki, Tailpig, MacGyver07, Deckiller, Mgnbar, Tetracube, Cspalletta, PBurns, Arthur Rubin, Th1rt3en,
CWenger, Fram, Geoﬀrey.landis, AndrewWTaylor, Attilios, Havocrazy, SmackBot, Honza Záruba, GoldenXuniversity, Vkyrt, Unyoyega,
Canthusus, Alex earlier account, Gilliam, Kevinalewis, Quinsareth, Oli Filth, Magicindark, Moshe Constantine Hassan Al-Silverburg,
Nbarth, Robth, MaxSem, Can't sleep, clown will eat me, Tamfang, Onorem, Koolone0, Jwy, Nakon, Snakeyes (usurped), Jbergquist,
Aaker, Dogears, Clicketyclack, Meetarnav, Accurizer, Bjankuloski06en~enwiki, 041744, Xenure, Hotblaster, Werdan7, Kazikame, Cerealkiller13, Animedude360, Nehrams2020, Iridescent, FVP, Use4d, Mimicat, Fsotrain09, Octane, Phoenixrod, Tawkerbot2, Robinhw,

200

CHAPTER 42. ZERO-DIMENSIONAL SPACE

CmdrObot, Lavateraguy, JasonHise, Zureks, Nadyes, PeanutCheeseBar, Arnavion, Skrapion, Skybon, Cydebot, Reywas92, Red Director,
Pedro Fonini, Robertinventor, Krm500, Ryan Gittins, Brink14, Aazn, JodyB, Nol888, Mailerdaemon, Cheveyo, JamesAM, Thijs!bot,
Ulnevets, GentlemanGhost, Protocoi, Pampas Cat, Mojo Hand, AgentPeppermint, D.H, AntiVandalBot, Luna Santin, Ryttu3k, Oatmealcookiemon, The Dan, Figma, Fusionshrimp, JAnDbot, Husond, Pipedreamergrey, Xeno, Hut 8.5, J-stan, Stuﬀchanger, Bennybp,
Bongwarrior, VoABot II, AuburnPilot, Littlewood~enwiki, Catgut, Jbav1278, JMyrleFuller, Glen, Wdﬂake, Donrad, Rettetast, Nono64,
Akronym, LedgendGamer, Asalt2233~enwiki, Tgeairn, Bogey97, Uncle Dick, Extransit, Thaurisil, Century0, Lantonov, Zoxxi, McSly,
Starnestommy, Supuhstar, LittleHow, NewEnglandYankee, Rominandreu, Richard Wolf VI, Juliancolton, Cometstyles, DH85868993,
MoForce, Homo logos, Xiahou, RJASE1, Xnuala, Hamzabahaa, Wikieditor06, BowToChris, Seldon1, Chaos5023, JohnBlackburne, Soliloquial, Veddan, Philip Trueman, Zidonuke, Anonymous Dissident, Crohnie, Sean D Martin, Berchin, IronMaidenRocks, Ian Strachan,
Thomas s. briggs, AgentCDE, Dmcq, Subh83, SieBot, RageGarden, Pi is 3.14159, Flyer22 Reborn, Paolo.dL, Prestonmag, Faradayplank,
Harry~enwiki, Yone Fernandes, Nick90210, Lightmouse, Hobartimus, Svick, Pinkadelica, ImageRemovalBot, ClueBot, The Thing That
Should Not Be, ArdClose, Njbh9, Mild Bill Hiccup, SuperHamster, Blanchardb, JohnTheCrow, Alexbot, Brews ohare, NuclearWarfare,
Arjayay, Razorﬂame, Frozen4322, Thfo, Error −128, Drippy knees, JDPhD, Questsong, Versus22, SoxBot III, Christianw7, XLinkBot,
Hotcrocodile, Zzubiri, WikHead, SilvonenBot, Alexius08, Jedi870, Addbot, Proofreader77, Theothersteve7, Jafeluv, Tcncv, Ronhjones,
Jncraton, Adam08, Glane23, Glass Sword, Favonian, Farmercarlos, Tassedethe, Pbryant7, Tide rolls, Teles, Zorrobot, Quantumobserver,
Luckas-bot, Roﬂcopter1000, Yobot, TaBOT-zerem, Sarrus, MassimoAr, Tempodivalse, AnomieBOT, Wickedmangroves, Turul2, AdjustShift, Citation bot, ArthurBot, Xqbot, Intelati, Blackknight7429, JFBGT, Ruy Pugliesi, Bradjuhasz, Papercutbiology, HedonismBot2911, Basket of Puppies, Locobot, Bigger digger, WaysToEscape, FrescoBot, 123456789riitta, 123456789gary, Sky Attacker, Majopius, A little insigniﬁcant, Åkebråke, DrilBot, Pinethicket, I dream of horses, JranZu, Atm2177, Daclyﬀ, LordxDeath, Jschnur, Btilm,
RandomStringOfCharacters, RockSolidCosmo, Double sharp, Puzl bustr, Vrenator, 4, Juana1990, DARTH SIDIOUS 2, Whisky drinker,
Swishaa218, Regancy42, Fredgds, WikitanvirBot, Gfoley4, Pekka.virta, Vish-aero, Dualitynature, GoingBatty, Vanished user zq46pw21,
4dimention, Slightsmile, Tommy2010, Wikipelli, Fafaerer, ZéroBot, Deehack, Azuris, Bamyers99, Brandmeister, L Kensington, Alborzagros, Tyughj2, MPLSboarder, Orange Suede Sofa, Brianumeda, DASHBotAV, ClueBot NG, Wcherowi, Joefromrandb, Vacation9, Bryce
Albe-Quenzer, Bakrnl, LaszloSimon, Frietjes, Fujicapesta, Helpful Pixie Bot, Ernest3.141, BG19bot, Waqsajaz, Longbyte1, Nospildoh,
Piguy101, Mark Arsten, Zeke, the Mad Horrorist, Puzzle314, Muthafucker, Snow Blizzard, Tesseract4d, Isacdaavid, CeraBot, Alkagl, NGC 2736, JYBot, Rainbowking5, Davidcdunne, Samwick6, Chakravarti1997, Hillbillyholiday, Uniquestman, 4-Dimensional0034,
Neve2004, Prokaryotes, Mrkicky, Newold123, Jackscd, Artist.poet, Loraof, Infernus 780, ChaoticDequix, Dalvarezso, Maddiemmm,
Ubernachten, Gustavo noise, HeidiShaban, Peshwavignesh and Anonymous: 595
• Fourth dimension in art Source: https://en.wikipedia.org/wiki/Fourth_dimension_in_art?oldid=689171734 Contributors: Michael Hardy,
Dimadick, Mandarax, DVdm, Christian75, Awien, Chiswick Chap, Lamro, TCO, Coldcreation, 7&6=thirteen, Chris857, MLWatts, BattyBot, Hillbillyholiday, Monkbot, Artist.poet, D-4597-aR and Anonymous: 2
• Gelfand–Kirillov dimension Source: https://en.wikipedia.org/wiki/Gelfand%E2%80%93Kirillov_dimension?oldid=672809607 Contributors: Michael Hardy, TakuyaMurata, Bearcat, Katharineamy, Yobot, FrescoBot, KonradVoelkel, Brad7777, Mark viking, Yolaf.TZ
and Anonymous: 1
• Global dimension Source: https://en.wikipedia.org/wiki/Global_dimension?oldid=675189725 Contributors: Silverﬁsh, Charles Matthews,
C12H22O11, Pearle, Oleg Alexandrov, Michael Slone, MalafayaBot, Rschwieb, JoergenB, STBot, Arcfrk, JackSchmidt, Addbot, Jim1138,
Sz-iwbot, Citation bot 1, Bomazi, ClueBot NG, Solomon7968, Deltahedron and Anonymous: 6
• Interdimensional Source: https://en.wikipedia.org/wiki/Interdimensional?oldid=633208097 Contributors: Michael Hardy, KFan II, Yobot,
Pwncoreshaman, WaysToEscape, A.amitkumar, Aurelius Lie and Anonymous: 3
• Isoperimetric dimension Source: https://en.wikipedia.org/wiki/Isoperimetric_dimension?oldid=625993034 Contributors: Michael Hardy,
Charles Matthews, Dbenbenn, Gadykozma, Oleg Alexandrov, Mathbot, Sodin, SmackBot, Ylloh, Josephorourke, .anacondabot, Anchor
Link Bot, BRicaud, Addbot, Yobot, Foobarnix, Xnn, Suslindisambiguator, Paolo Lipparini and Anonymous: 1
• Kaplan–Yorke conjecture Source: https://en.wikipedia.org/wiki/Kaplan%E2%80%93Yorke_conjecture?oldid=650537149 Contributors: Michael Hardy, Bearcat, Giftlite, TheCatalyst31, Rankersbo, Alvin Seville, Gemaze and BG19bot
• Kodaira dimension Source: https://en.wikipedia.org/wiki/Kodaira_dimension?oldid=647715758 Contributors: Michael Hardy, Charles
Matthews, Giftlite, R.e.b., Colonies Chris, Nick Number, David Eppstein, DarknessBot, WereSpielChequers, Marsupilamov, Addbot,
Ozob, Luckas-bot, Yobot, AnomieBOT, Omnipaedista, FrescoBot, RedBot, Trappist the monk, Uni.Liu, BTotaro, BG19bot, Brad7777,
ChrisGualtieri, Noellapin, Deltahedron, Enyokoyama, K9re11 and Anonymous: 6
• Krull dimension Source: https://en.wikipedia.org/wiki/Krull_dimension?oldid=688043442 Contributors: AxelBoldt, Zundark, Toby
Bartels, Michael Hardy, TakuyaMurata, Charles Matthews, Giftlite, Fropuﬀ, Dan Gardner, Vivacissamamente, YixilTesiphon, R.e.b.,
ChiLlBeserker, Gaius Cornelius, Gwaihir, Light current, JahJah, SmackBot, MalafayaBot, Tesseran, Joerg Winkelmann~enwiki, Cronholm144, Krasnoludek, CBM, Gihanuk, Ntsimp, Hardmath, Vanish2, Policron, LokiClock, RiverStyx23, SieBot, YonaBot, Mathemajor,
Marsupilamov, MystBot, Addbot, Lightbot, Legobot, Luckas-bot, TaBOT-zerem, KamikazeBot, Clare4004, Onzie9, Junior Wrangler,
EmausBot, ZéroBot, AvicAWB, D.Lazard, Danramras, BTotaro, MaximalIdeal, Solomon7968, ChrisGualtieri, Mark viking, Ozgunner,
AJeiEM and Anonymous: 16
• Matroid rank Source: https://en.wikipedia.org/wiki/Matroid_rank?oldid=675336702 Contributors: Mack2, David Eppstein and Zroe1029
• Negative-dimensional space Source: https://en.wikipedia.org/wiki/Negative-dimensional_space?oldid=673554250 Contributors: Michael
Hardy, Ogerard, Lamro, Northernhenge and Anonymous: 1
• Nonlinear dimensionality reduction Source: https://en.wikipedia.org/wiki/Nonlinear_dimensionality_reduction?oldid=668904482 Contributors: Edward, Michael Hardy, Kku, MartinHarper, Charles Matthews, Guaka, Catbar, T0m, Lawrennd, Mukerjee, Andreas Kaufmann, Forderud, Oleg Alexandrov, Firsfron, Jfr26, Male1979, Qwertyus, Rjwilmsi, Gmelli, Mathbot, Choess, Gussisaurio, Kri, Wavelength, Gaius Cornelius, Kilianw, SmackBot, Mcld, Memming, Yannick Copin, TNeloms, Shorespirit, Morgaladh, AnAj, Coﬀee2theorems,
Dfalcantara, A3nm, STBot, Salih, Trondarild, Jacobsn, Pjoef, Fragrant, Melcombe, Headlessplatter, Agor153, 1ForTheMoney, JamesXinzhiLi, Arbitrarily0, Yobot, Ziyuang, FrescoBot, X7q, MGA73bot, Rc3002, Ciuncun, John of Reading, Ambarish.jash, Ida Shaw,
Aria802, Phrank36, Zephyrus Tavvier, Foreﬁnger2, Tatome, WikiMSL, Laughsinthestocks, Kaijiang ustc, BG19bot, Tommyboucher,
Sunjigang1965, ChrisGualtieri, Ljuvela, OhGodItsSoAmazing, Monkbot, Siddhantmanocha, Jcnebel and Anonymous: 64
• One-dimensional space Source: https://en.wikipedia.org/wiki/One-dimensional_space?oldid=673381431 Contributors: Mporter, Tomruen, Wtmitchell, Red Slash, Sardanaphalus, RDBury, R'n'B, JohnBlackburne, Lamro, Bobathon71, Addbot, Amirobot, JackieBot,
Xqbot, HRoestBot, Double sharp, 4, Hhhippo, ZéroBot, Rocketrod1960, ClueBot NG, Lanthanum-138, BG19bot, IluvatarBot, Vanished user lt94ma34le12, JYBot, Jjbernardiscool, TCMemoire, Cranberry Products, KasparBot and Anonymous: 9

42.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

201

• Ordinate Source: https://en.wikipedia.org/wiki/Ordinate?oldid=667025598 Contributors: Zundark, Matusz, Michael Hardy, Grape~enwiki,
Hyacinth, Optim, Gandalf61, Edcolins, Musiphil, Oleg Alexandrov, Linas, Polyparadigm, Maxal, Allens, SmackBot, BiT, Octahedron80,
R'n'B, LizardJr8, Käptn Weltall, Addbot, Theraven502, Twri, Smallman12q, Double sharp, Solomonfromﬁnland, Stringybark, Dearhumanity, Aupif, ClueBot NG, Wwjd2012, Mguggis, Bryanrutherford0 and Anonymous: 19
• Regular sequence Source: https://en.wikipedia.org/wiki/Regular_sequence?oldid=597641160 Contributors: Charles Matthews, Berjoh,
Crust, Oleg Alexandrov, R.e.b., The Rambling Man, Typometer, Arcfrk, Jsbeder, Arbitrarily0, D.Lazard, BTotaro, BG19bot, Recursive
mindloop and Anonymous: 8
• Relative canonical model Source: https://en.wikipedia.org/wiki/Relative_canonical_model?oldid=492228791 Contributors: Michael
Hardy, Charles Matthews, Arthur Rubin, SmackBot, CBM, Pjoef, Favonian, Yobot, Createangelos, FrescoBot, Deltahedron and Anonymous: 5
• Relative dimension Source: https://en.wikipedia.org/wiki/Relative_dimension?oldid=634231621 Contributors: Jitse Niesen, Nbarth,
CBM, Jj137, David Eppstein, R'n'B, Philip Trueman, Brad7777 and Anonymous: 1
• Schauder dimension Source: https://en.wikipedia.org/wiki/Schauder_basis?oldid=671861507 Contributors: AxelBoldt, Michael Hardy,
TakuyaMurata, Charles Matthews, Jitse Niesen, Aetheling, Tobias Bergemann, Giftlite, BenFrantzDale, Lethe, Rich Farmbrough, Sam
Derbyshire, Linas, Rjwilmsi, R.e.b., Bgwhite, Wavelength, Henning Makholm, Ulner, Michael Kinyon, JHunterJ, Headbomb, Forgetfulfunctor, Vanish2, Jay Gatsby, Gamesou, AlleborgoBot, SieBot, Mild Bill Hiccup, Brews ohare, Addbot, DOI bot, LaaknorBot, Yobot,
Citation bot, Bdmy, Constructive editor, Trappist the monk, 777sms, RomDem, Chricho, BG19bot, Anaxagore~enwiki, Mgkrupa and
Anonymous: 11
• Seven-dimensional space Source: https://en.wikipedia.org/wiki/Seven-dimensional_space?oldid=655918492 Contributors: Dominus,
Mporter, Tomruen, Algebraist, Wavelength, Arthur Rubin, Colonies Chris, Tamfang, Sammy1339, Titus III, Pentascape, ShelfSkewed,
Myasuda, R'n'B, JohnBlackburne, Jdcrutch, JL-Bot, Niceguyedc, Muhandes, TimothyRias, Addbot, Favonian, Amirobot, AnomieBOT,
Xqbot, Foobarnix, Double sharp, 4, Jfmantis, GoingBatty, Chricho, ClueBot NG, Lanthanum-138, BG19bot, Itc editor2, KHEname,
Loraof and Anonymous: 6
• Six-dimensional space Source: https://en.wikipedia.org/wiki/Six-dimensional_space?oldid=674547053 Contributors: Michael Hardy,
Ixfd64, Mporter, Tomruen, Chris Howard, Rgdboer, Jheald, Rjwilmsi, R.e.b., Srleﬄer, Wavelength, Arthur Rubin, Allens, SmackBot, Colonies Chris, Tamfang, Salamurai, Newone, Eastlaw, CRGreathouse, Myasuda, Cydebot, Mato, D.H, JohnBlackburne, Lamro,
YohanN7, Mild Bill Hiccup, Robert Skyhawk, Muhandes, Brews ohare, Cenarium, TimothyRias, WikHead, Addbot, Jkasd, Yobot, Xqbot,
, Jschnur, Double sharp, Sumone10154, 4, GoingBatty, Solarra, Alborzagros, ClueBot NG, Bibcode Bot, Liefs13, Solomon7968,
Sixdimensiondesign, OCCullens, Sahanded and Anonymous: 21
• String theory Source: https://en.wikipedia.org/wiki/String_theory?oldid=689244912 Contributors: AxelBoldt, Sodium, Mav, Bryan
Derksen, Zundark, The Anome, Tarquin, Taw, Eean, Malcolm Farmer, Hephaestos, Olivier, Drseudo, Stevertigo, Spiﬀ~enwiki, Edward, PhilipMW, Michael Hardy, Bewildebeast, Dante Alighieri, Gabbe, Graue, Tgeorgescu, Mcarling, CesarB, Looxix~enwiki, Ahoerstemeier, Theresa knott, Suisui, Angela, Den fjättrade ankan~enwiki, Jdforrester, Julesd, Salsa Shark, Schneelocke, Charles Matthews,
Timwi, Bemoeial, Jitse Niesen, 4lex, Greenrd, ErikStewart, Furrykef, Saltine, Phys, Omegatron, Bevo, Topbanana, Trent, Nufy8, Robbot, Craig Stuntz, Fredrik, Chris 73, R3m0t, COGDEN, Mirv, Wjhonson, Sverdrup, Academic Challenger, DHN, Hadal, Khlo, ElBenevolente, HaeB, Tobias Bergemann, Giftlite, DocWatson42, Christopher Parham, Awolf002, Mporter, Amorim Parga, Mikez, Harp,
Kim Bruning, Tom harrison, Ferkelparade, Leﬂyman, Fropuﬀ, No Guru, Anville, Moyogo, Curps, Pashute, Nomad~enwiki, Mboverload, Solipsist, SWAdair, DemonThing, Wmahan, Btphelps, MSTCrow, Decoy, Chowbok, Gadﬁum, Steuard, Pgan002, Quadell, Carandol~enwiki, Antandrus, Beland, JoJan, Khaosworks, Tothebarricades.tk, Thincat, Tomruen, Shidobu, Icairns, Lumidek, NoPetrol, Avihu,
Fanghong~enwiki, Trevor MacInnis, Lacrimosus, Zro, Mike Rosoft, D6, Urvabara, Felix Wan, Jkl, Discospinster, ElTyrant, Rich Farmbrough, Rhobite, Pjacobi, Alien life form, Vapour, Silence, Kzzl, LindsayH, Mani1, Pavel Vozenilek, Paul August, Bender235, Kjoonlee, Mashford, Kelvinc, Perlman10s, Panu~enwiki, Brian0918, Dpotter, Livajo, El C, Laurascudder, Shanes, Zegoma beach, RoyBoy,
Causa sui, Bobo192, Directorstratton, Janna Isabot, Smalljim, John Vandenberg, Flxmghvgvk, I9Q79oL78KiL0QTFHgyc, Physicistjedi,
Bongoo, 4v4l0n42, Merope, Geschichte, Linuxlad, Phils, Merenta, Alansohn, Gary, JYolkowski, Enirac Sum, Ryanmcdaniel, Arthena,
Borisblue, Rd232, Plumbago, Axl, R Calvete, Lightdarkness, Kocio, Bart133, Wtmitchell, Isaac, Tycho, Cal 1234, Fadereu, CloudNine,
Sciurinæ, Computerjoe, Kusma, DV8 2XL, Pwqn, Gene Nygaard, Ringbang, Ceyockey, Falcorian, Bobrayner, Joriki, Mel Etitis, Linas,
BillC, Jacobolus, HFarmer, Before My Ken, Netdragon, MONGO, GeorgeOrr, Mpatel, Bbatsell, GregorB,
, Joke137, Christopher Thomas, Dysepsion, GSlicer, Jan.bannister, Graham87, Magister Mathematicae, Hillbrand, BD2412, Elvey, Galwhaa, Raymond Hill,
JIP, RxS, Athelwulf, Edison, Sjakkalle, Rjwilmsi, Xgamer4, Jake Wartenberg, Arabani, MarSch, TheRingess, Jmcc150, Aero66, Crazynas, Juan Marquez, R.e.b., Bubba73, DoubleBlue, Zelos, AlisonW, Asafavi, Lionelbrits, Conoriﬁc, Zunz, Mathbot, Crazycomputers,
RexNL, Gurch, Algri, TeaDrinker, Zifnabxar, XAXISx, Erik4, Phoenix2~enwiki, Antimatter15, Ggb667, Chobot, Visor, DVdm, Mhking, VolatileChemical, Bgwhite, Algebraist, Ben Tibbetts, YurikBot, Ugha, Wavelength, Borgx, NuclearFusion~enwiki, Angus Lepper,
Hairy Dude, Jimp, Hillman, Cyferx, Wolfmankurd, Pip2andahalf, RussBot, Moronoman, Crazytales, Pippo2001, Bhny, Pigman, SpuriousQ, Branman515, Stephenb, Gaius Cornelius, Eleassar, Bovineone, Cheesus, Shanel, NawlinWiki, Tong~enwiki, Mike18xx, SCZenz,
Cleared as ﬁled, Bdiah, Pym98, SColombo, Haemo, FF2010, Closedmouth, Reyk, Brina700, Chris Brennan, Vicarious, Brianlucas, Geoﬀrey.landis, Hitchhiker89, Spliﬀy, Pred, ArielGold, Roy Fultun, Ilmari Karonen, Katieh5584, Pentasyllabic, Lunch, DVD R W, WikiFew, That Guy, From That Show!, Street Scholar, AndrewWTaylor, QSquared, Sardanaphalus, Vanka5, MacsBug, Hvitlys, SmackBot,
Kurochka, Zazaban, Tom Lougheed, Prodego, KnowledgeOfSelf, Hydrogen Iodide, Melchoir, Vald, Skrewtape, Atomota, Canthusus,
GaeusOctavius, Cool3, Andyvn22, Skizzik, RobertM525, Dauto, Bluebot, SSJ 5, Keegan, Aidan Croft, Thumperward, Oli Filth, Silly
rabbit, Timneu22, SchﬁftyThree, Moshe Constantine Hassan Al-Silverburg, Complexica, Rediahs, RayAYang, Aero77, Adamstevenson,
Ikiroid, Epastore, Baronnet, Ned Scott, Sbharris, Colonies Chris, Konstable, Sct72, Scwlong, Can't sleep, clown will eat me, Timothy Clemans, Onorem, Neilanderson, EvelinaB, TKD, KerathFreeman, Addshore, UU, The tooth, Pepsidrinka, Somebody2292, --=The
Doctor=--, Fuhghettaboutit, Cybercobra, Irish Souﬄe, Nakon, Jdlambert, James McNally, MichaelBillington, Lostart, Insineratehymn,
Drphilharmonic, SpiderJon, DMacks, Ihatetoregister, Where, Michael IFA, Yevgeny Kats, Vasiliy Faronov, Byelf2007, Angela26, Visium, Rory096, Zymurgy, Harryboyles, Mdl53711, T-dot, Titus III, Ergative rlt, MagnaMopus, UberCryxic, Vgy7ujm, Linnell, Mgiganteus1, Nonsuch, IronGargoyle, Ckatz, DoItAgain, AstroGod, Kirbytime, Jimbo Mahoney, FredrickS, Invisifan, Ryulong, Ryanjunk,
MathStuf, Mike Doughney, Norm mit, Hindol, Dan Gluck, Huntscorpio, Iridescent, K, Sunoco, You? Me? Us?, CzarB, Rabinzkaman, JoeBot, Lottamiata, Tony Fox, Vrkaul, Torrazzo, Gil Gamesh, Areldyb, Courcelles, Tawkerbot2, Gebrah, Shamvil, DKqwerty,
Lbr123, Harold f, Heqs, Devourer09, Duduong, Sarvagnya, Dewayne76, JForget, Cg-realms, InvisibleK, CRGreathouse, CmdrObot,
Earthlyreason, Van helsing, Olaf Davis, CBM, Rawling, Jibal, Witten Is God, Nunquam Dormio, Giko, KnightLago, Thubsch, Leujohn, SlashDot, TheTito, Karenjc, Myasuda, Emarv, Cydebot, Gmusser, Gogo Dodo, Jkokavec, Kahananite, Quajafrie, Michael C

202

CHAPTER 42. ZERO-DIMENSIONAL SPACE

Price, Doug Weller, DumbBOT, Narayanese, AlphaNumeric, SRoughsedge, Vanished User jdksfajlasd, Woland37, Zalgo, Daniel Olsen,
UberScienceNerd, Bkazaz, DJBullﬁsh, Thijs!bot, Epbr123, Rwmnau, Babemachine, Pimpin101, Mbell, O, Faigl.ladislav, Ucanlookitup, Andyjsmith, Headbomb, Tcturner2002, Marek69, Brahmajnani, Arthurcprado~enwiki, Y.t., D3gtrd, Babemonkey, Dark dude,
Duncan McB, EdJohnston, MichaelMaggs, Ancientanubis, Natalie Erin, Hempfel, Jomoal99, Mmortal03, Mentiﬁsto, Geekdom04, AntiVandalBot, Luna Santin, Seaphoto, Ed270791, Opelio, Doc Tropics, David136a, NithinBekal, Dotdotdotdash, Helicoptor, Poshzombie, MontanNito, Dylan Lake, Maximilian77, Shlomi Hillel, Db63376, SamIAmNot, Knotwork, Res2216ﬁrestar, Superior IQ Genius,
MER-C, Andonic, Sitethief, 100110100, TallulahBelle, Nestamachine, Savant13, Daynightrader, Goldenglove, Charibdis, Acroterion,
Ophion, Aigisthos, Editmyhandman, Aruben537, Magioladitis, WolfmanSF, Bongwarrior, VoABot II, Yandman, JamesBWatson, ‫باسم‬,
Qutt, Jespinos, Kevinmon, Aka042, Froid, DAGwyn, Catgut, Panser Born, Ensign beedrill, Perspectival, JJ Harrison, Dirac66, Justanother, Aziz1005, Cpl Syx, ChazBeckett, Teardrop ontheﬁre, WLU, Stephen shenker, Robin S, SkepticVK, Joshua Davis, Mkroh, B9
hummingbird hovering, S3000, Hdt83, MartinBot, FlieGerFaUstMe262, Ytomem, Shimwell, Arjun01, KrishSundaresan, Anaxial, Jay
Litman, Alexcalamaro, Andrej.westermann, Smokizzy, LedgendGamer, Cyrus Andiron, Peteryoung144, Tgeairn, Artaxiad, HEL, AlphaEta, J.delanoy, AstroHurricane001, Maurice Carbonaro, Yonidebot, Morris729, M C Y 1008, 69gangsta420, It Is Me Here, Shawn
in Montreal, Janus Shadowsong, Bailo26, Fredsie, Madagaskar07, Duchesserin, AntiSpamBot, CHIAGEHYANG, Chiswick Chap, Watsup1313, Belovedfreak, HaloInverse, NewEnglandYankee, Scott1329m, Thesis4Eva, Policron, Jrcla2, WJBscribe, Rnricklefs, Jamesofur, Eyelidlessness, Jonnyk aus, Kvdveer, JavierMC, Izno, Xiahou, CardinalDan, Sheliak, HamatoKameko, Malik Shabazz, Concertmusic, JohnBlackburne, JustinHagstrom, Fences and windows, Wooba doob, Philip Trueman, DoorsAjar, HowardFrampton, TXiKiBoT,
Zidonuke, Red Act, Kriak, Calwiki, Technopat, Hqb, Andrius.v, Anonymous Dissident, Crohnie, AlysTarr, Qxz, Vanished user ikijeirw34iuaeolaseriﬃc, Impunv, Seraphim, Martin451, Don4of4, ABigGreenHippo, Huperphuﬀ, LeaveSleaves, Kaenneth, StringyGuy,
Maxim, Erth64net, Meters, Lamro, Rickstauduhar, Enviroboy, Turgan, Anna512, PhysPhD, Northfox, NPguy, Matthew Sanders, Luke
Walkerson, Newbyguesses, MissMJ, SieBot, Escher26, J.A.Ireland, BA (IHPST), 4wajzkd02, Robdunst, Dreamafter, Pallab1234, Dbelange, MTHarden, Lemonﬂash, Kylemew, Yintan, GlassCobra, Wpegden, Likebox, Flyer22 Reborn, Exert, ProGeek314, Arbor to
SJ, Babawhitemoose, Caidh, Dhatﬁeld, Audree, Oxymoron83, Pretty Green, Weaselstomp, Manway, Alex.muller, Taco Manipulator,
Tschach, Manheat84, Anchor Link Bot, Mikebernstein, ImperialismGo, Nergaal, Ionﬁeld, Ayleuss, Sh4wz0r, Naturespace, Martarius, Phyte, ClueBot, The Thing That Should Not Be, String4d, Illusion96, Polyamorph, Mpd1989, Alexdeburca18, Wiggl3sLimited,
Excirial, Kjramesh, Jusdafax, Resoru, WikiZorro, Eeekster, Verum~enwiki, Tamaratrouts, Gtstricky, Humanino, Brews ohare, NuclearWarfare, Cenarium, Arjayay, Razorﬂame, Scoobey, BOTarate, Sideswiper, Thingg, Capudo, BVBede, Versus22, Introductory adverb clause, MelonBot, SoxBot III, Egmontaz, Notpayingthepsychiatrist, DumZiBoT, BahTab, TimothyRias, Aj00200, Reaperfromhell,
Dunkaroo207, XLinkBot, AlexGWU, Impshum, Saeed.Veradi, Little Mountain 5, Guy392, David424, Truthnlove, Qweeveen, Tayste,
Addbot, Steven66s, Denali134, Elemented9, Varrey280303, Eric Drexler, Some jerk on the Internet, Fizzycyst, Uruk2008, DOI bot,
Jojhutton, AngryBacon, Captain-tucker, Auspex1729, Kongr43gpen, Fgnievinski, Rhetoric Of A Sophist, Ronhjones, CanadianLinuxUser, Cst17, Download, Glane23, Bassbonerocks, Chzz, Favonian, Kronix35, LinkFA-Bot, Udugunit, Aktsu, Tassedethe, Numbo3bot, Anpecota, Tide rolls, HerpesVirus, SDJ, OlEnglish, Scourge of God, Davidmedlar, Couldbenoway66, Yobot, Maxdamantus, Terrisknickers, Kartano, TaBOT-zerem, Julia W, Unique and proud of it, FireMouseHQ, Terriﬁctriﬃd, ArchonMagnus, CinchBug, Synchronism, AnomieBOT, Cleeseheb, 1exec1, Charlesvi, Bigdaddy4x4, Gitman4, Jim1138, IRP, Mintrick, Drweetmola, Ornamentalone,
M00npirate, Gautam10, Csigabi, Poli-Psy, Materialscientist, 90 Auto, Citation bot, Teleprinter Sleuth, Vuerqex, Twri, Frankenpuppy,
Fuzzy Bob Saget, DirlBot, Georgepowell2008, Heidisql, Cureden, Ekwos, Capricorn42, Gensanders, NFD9001, Anna Frodesiak, Tomwsulcer, A23649, Pra1998, Coretheapple, Ruy Pugliesi, Jagbag2, Vandalism destroyer, Ab1, Omnipaedista, Bandit5005, Shirik, RibotBOT, Waleswatcher, Saalstin, Amaury, Aaron35510, Caz34, Doulos Christos, Sewblon, Born Gay, Capricorn24, SchnitzelMannGreek,
A. di M., SpacePyjamas, Kierkkadon, A.amitkumar, Dougofborg, StringLove, Nobelprizewinner, Astiburg, FrescoBot, Fortdj33, Paine
Ellsworth, Goodbye Galaxy, HJ Mitchell, Steve Quinn, Vhann, Kwiki, Xhaoz, Citation bot 1, Batong, Gil987, Pinethicket, I dream
of horses, Tallboyhoops1991, Three887, Steveo27ﬁve, RedBot, Sardinita, Serols, Vhsatheeshkumar, Swisstingle, DeletionUK, Reconsider the static, IVAN3MAN, Remingtonhill1, Orenburg1, Coltonhs, Willy Weazley, Smamaret, Bethovenn, Dinamik-bot, Dc987, Oswaldo Zapata, Egemont, Syebo, Alaithiran, Reaper Eternal, Seahorseruler, Ybungalobill, Quaker phil, Specs112, Dr. Aakash Patel,
Tbhotch, StormbringerUK, Minimac, Mathgenius3141592, Keegscee, Omgwaﬀels, Mick le pick, Solancel, Aznhero3793, Dwielark,
Afteread, Enauspeaker, EmausBot, MaooaM, Immunize, Az29, Milkocookie, Faolin42, Fotoni, RA0808, RenamedUser01302013, 8digits, Yukiseaside, Slightsmile, Tommy2010, Winner 42, Wikipelli, JonezyKiDx, Joe Gazz84, ZéroBot, Timeitsways, John Cline, Cogiati, Quaqa, Chrispaps2413, Nasulikid, Vollrath2323, Benjamin1414141414141414, Arbnos, Green Lane, A930913, Bamyers99, Azeraphale, H3llBot, Encyclopadia, Danga1988, Ollainen, PoisonGM, Wayne Slam, OnePt618, Knome335, L Kensington, Lulzprotuns,
Kranix, Rpcappello, Maschen, Vastly~enwiki, Donner60, CatFiggy, CountMacula, Orange Suede Sofa, Etov, M1k3 101, Bill william
compton, Wakabaloola, TERBAFAN, Nickslspride34, NeuralLotus, Isocliﬀ, Brechbill123, Xanchester, ClueBot NG, Martti Muukkonen, KagakuKyouju, Jeﬀ Song, This lousy T-shirt, Satellizer, Name Omitted, Marcdean123, Wiki incorp, Frietjes, O.Koslowski, Alexdamaino9, Dream of Nyx, Blackhall616, Widr, Sashhere, WikiPuppies, Stu181, T00g00d96, Pluma, Storm.sarup, Helpful Pixie Bot,
Manzeet, Waﬄeboy36, HMSSolent, Mikeshelton1, Bibcode Bot, 2001:db8, Phillip.phillipson, Hoaxinator, Lowercase sigmabot, Thor
cherubim, Mrshabam, Nishch, Flowerhat15, AvocatoBot, Housegeek224, MahRanch, Benzband, Altaïr, Benhenchdickthomas, Shreyakstring, Sweaty maori sphincter, DaFalk, Dsabo74, Ratanmaitra, MM4EVAH, Steven.w.kowalski, Minsbot, JGallardo2600, Dylanlatham,
Myfriendganesha, OCCullens, Likeaboss189, Sean271293, LinusE8, BattyBot, Several Pending, Aldrich2122, CommanderMoka, The
Illusive Man, ChrisGualtieri, KoalamaN2, Trevorkid45, Catsloveit07, Alex Modzz, Rustyjamsen, Goh ryangoh, Dexbot, Exolius, Hilander316, Alman1234321, SuperCalzer, LightandDark2000, MeekMelange, BQND, Cdarrai1, Kephir, TheMonkeyboy524, Michael
Anon, TwoTwoHello, Mattfat8, Lugia2453, Anruy, Rachel weld, Jamesx12345, AHusain314, BossEditors, Hillbillyholiday, Joeinwiki,
Mattninja, Theshadow444, Asaa82, Jakemarz197, Kzhang1025, Epicgenius, Spongbob456789, , TestMaster, Ianreisterariola, GrapperJ, Makeitnasty, Moemajdi, I am One of Many, NualaIvy, BAZINGASS, St3fanPC, Eyesnore, Isaac grozd, Jordanissexyaf1999,
Baruch6525, Mosbruckercj, Ihatedirac2k13, Jonamithy121314, 123physicsquantum, Jt198, RaphaelQS, HeyJude70, AParker628, DimReg, A.k.blaze1, Joshuk, Zenibus, Nianoobasik, Ihelpapplen, Gamo To Apoel, SacredLabyrinth, Ginsuloft, Vampre1122, Dimension10,
Howard Wolowitz, AddWittyNameHere, Polytope24, Elysion, Tutun12$, Longerboats5, SimonWombat8, Konveyor Belt, Vtank54, Micheal545,
Hck24, Caliae19, Hexaﬁsh, Simpick, TheRealTheKoi, Bballbro62, Monkbot, ArmyPath, TheQ Editor, Jtsmith098, Joshmiller1, Hanseer360,
XXvPIEvXx, Dbennett 24, Ghikpenos, Nick65633, Saundra03, Thehippothatknows, Sewwgers, Teelaskeletor, Cirksena, Balockaye1234,
PloppyDoo, Yesufu29, Lumpy2k14, Podayeruma, Abstract92, Sbenﬁel, Monkman2k4, Swegwegdgfyetkfoﬀkkfkfkv, John95541234,
Poopman224, ScrapIronIV, Tetra quark, GeneralizationsAreBad, Shivansh2014n, KasparBot, SHUCKYLUCKY, Fabiotheoto, FartGoblin, Joca potato, Joshcool246, Theoretical Physisist4444, Reg7d88 and Anonymous: 1553
• Two-dimensional space Source: https://en.wikipedia.org/wiki/Two-dimensional_space?oldid=682235655 Contributors: Dino, Bevo,
Mporter, Kri, Wavelength, RDBury, Incnis Mrsi, Kjkjava, Newone, JAnDbot, Magioladitis, R'n'B, JohnBlackburne, Dmcq, Mitch Ames,
Addbot, Luckas-bot, Yobot, AnomieBOT, Xqbot, GrouchoBot, Frosted14, Gire 3pich2005, Double sharp, 4, EmausBot, Jpvandijk,

42.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

203

Tijfo098, Rezabot, மதனாஹரன், Microextruders, IluvatarBot, JYBot, Aymankamelwiki, Saehry, Brirush, Suelru, Monmonmon098,
Loraof and Anonymous: 15
• VC dimension Source: https://en.wikipedia.org/wiki/VC_dimension?oldid=688214810 Contributors: The Anome, Mark Durst, Michael
Hardy, Dcljr, LouI, BAxelrod, Hike395, Charles Matthews, Dcoetzee, Dmytro, Pgan002, APH, Gene s, Icairns, Neko-chan, Teorth,
Rajah, Woohookitty, Sujith, Anindya, MithrandirMage, Crasshopper, Rofti, Repied, David s graﬀ, CBM, David Eppstein, Stassa, LordAnubisBOT, Nadiatalent, Ubermammal, Melcombe, Dtunkelang, Tsourakakis, Ost316, Addbot, Download, Luckas-bot, Tanoshimi,
AnomieBOT, SvartMan, D'ohBot, MGA73bot, MondalorBot, Dinamik-bot, Pbasista, Detian, Vrmpx, Chire, ClueBot NG, FritzFasbinder, Jochen Burghardt, Fedelis4198, Velvel2 and Anonymous: 36
• Zero-dimensional space Source: https://en.wikipedia.org/wiki/Zero-dimensional_space?oldid=676641292 Contributors: The Anome,
Dominus, Tobias Bergemann, Mporter, Paul August, Vipul, Linas, YurikBot, Trovatore, Kompik, Arthur Rubin, That Guy, From That
Show!, SmackBot, Incnis Mrsi, Melchoir, Vina-iwbot~enwiki, Cesium 133, Stotr~enwiki, P2005t, Dp462090, Cydebot, Ntsimp, R'n'B,
Squad51, Trumpet marietta 45750, Anonymous Dissident, Plclark, Lamro, SieBot, Mojoworker, Mitch Ames, Addbot, Numbo3-bot,
Luckas-bot, Yobot, 4th-otaku, Citation bot, IVAN3MAN, Dr. John D. McCarthy, Drusus 0, D.Lazard, ClueBot NG, Helpful Pixie Bot,
ChrisGualtieri, Mark viking, Jjbernardiscool, Noyster and Anonymous: 10

42.5.2

Images

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