Dimension

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Dimension
From Wikipedia, the free encyclopedia

Contents
1

Dimension

1

1.1

In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.1

Dimension of a vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.2

Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.3

Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.1.4

Krull dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.1.5

Lebesgue covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.1.6

Inductive dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.1.7

Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.1.8

Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

In physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2.1

Spatial dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2.2

Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2.3

Additional dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Networks and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

In literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.5

In philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.6

More dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.7

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.7.1

6

1.2

2

Topics by dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.8

References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.9

Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Extended real number line

9

2.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1.1

Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1.2

Measure and integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2

Order and topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.3

Arithmetic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.4

Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.5

Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.6

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

i

ii

CONTENTS
2.7

3

4

5

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Interval (mathematics)

12

3.1

Notations for intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.1.1

Including or excluding endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.1.2

Infinite endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3.1.3

Integer intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3.2

Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3.3

Classification of intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

3.3.1

Intervals of the extended real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

3.4

Properties of intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

3.5

Dyadic intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.6

Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.6.1

Multi-dimensional intervals

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.6.2

Complex intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.7

Topological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.8

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

3.9

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

3.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

Line (geometry)

17

4.1

Definitions versus descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

4.2

Ray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

4.3

Euclidean geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

4.3.1

Cartesian plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

4.3.2

Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

4.3.3

Vector equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

4.3.4

Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

4.3.5

Types of lines

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

4.4

Projective geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.5

Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

4.6

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

4.7

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

4.8

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

4.9

External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

Point (geometry)

26

5.1

Points in Euclidean geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

5.2

Dimension of a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

5.2.1

Vector space dimension

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

5.2.2

Topological dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

5.2.3

Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

CONTENTS

iii

5.3

Geometry without points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

5.4

Point masses and the Dirac delta function

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

5.5

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

5.6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

5.7

External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.8

Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.8.1

Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.8.2

Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.8.3

Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

Chapter 1

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.
1

2

CHAPTER 1. DIMENSION

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.

1.1 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.

1.1.1

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.

1.1.2

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.

1.1. IN MATHEMATICS

1.1.3

3

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]

1.1.4

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.

1.1.5

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".

1.1.6

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.

4

1.1.7

CHAPTER 1. DIMENSION

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]

1.1.8

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.

1.2 In physics
1.2.1

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.)

1.2.2

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.

1.2.3

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]

1.3. NETWORKS AND DIMENSION

5

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.

1.3 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.

1.4 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.

6

CHAPTER 1. DIMENSION

1.5 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.

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

1.7 See also
1.7.1

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)

1.8. REFERENCES

7

• 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

1.8 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

8

CHAPTER 1. DIMENSION

[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.

1.9 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.
• Michio Kaku, Hyperspace, a Scientific Odyssey Through the 10th Dimension, Oxford University Press: 1994.

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

Chapter 2

Extended real number line
“Positive infinity” redirects here. For the band, see Positive Infinity.
In mathematics, the affinely extended real number system is obtained from the real number system R by adding
two elements: +∞ and −∞ (read as positive infinity and negative infinity respectively). These new elements are not
real numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in
the theory of measure and integration. The affinely extended real number system is denoted R or [−∞, +∞] or R U
{−∞, +∞}.
When the meaning is clear from context, the symbol +∞ is often written simply as ∞.

2.1 Motivation
2.1.1

Limits

We often wish to describe the behavior of a function f(x), as either the argument x or the function value f(x) gets
“very big” in some sense. For example, consider the function

f (x) = x−2 .
The graph of this function has a horizontal asymptote at y = 0. Geometrically, as we move farther and farther to the
right along the x-axis, the value of 1/x2 approaches 0. This limiting behavior is similar to the limit of a function at a
real number, except that there is no real number to which x approaches.
By adjoining the elements +∞ and −∞ to R, we allow a formulation of a “limit at infinity” with topological properties
similar to those for R.
To make things completely formal, the Cauchy sequences definition of R allows us to define +∞ as the set of all
sequences of rationals which, for any K>0, from some point on exceed K. We can define −∞ similarly.

2.1.2

Measure and integration

In measure theory, it is often useful to allow sets which have infinite measure and integrals whose value may be
infinite.
Such measures arise naturally out of calculus. For example, in assigning a measure to R that agrees with the usual
length of intervals, this measure must be larger than any finite real number. Also, when considering infinite integrals,
such as

1



dx
x
9

10

CHAPTER 2. EXTENDED REAL NUMBER LINE

the value “infinity” arises. Finally, it is often useful to consider the limit of a sequence of functions, such as
{
2n(1 − nx), if 0 ≤ x ≤ n1
fn (x) =
0,
if n1 < x ≤ 1
Without allowing functions to take on infinite values, such essential results as the monotone convergence theorem and
the dominated convergence theorem would not make sense.

2.2 Order and topological properties
The affinely extended real number system turns into a totally ordered set by defining −∞ ≤ a ≤ +∞ for all a. This
order has the desirable property that every subset has a supremum and an infimum: it is a complete lattice.
This induces the order topology on R. In this topology, a set U is a neighborhood of +∞ if and only if it contains a
set {x : x > a} for some real number a, and analogously for the neighborhoods of −∞. R is a compact Hausdorff
space homeomorphic to the unit interval [0, 1]. Thus the topology is metrizable, corresponding (for a given homeomorphism) to the ordinary metric on this interval. There is no metric which is an extension of the ordinary metric
on R.
With this topology the specially defined limits for x tending to +∞ and −∞, and the specially defined concepts of
limits equal to +∞ and −∞, reduce to the general topological definitions of limits.

2.3 Arithmetic operations
The arithmetic operations of R can be partially extended to R as follows:

a + ∞ = +∞ + a = +∞,
a − ∞ = −∞ + a = −∞,

a ̸= −∞
a ̸= +∞

a · (±∞) = ±∞ · a = ±∞, a ∈ (0, +∞]
a · (±∞) = ±∞ · a = ∓∞, a ∈ [−∞, 0)
a
= 0,
a∈R
±∞
±∞
= ±∞, a ∈ (0, +∞)
a
±∞
= ∓∞, a ∈ (−∞, 0)
a
For exponentiation, see Exponentiation#Limits of powers. Here, "a + ∞" means both "a + (+∞)" and "a − (−∞)",
while "a − ∞" means both "a − (+∞)" and "a + (−∞)".
The expressions ∞ − ∞, 0 × (±∞) and ±∞ / ±∞ (called indeterminate forms) are usually left undefined. These rules
are modeled on the laws for infinite limits. However, in the context of probability or measure theory, 0 × (±∞) is
often defined as 0.
The expression 1/0 is not defined either as +∞ or −∞, because although it is true that whenever f(x) → 0 for a
continuous function f(x) it must be the case that 1/f(x) is eventually contained in every neighborhood of the set {−∞,
+∞}, it is not true that 1/f(x) must tend to one of these points. An example is f(x) = (sin x)/x (as x goes to infinity).
(The modulus | 1/f(x) |, nevertheless, does approach +∞.)

2.4 Algebraic properties
With these definitions R is not a field, nor a ring, and not even a group or semigroup. However, it still has several
convenient properties:

2.5. MISCELLANEOUS

11

• a + (b + c) and (a + b) + c are either equal or both undefined.
• a + b and b + a are either equal or both undefined.
• a × (b × c) and (a × b) × c are either equal or both undefined.
• a × b and b × a are either equal or both undefined
• a × (b + c) and (a × b) + (a × c) are equal if both are defined.
• if a ≤ b and if both a + c and b + c are defined, then a + c ≤ b + c.
• if a ≤ b and c > 0 and both a × c and b × c are defined, then a × c ≤ b × c.
In general, all laws of arithmetic are valid in R as long as all occurring expressions are defined.

2.5 Miscellaneous
Several functions can be continuously extended to R by taking limits. For instance, one defines exp(−∞) = 0, exp(+∞)
= +∞, ln(0) = −∞, ln(+∞) = +∞ etc.
Some discontinuities may additionally be removed. For example, the function 1/x2 can be made continuous (under
some definitions of continuity) by setting the value to +∞ for x = 0, and 0 for x = +∞ and x = −∞. The function
1/x can not be made continuous because the function approaches −∞ as x approaches 0 from below, and +∞ as x
approaches 0 from above.
Compare the real projective line, which does not distinguish between +∞ and −∞. As a result, on one hand a function
may have limit ∞ on the real projective line, while in the affinely extended real number system only the absolute value
of the function has a limit, e.g. in the case of the function 1/x at x = 0. On the other hand
limx→−∞ f (x) and limx→+∞ f (x)
correspond on the real projective line to only a limit from the right and one from the left, respectively, with the full
limit only existing when the two are equal. Thus ex and arctan(x) cannot be made continuous at x = ∞ on the real
projective line.

2.6 See also
• Real projective line, which adds a single, unsigned infinity to the real number line.
• Division by zero
• Extended complex plane
• Improper integral
• Series (mathematics)

2.7 References
• Aliprantis, Charalambos D.; Burkinshaw, Owen (1998), Principles of Real Analysis (3rd ed.), San Diego, CA:
Academic Press, Inc., p. 29, ISBN 0-12-050257-7, MR 1669668
• David W. Cantrell, “Affinely Extended Real Numbers”, MathWorld.

Chapter 3

Interval (mathematics)
This article is about intervals of real numbers and other totally ordered sets. For the most general definition, see
partially ordered set. For other uses, see Interval (disambiguation).
In mathematics, an (real) interval is a set of real numbers with the property that any number that lies between two
numbers in the set is also included in the set. For example, the set of all numbers x satisfying 0 ≤ x ≤ 1 is an interval
which contains 0 and 1, as well as all numbers between them. Other examples of intervals are the set of all real
numbers R , the set of all negative real numbers, and the empty set.
Real intervals play an important role in the theory of integration, because they are the simplest sets whose “size” or
“measure” or “length” is easy to define. The concept of measure can then be extended to more complicated sets of
real numbers, leading to the Borel measure and eventually to the Lebesgue measure.
Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides
guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical approximations,
and arithmetic roundoff.
Intervals are likewise defined on an arbitrary totally ordered set, such as integers or rational numbers. The notation
of integer intervals is considered in the special section below.

3.1 Notations for intervals
The interval of numbers between a and b, including a and b, is often denoted [a, b]. The two numbers are called the
endpoints of the interval. In countries where numbers are written with a decimal comma, a semicolon may be used
as a separator, to avoid ambiguity.

3.1.1

Including or excluding endpoints

To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either
replaced with a parenthesis, or reversed. Both notations are described in International standard ISO 31-11. Thus, in
set builder notation,
(a, b) = ]a, b[ = {x ∈ R | a < x < b},
[a, b) = [a, b[ = {x ∈ R | a ≤ x < b},
(a, b] = ]a, b] = {x ∈ R | a < x ≤ b},
[a, b] = [a, b] = {x ∈ R | a ≤ x ≤ b}.
Note that (a, a), [a, a), and (a, a] each represents the empty set, whereas [a, a] denotes the set {a}. When a > b, all
four notations are usually taken to represent the empty set.
Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation
(a, b) is often used to denote an ordered pair in set theory, the coordinates of a point or vector in analytic geometry
12

3.2. TERMINOLOGY

13

and linear algebra, or (sometimes) a complex number in algebra. That is why Bourbaki introduced the notation ]a,
b[ to denote the open interval.[1] The notation [a, b] too is occasionally used for ordered pairs, especially in computer
science.
Some authors use ]a, b[ to denote the complement of the interval (a, b); namely, the set of all real numbers that are
either less than or equal to a, or greater than or equal to b.

3.1.2

Infinite endpoints

In both styles of notation, one may use an infinite endpoint to indicate that there is no bound in that direction.
Specifically, one may use a = −∞ or b = +∞ (or both). For example, (0, +∞) is the set of positive real numbers also
written ℝ+ , and (−∞, +∞) is the set of real numbers ℝ.
The extended real number line includes −∞ and +∞ as elements. The notations [−∞, b] , [−∞, b) , [a, +∞] , and (a,
+∞] may be used in this context. For example (−∞, +∞] means the extended real numbers excluding only −∞.

3.1.3

Integer intervals

The notation [a .. b] when a and b are integers, or {a .. b}, or just a .. b is sometimes used to indicate the interval
of all integers between a and b, including both. This notation is used in some programming languages; in Pascal, for
example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of
valid indices of an array.
An integer interval that has a finite lower or upper endpoint always includes that endpoint. Therefore, the exclusion
of endpoints can be explicitly denoted by writing a .. b − 1 , a + 1 .. b , or a + 1 .. b − 1. Alternate-bracket notations
like [a .. b) or [a .. b[ are rarely used for integer intervals.

3.2 Terminology
An open interval does not include its endpoints, and is indicated with parentheses. For example (0,1) means greater
than 0 and less than 1. A closed interval includes its endpoints, and is denoted with square brackets. For example
[0,1] means greater than or equal to 0 and less than or equal to 1.
A degenerate interval is any set consisting of a single real number. Some authors include the empty set in this
definition. A real interval that is neither empty nor degenerate is said to be proper, and has infinitely many elements.
An interval is said to be left-bounded or right-bounded if there is some real number that is, respectively, smaller
than or larger than all its elements. An interval is said to be bounded if it is both left- and right-bounded; and is said
to be unbounded otherwise. Intervals that are bounded at only one end are said to be half-bounded. The empty
set is bounded, and the set of all reals is the only interval that is unbounded at both ends. Bounded intervals are also
commonly known as finite intervals.
Bounded intervals are bounded sets, in the sense that their diameter (which is equal to the absolute difference between
the endpoints) is finite. The diameter may be called the length, width, measure, or size of the interval. The size of
unbounded intervals is usually defined as +∞, and the size of the empty interval may be defined as 0 or left undefined.
The centre (midpoint) of bounded interval with endpoints a and b is (a + b)/2, and its radius is the half-length |a −
b|/2. These concepts are undefined for empty or unbounded intervals.
An interval is said to be left-open if and only if it has no minimum (an element that is smaller than all other elements);
right-open if it has no maximum; and open if it has both properties. The interval [0,1) = {x | 0 ≤ x < 1}, for example,
is left-closed and right-open. The empty set and the set of all reals are open intervals, while the set of non-negative
reals, for example, is a right-open but not left-open interval. The open intervals coincide with the open sets of the
real line in its standard topology.
An interval is said to be left-closed if it has a minimum element, right-closed if it has a maximum, and simply closed
if it has both. These definitions are usually extended to include the empty set and to the (left- or right-) unbounded
intervals, so that the closed intervals coincide with closed sets in that topology.
The interior of an interval I is the largest open interval that is contained in I; it is also the set of points in I which are
not endpoints of I. The closure of I is the smallest closed interval that contains I; which is also the set I augmented

14

CHAPTER 3. INTERVAL (MATHEMATICS)

with its finite endpoints.
For any set X of real numbers, the interval enclosure or interval span of X is the unique interval that contains X
and does not properly contain any other interval that also contains X.

3.3 Classification of intervals
The intervals of real numbers can be classified into eleven different types, listed below; where a and b are real numbers,
with a < b :
empty: [b, a] = (a, a) = [a, a) = (a, a] = {} = ∅
degenerate: [a, a] = {a}
proper and bounded:
(a, b) = {x | a < x < b}
[a, b] = {x | a ≤ x ≤ b}
[a, b) = {x | a ≤ x < b}
(a, b] = {x | a < x ≤ b}
left-bounded and right-unbounded:
(a, ∞) = {x | x > a}
[a, ∞) = {x | x ≥ a}
left-unbounded and right-bounded:
(−∞, b) = {x | x < b}
(−∞, b] = {x | x ≤ b}
unbounded at both ends: (−∞, +∞) = R

3.3.1

Intervals of the extended real line

In some contexts, an interval may be defined as a subset of the extended real numbers, the set of all real numbers
augmented with −∞ and +∞.
In this interpretation, the notations [−∞, b] , [−∞, b) , [a, +∞] , and (a, +∞] are all meaningful and distinct. In
particular, (−∞, +∞) denotes the set of all ordinary real numbers, while [−∞, +∞] denotes the extended reals.
This choice affects some of the above definitions and terminology. For instance, the interval (−∞, +∞) = R is closed
in the realm of ordinary reals, but not in the realm of the extended reals.

3.4 Properties of intervals
The intervals are precisely the connected subsets of R . It follows that the image of an interval by any continuous
function is also an interval. This is one formulation of the intermediate value theorem.
The intervals are also the convex subsets of R . The interval enclosure of a subset X ⊆ R is also the convex hull of
X.
The intersection of any collection of intervals is always an interval. The union of two intervals is an interval if and
only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other
(e.g., (a, b) ∪ [b, c] = (a, c] ).
If R is viewed as a metric space, its open balls are the open bounded sets (c + r, c − r), and its closed balls are the
closed bounded sets [c + r, c − r].

3.5. DYADIC INTERVALS

15

Any element x of an interval I defines a partition of I into three disjoint intervals I1 , I2 , I3 : respectively, the elements
of I that are less than x, the singleton [x, x] = {x} , and the elements that are greater than x. The parts I1 and I3 are
both non-empty (and have non-empty interiors) if and only if x is in the interior of I. This is an interval version of
the trichotomy principle.

3.5 Dyadic intervals
A dyadic interval is a bounded real interval whose endpoints are 2jn and j+1
2n , where j and n are integers. Depending
on the context, either endpoint may or may not be included in the interval.
Dyadic intervals have the following properties:
• The length of a dyadic interval is always an integer power of two.
• Each dyadic interval is contained in exactly one dyadic interval of twice the length.
• Each dyadic interval is spanned by two dyadic intervals of half the length.
• If two open dyadic intervals overlap, then one of them is a subset of the other.
The dyadic intervals consequently have a structure that reflects that of an infinite binary tree.
Dyadic intervals are relevant to several areas of numerical analysis, including adaptive mesh refinement, multigrid
methods and wavelet analysis. Another way to represent such a structure is p-adic analysis (for p = 2).[2]

3.6 Generalizations
3.6.1

Multi-dimensional intervals

In many contexts, an n -dimensional interval is defined as a subset of Rn that is the Cartesian product of n intervals,
I = I1 × I2 × · · · × In , one on each coordinate axis.
For n = 2 , this generally defines a rectangle whose sides are parallel to the coordinate axes; for n = 3 , it defines an
axis-aligned rectangular box.
A facet of such an interval I is the result of replacing any non-degenerate interval factor Ik by a degenerate interval
consisting of a finite endpoint of Ik . The faces of I comprise I itself and all faces of its facets. The corners of I
are the faces that consist of a single point of Rn .

3.6.2

Complex intervals

Intervals of complex numbers can be defined as regions of the complex plane, either rectangular or circular.[3]

3.7 Topological algebra
Intervals can be associated with points of the plane and hence regions of intervals can be associated with regions of
the plane. Generally, an interval in mathematics corresponds to an ordered pair (x,y) taken from the direct product R
× R of real numbers with itself. Often it is assumed that y > x. For purposes of mathematical structure, this restriction
is discarded,[4] and “reversed intervals” where y − x < 0 are allowed. Then the collection of all intervals [x,y] can be
identified with the topological ring formed by the direct sum of R with itself where addition and multiplication are
defined component-wise.
The direct sum algebra (R ⊕ R, +, ×) has two ideals, { [x,0] : x ∈ R } and { [0,y] : y ∈ R }. The identity element
of this algebra is the condensed interval [1,1]. If interval [x,y] is not in one of the ideals, then it has multiplicative
inverse [1/x, 1/y]. Endowed with the usual topology, the algebra of intervals forms a topological ring. The group of
units of this ring consists of four quadrants determined by the axes, or ideals in this case. The identity component of
this group is quadrant I.

16

CHAPTER 3. INTERVAL (MATHEMATICS)

Every interval can be considered a symmetric interval around its midpoint. In a reconfiguration published in 1956
by M Warmus, the axis of “balanced intervals” [x, −x] is used along with the axis of intervals [x,x] that reduce to a
point. Instead of the direct sum R ⊕ R , the ring of intervals has been identified[5] with the split-complex number
plane by M. Warmus and D. H. Lehmer through the identification
z = (x + y)/2 + j (x − y)/2.
This linear mapping of the plane, which amounts of a ring isomorphism, provides the plane with a multiplicative
structure having some analogies to ordinary complex arithmetic, such as polar decomposition.

3.8 See also
• Inequality
• Interval graph
• Interval finite element

3.9 References
[1] http://hsm.stackexchange.com/a/193
[2] Kozyrev, Sergey (2002). “Wavelet theory as p-adic spectral analysis”. Izvestiya RAN. Ser. Mat. 66 (2): 149–158.
doi:10.1070/IM2002v066n02ABEH000381. Retrieved 2012-04-05.
[3] Complex interval arithmetic and its applications, Miodrag Petković, Ljiljana Petković, Wiley-VCH, 1998, ISBN 978-3527-40134-5
[4] Kaj Madsen (1979) Review of “Interval analysis in the extended interval space” by Edgar Kaucher from Mathematical
Reviews
[5] D. H. Lehmer (1956) Review of “Calculus of Approximations” from Mathematical Reviews

• T. Sunaga, “Theory of interval algebra and its application to numerical analysis”, In: Research Association of
Applied Geometry (RAAG) Memoirs, Ggujutsu Bunken Fukuy-kai. Tokyo, Japan, 1958, Vol. 2, pp. 29–46
(547-564); reprinted in Japan Journal on Industrial and Applied Mathematics, 2009, Vol. 26, No. 2-3, pp.
126–143.

3.10 External links
• A Lucid Interval by Brian Hayes: An American Scientist article provides an introduction.
• Interval Notation Basics
• Interval computations website
• Interval computations research centers
• Interval Notation by George Beck, Wolfram Demonstrations Project.
• Weisstein, Eric W., “Interval”, MathWorld.

Chapter 4

Line (geometry)
Not to be confused with Curved line.
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e.,

y
4
3
2
1

x
−4

−3

−2

−1

0

1

2

3

4

−1
−2
−3

y=2.0x+1
y=0.5x−1
y=0.5x+1

−4

The red and blue lines on this graph have the same slope (gradient); the red and green lines have the same y-intercept (cross the
y-axis at the same place).

17

18

CHAPTER 4. LINE (GEOMETRY)

A representation of one line segment.

having no curvature) with negligible width and depth. Lines are an idealization of such objects. Until the seventeenth
century, lines were defined like this: “The [straight or curved] line is the first species of quantity, which has only one
dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which
[…] will leave from its imaginary moving some vestige in length, exempt of any width. […] The straight line is that
which is equally extended between its points”[1]
Euclid described a line as “breadthless length” which “lies equally with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed the geometry, which is now called
Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of nineteenth
century (such as non-Euclidean, projective and affine geometry).
In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry
is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose
coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be
an independent object, distinct from the set of points which lie on it.
When a geometry is described by a set of axioms, the notion of a line is usually left undefined (a so-called primitive
object). The properties of lines are then determined by the axioms which refer to them. One advantage to this
approach is the flexibility it gives to users of the geometry. Thus in differential geometry a line may be interpreted
as a geodesic (shortest path between points), while in some projective geometries a line is a 2-dimensional vector
space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for
example, permits physicists to think of the path of a light ray as being a line.
A line segment is a part of a line that is bounded by two distinct end points and contains every point on the line
between its end points. Depending on how the line segment is defined, either of the two end points may or may not
be part of the line segment. Two or more line segments may have some of the same relationships as lines, such as
being parallel, intersecting, or skew, but unlike lines they may be none of these, if they are coplanar and either do not
intersect or are collinear.

4.1 Definitions versus descriptions
All definitions are ultimately circular in nature since they depend on concepts which must themselves have definitions,
a dependence which can not be continued indefinitely without returning to the starting point. To avoid this vicious
circle certain concepts must be taken as primitive concepts; terms which are given no definition.[2] In geometry, it
is frequently the case that the concept of line is taken as a primitive.[3] In those situations where a line is a defined
concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is
a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy.
In a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too
abstract to be dealt with. In this circumstance it is possible that a description or mental image of a primitive notion
is provided to give a foundation to build the notion on which would formally be based on the (unstated) axioms.
Descriptions of this type may be referred to, by some authors, as definitions in this informal style of presentation.
These are not true definitions and could not be used in formal proofs of statements. The “definition” of line in Euclid’s
Elements falls into this category.[4] Even in the case where a specific geometry is being considered (for example,
Euclidean geometry), there is no generally accepted agreement among authors as to what an informal description of
a line should be when the subject is not being treated formally.

4.2 Ray
Given a line and any point A on it, we may consider A as decomposing this line into two parts. Each such part is
called a ray (or half-line) and the point A is called its initial point. The point A is considered to be a member of the

4.3. EUCLIDEAN GEOMETRY

19

ray.[5] Intuitively, a ray consists of those points on a line passing through A and proceeding indefinitely, starting at A,
in one direction only along the line. However, in order to use this concept of a ray in proofs a more precise definition
is required.
Given distinct points A and B, they determine a unique ray with initial point A. As two points define a unique line,
this ray consists of all the points between A and B (including A and B) and all the points C on the line through A and B
such that B is between A and C.[6] This is, at times, also expressed as the set of all points C such that A is not between
B and C.[7] A point D, on the line determined by A and B but not in the ray with initial point A determined by B, will
determine another ray with initial point A. With respect to the AB ray, the AD ray is called the opposite ray.

A

B

C

Ray

Thus, we would say that two different points, A and B, define a line and a decomposition of this line into the disjoint
union of an open segment (A, B) and two rays, BC and AD (the point D is not drawn in the diagram, but is to the left
of A on the line AB). These are not opposite rays since they have different initial points.
In Euclidean geometry two rays with a common endpoint form an angle.
The definition of a ray depends upon the notion of betweenness for points on a line. It follows that rays exist only for
geometries for which this notion exists, typically Euclidean geometry or affine geometry over an ordered field. On
the other hand, rays do not exist in projective geometry nor in a geometry over a non-ordered field, like the complex
numbers or any finite field.
In topology, a ray in a space X is a continuous embedding R+ → X. It is used to define the important concept of end
of the space.

4.3 Euclidean geometry
See also: Euclidean geometry
When geometry was first formalised by Euclid in the Elements, he defined a general line (straight or curved) to be
“breadthless length” with a straight line being a line “which lies evenly with the points on itself”.[8] These definitions
serve little purpose since they use terms which are not, themselves, defined. In fact, Euclid did not use these definitions
in this work and probably included them just to make it clear to the reader what was being discussed. In modern
geometry, a line is simply taken as an undefined object with properties given by axioms,[9] but is sometimes defined
as a set of points obeying a linear relationship when some other fundamental concept is left undefined.
In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (Euclid’s original axioms contained various
flaws which have been corrected by modern mathematicians),[10] a line is stated to have certain properties which relate
it to other lines and points. For example, for any two distinct points, there is a unique line containing them, and any
two distinct lines intersect in at most one point.[11] In two dimensions, i.e., the Euclidean plane, two lines which do
not intersect are called parallel. In higher dimensions, two lines that do not intersect are parallel if they are contained
in a plane, or skew if they are not.
Any collection of finitely many lines partitions the plane into convex polygons (possibly unbounded); this partition is
known as an arrangement of lines.

4.3.1

Cartesian plane

Main article: Linear equation
Lines in a Cartesian plane or, more generally, in affine coordinates, can be described algebraically by linear equations.
In two dimensions, the equation for non-vertical lines is often given in the slope-intercept form:

20

CHAPTER 4. LINE (GEOMETRY)

y = mx + b
where:
m is the slope or gradient of the line.
b is the y-intercept of the line.
x is the independent variable of the function y = f(x).
The slope of the line through points A(xₐ, yₐ) and B(x , y ), when xₐ ≠ x , is given by m = (y − yₐ)/(x − xₐ) and the
equation of this line can be written y = m(x − xₐ) + yₐ.
In R2 , every line L (including vertical lines) is described by a linear equation of the form
L = {(x, y) | ax + by = c}
with fixed real coefficients a, b and c such that a and b are not both zero. Using this form, vertical lines correspond
to the equations with b = 0.
There are many variant ways to write the equation of a line which can all be converted from one to another by algebraic
manipulation. These forms (see Linear equation for other forms) are generally named by the type of information (data)
about the line that is needed to write down the form. Some of the important data of a line is its slope, x-intercept,
known points on the line and y-intercept.
The equation of the line passing through two different points P0 = (x0 , y0 ) and P1 = (x1 , y1 ) may be written as
(y − y0 )(x1 − x0 ) = (y1 − y0 )(x − x0 )
If x0 ≠ x1 , this equation may be rewritten as
y = (x − x0 )

y1 − y0
+ y0
x1 − x0

or
y1 − y0
x1 y0 − x0 y1
+
.
x1 − x0
x1 − x0
In three dimensions, lines can not be described by a single linear equation, so they are frequently described by
parametric equations:

y=x

x = x0 + at
y = y0 + bt
z = z0 + ct
where:
x, y, and z are all functions of the independent variable t which ranges over the real numbers.
(x0 , y0 , z0 ) is any point on the line.
a, b, and c are related to the slope of the line, such that the vector (a, b, c) is parallel to the line.
They may also be described as the simultaneous solutions of two linear equations
a1 x + b1 y + c1 z − d1 = 0
a2 x + b2 y + c2 z − d2 = 0
such that (a1 , b1 , c1 ) and (a2 , b2 , c2 ) are not proportional (the relations a1 = ta2 , b1 = tb2 , c1 = tc2 imply t = 0).
This follows since in three dimensions a single linear equation typically describes a plane and a line is what is common
to two distinct intersecting planes.

4.3. EUCLIDEAN GEOMETRY

21

Normal form
The normal segment for a given line is defined to be the line segment drawn from the origin perpendicular to the line.
This segment joins the origin with the closest point on the line to the origin. The normal form of the equation of a
straight line on the plane is given by:

y sin θ + x cos θ − p = 0,
where θ is the angle of inclination of the normal segment (the oriented angle from the unit vector of the x axis to this
segment), and p is the (positive) length of the normal segment. The normal form can be derived from the general
form by dividing all of the coefficients by

|c| √ 2
a + b2 .
−c
This form is also called the Hesse normal form,[12] after the German mathematician Ludwig Otto Hesse.
Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite
parameters, θ and p, to be specified. Note that if p > 0, then θ is uniquely defined modulo 2π. On the other hand,
if the line is through the origin (c = 0, p = 0), one drops the |c|/(−c) term to compute sinθ and cosθ, and θ is only
defined modulo π.

4.3.2

Polar coordinates

In polar coordinates on the Euclidean plane the slope-intercept form of the equation of a line is expressed as:

r=

mr cos θ + b
,
sin θ

where m is the slope of the line and b is the y-intercept. When θ = 0 the graph will be undefined. The equation can
be rewritten to eliminate discontinuities in this manner:

r sin θ = mr cos θ + b.
In polar coordinates on the Euclidean plane, the intercept form of the equation of a line that is non-horizontal, nonvertical, and does not pass through pole may be expressed as,

r=

cos θ
xo

1
+

sin θ
yo

where xo and yo represent the x and y intercepts respectively. The above equation is not applicable for vertical and
horizontal lines because in these cases one of the intercepts does not exist. Moreover, it is not applicable on lines
passing through the pole since in this case, both x and y intercepts are zero (which is not allowed here since xo and
yo are denominators). A vertical line that doesn't pass through the pole is given by the equation

r cos θ = xo .

22

CHAPTER 4. LINE (GEOMETRY)

Similarly, a horizontal line that doesn't pass through the pole is given by the equation

r sin θ = yo .
The equation of a line which passes through the pole is simply given as:

θ=m
where m is the slope of the line.

4.3.3

Vector equation

The vector equation of the line through points A and B is given by r = OA + λAB (where λ is a scalar).
If a is vector OA and b is vector OB, then the equation of the line can be written: r = a + λ(b − a).
A ray starting at point A is described by limiting λ. One ray is obtained if λ ≥ 0, and the opposite ray comes from λ
≤ 0.

4.3.4

Euclidean space

In three-dimensional space, a first degree equation in the variables x, y, and z defines a plane, so two such equations,
provided the planes they give rise to are not parallel, define a line which is the intersection of the planes. More
generally, in n-dimensional space n−1 first-degree equations in the n coordinate variables define a line under suitable
conditions.
In more general Euclidean space, Rn (and analogously in every other affine space), the line L passing through two
different points a and b (considered as vectors) is the subset

L = {(1 − t) a + t b | t ∈ R}
The direction of the line is from a (t = 0) to b (t = 1), or in other words, in the direction of the vector b − a. Different
choices of a and b can yield the same line.
Collinear points
Main article: Collinearity
Three points are said to be collinear if they lie on the same line. Three points usually determine a plane, but in the
case of three collinear points this does not happen.
In affine coordinates, in n-dimensional space the points X=(x1 , x2 , ..., x ), Y=(y1 , y2 , ..., y ), and Z=(z1 , z2 , ..., z )
are collinear if the matrix

1
1
1

x1
y1
z1

x2
y2
z2

...
...
...


xn
yn 
zn

has a rank less than 3. In particular, for three points in the plane (n = 2), the above matrix is square and the points
are collinear if and only if its determinant is zero.
Equivalently for three points in a plane, the points are collinear if and only if the slope between one pair of points
equals the slope between any other pair of points (in which case the slope between the remaining pair of points will

4.3. EUCLIDEAN GEOMETRY

23

equal the other slopes). By extension, k points in a plane are collinear if and only if any (k–1) pairs of points have
the same pairwise slopes.
In Euclidean geometry, the Euclidean distance d(a,b) between two points a and b may be used to express the collinearity between three points by:[13][14]
The points a, b and c are collinear if and only if d(x,a) = d(c,a) and d(x,b) = d(c,b) implies x=c.
However there are other notions of distance (such as the Manhattan distance) for which this property is not true.
In the geometries where the concept of a line is a primitive notion, as may be the case in some synthetic geometries,
other methods of determining collinearity are needed.

4.3.5

Types of lines

In a sense,[15] all lines in Euclidean geometry are equal, in that, without coordinates, one can not tell them apart from
one another. However, lines may play special roles with respect to other objects in the geometry and be divided into
types according to that relationship. For instance, with respect to a conic (a circle, ellipse, parabola, or hyperbola),
lines can be:
• tangent lines, which touch the conic at a single point;
• secant lines, which intersect the conic at two points and pass through its interior;
• exterior lines, which do not meet the conic at any point of the Euclidean plane; or
• a directrix, whose distance from a point helps to establish whether the point is on the conic.
In the context of determining parallelism in Euclidean geometry, a transversal is a line that intersects two other lines
that may or not be parallel to each other.
For more general algebraic curves, lines could also be:
• i-secant lines, meeting the curve in i points counted without multiplicity, or
• asymptotes, which a curve approaches arbitrarily closely without touching it.
With respect to triangles we have:
• the Euler line,
• the Simson lines, and
• central lines.
For a convex quadrilateral with at most two parallel sides, the Newton line is the line that connects the midpoints of
the two diagonals.
For a hexagon with vertices lying on a conic we have the Pascal line and, in the special case where the conic is a pair
of lines, we have the Pappus line.
Parallel lines are lines in the same plane that never cross. Intersecting lines share a single point in common. Coincidental lines coincide with each other—every point that is on either one of them is also on the other.
Perpendicular lines are lines that intersect at right angles.
In three-dimensional space, skew lines are lines that are not in the same plane and thus do not intersect each other.

24

CHAPTER 4. LINE (GEOMETRY)

4.4 Projective geometry
Main article: Projective geometry
In many models of projective geometry, the representation of a line rarely conforms to the notion of the “straight
curve” as it is visualised in Euclidean geometry. In elliptic geometry we see a typical example of this.[16] In the
spherical representation of elliptic geometry, lines are represented by great circles of a sphere with diametrically
opposite points identified. In a different model of elliptic geometry, lines are represented by Euclidean planes passing
through the origin. Even though these representations are visually distinct, they satisfy all the properties (such as, two
points determining a unique line) that make them suitable representations for lines in this geometry.

4.5 Geodesics
Main article: geodesics
The “shortness” and “straightness” of a line, interpreted as the property that the distance along the line between any
two of its points is minimized (see triangle inequality), can be generalized and leads to the concept of geodesics in
metric spaces.

4.6 See also
• Line coordinates
• Line segment
• Curve
• Locus
• Distance from a point to a line
• Distance between two lines
• Affine function
• Incidence (geometry)
• Plane (geometry)
• Rectilinear

4.7 Notes
[1] In (rather old) French: “La ligne est la première espece de quantité, laquelle a tant seulement une dimension à sçavoir
longitude, sans aucune latitude ni profondité, & n'est autre chose que le flux ou coulement du poinct, lequel […] laissera
de son mouvement imaginaire quelque vestige en long, exempt de toute latitude. […] La ligne droicte est celle qui est
également estenduë entre ses poincts.” Pages 7 and 8 of Les quinze livres des éléments géométriques d'Euclide Megarien,
traduits de Grec en François, & augmentez de plusieurs figures & demonstrations, avec la corrections des erreurs commises
és autres traductions, by Pierre Mardele, Lyon, MDCXLV (1645).
[2] Coxeter 1969, pg. 4
[3] Faber 1983, pg. 95
[4] Faber 1983, pg. 95
[5] On occasion we may consider a ray without its initial point. Such rays are called open rays, in contrast to the typical ray
which would be said to be closed.

4.8. REFERENCES

25

[6] Wylie, Jr. 1964, pg. 59, Definition 3
[7] Pedoe 1988, pg. 2
[8] Faber, Appendix A, p. 291.
[9] Faber, Part III, p. 95.
[10] Faber, Part III, p. 108.
[11] Faber, Appendix B, p. 300.
[12] Bôcher, Maxime (1915), Plane Analytic Geometry: With Introductory Chapters on the Differential Calculus, H. Holt, p. 44.
[13] Alessandro Padoa, Un nouveau système de définitions pour la géométrie euclidienne, International Congress of Mathematicians, 1900
[14] Bertrand Russell, The Principles of Mathematics, p.410
[15] Technically, the collineation group acts transitively on the set of lines.
[16] Faber, Part III, p. 108.

4.8 References
• Coxeter, H.S.M (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons, ISBN 0-47118283-4
• Faber, Richard L. (1983). Foundations of Euclidean and Non-Euclidean Geometry. New York: Marcel Dekker.
ISBN 0-8247-1748-1.
• Pedoe, Dan (1988), Geometry: A Comprehensive Course, Mineola, NY: Dover, ISBN 0-486-65812-0
• Wylie, Jr., C. R. (1964), Foundations of Geometry, New York: McGraw-Hill, ISBN 0-07-072191-2

4.9 External links
• Hazewinkel, Michiel, ed. (2001), “Line (curve)", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608010-4
• Weisstein, Eric W., “Line”, MathWorld.
• Equations of the Straight Line at Cut-the-Knot
• Citizendium

Chapter 5

Point (geometry)
In modern mathematics, a point refers usually to an element of some set called a space.
More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built. Being
a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is
defined only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not have
any length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a point
is meant to capture the notion of a unique location in Euclidean space.

5.1 Points in Euclidean geometry
Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclid
originally defined the point as “that which has no part”. In two-dimensional Euclidean space, a point is represented
by an ordered pair (x, y) of numbers, where the first number conventionally represents the horizontal and is often
denoted by x, and the second number conventionally represents the vertical and is often denoted by y. This idea is
easily generalized to three-dimensional Euclidean space, where a point is represented by an ordered triplet (x, y, z)
with the additional third number representing depth and often denoted by z. Further generalizations are represented
by an ordered tuplet of n terms, (a1 , a2 , … , an) where n is the dimension of the space in which the point is located.
Many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points; As an example, a line is an infinite set of points of the form
L={(a1 ,a2 ,...an )|a1 c1 +a2 c2 +...an cn =d} , where c1 through cn and d are constants and n is the dimension of the space.
Similar constructions exist that define the plane, line segment and other related concepts. By the way, a degenerate
line segment consists of only one point.
In addition to defining points and constructs related to points, Euclid also postulated a key idea about points; he
claimed that any two points can be connected by a straight line. This is easily confirmed under modern expansions
of Euclidean geometry, and had lasting consequences at its introduction, allowing the construction of almost all the
geometric concepts of the time. However, Euclid’s postulation of points was neither complete nor definitive, as he
occasionally assumed facts about points that did not follow directly from his axioms, such as the ordering of points
on the line or the existence of specific points. In spite of this, modern expansions of the system serve to remove these
assumptions.

5.2 Dimension of a point
There are several inequivalent definitions of dimension in mathematics. In all of the common definitions, a point is
0-dimensional.
26

5.2. DIMENSION OF A POINT

27

2

1

-2

-1

1

2

-1

-2
A finite set of points (blue) in two-dimensional Euclidean space.

5.2.1

Vector space dimension

Main article: Dimension (vector space)
The dimension of a vector space is the maximum size of a linearly independent subset. In a vector space consisting
of a single point (which must be the zero vector 0), there is no linearly independent subset. The zero vector is not
itself linearly independent, because there is a non trivial linear combination making it zero: 1 · 0 = 0 .

5.2.2

Topological dimension

Main article: Lebesgue covering dimension
The topological dimension of a topological space X is defined to be the minimum value of n, such that every finite
open cover A of X admits a finite open cover B of X which refines A in which no point is included in more than n+1
elements. If no such minimal n exists, the space is said to be of infinite covering dimension.
A point is zero-dimensional with respect to the covering dimension because every open cover of the space has a
refinement consisting of a single open set.

28

CHAPTER 5. POINT (GEOMETRY)

5.2.3

Hausdorff dimension

Let X be a metric space. If S ⊂ X and d ∈ [0, ∞), the d-dimensional Hausdorff content of S is the infimum of the
set of numbers δ ≥ 0 such ∑
that there is some (indexed) collection of balls {B(xi , ri ) : i ∈ I} covering S with ri > 0
for each i ∈ I that satisfies i∈I rid < δ .
The Hausdorff dimension of X is defined by

d
dimH (X) := inf{d ≥ 0 : CH
(X) = 0}.

A point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius.

5.3 Geometry without points
Although the notion of a point is generally considered fundamental in mainstream geometry and topology, there are
some systems that forgo it, e.g. noncommutative geometry and pointless topology. A “pointless” or “pointfree” space
is defined not as a set, but via some structure (algebraic or logical respectively) which looks like a well-known function
space on the set: an algebra of continuous functions or an algebra of sets respectively. More precisely, such structures
generalize well-known spaces of functions in a way that the operation “take a value at this point” may not be defined.
A further tradition starts from some books of A. N. Whitehead in which the notion of region is assumed as a primitive
together with the one of inclusion or connection.

5.4 Point masses and the Dirac delta function
Main article: Dirac delta function
Often in physics and mathematics, it is useful to think of a point as having non-zero mass or charge (this is especially
common in classical electromagnetism, where electrons are idealized as points with non-zero charge). The Dirac
delta function, or δ function, is (informally) a generalized function on the real number line that is zero everywhere
except at zero, with an integral of one over the entire real line.[1][2][3] The delta function is sometimes thought of as
an infinitely high, infinitely thin spike at the origin, with total area one under the spike, and physically represents an
idealized point mass or point charge.[4] It was introduced by theoretical physicist Paul Dirac. In the context of signal
processing it is often referred to as the unit impulse symbol (or function).[5] Its discrete analog is the Kronecker
delta function which is usually defined on a finite domain and takes values 0 and 1.

5.5 See also
• Accumulation point
• Affine space
• Boundary point
• Critical point
• Cusp
• Foundations of geometry
• Position (geometry)
• Pointwise
• Singular point of a curve

5.6. REFERENCES

29

5.6 References
[1] Dirac 1958, §15 The δ function, p. 58
[2] Gel'fand & Shilov 1968, Volume I, §§1.1, 1.3
[3] Schwartz 1950, p. 3
[4] Arfken & Weber 2000, p. 84
[5] Bracewell 1986, Chapter 5

• Clarke, Bowman, 1985, "Individuals and Points," Notre Dame Journal of Formal Logic 26: 61–75.
• De Laguna, T., 1922, “Point, line and surface as sets of solids,” The Journal of Philosophy 19: 449–61.
• Gerla, G., 1995, "Pointless Geometries" in Buekenhout, F., Kantor, W. eds., Handbook of incidence geometry:
buildings and foundations. North-Holland: 1015–31.
• Whitehead A. N., 1919. An Enquiry Concerning the Principles of Natural Knowledge. Cambridge Univ. Press.
2nd ed., 1925.
• --------, 1920. The Concept of Nature. Cambridge Univ. Press. 2004 paperback, Prometheus Books. Being
the 1919 Tarner Lectures delivered at Trinity College.
• --------, 1979 (1929). Process and Reality. Free Press.

5.7 External links
• Definition of Point with interactive applet
• Points definition pages, with interactive animations that are also useful in a classroom setting. Math Open
Reference
• Point at PlanetMath.org.
• Weisstein, Eric W., “Point”, MathWorld.

30

CHAPTER 5. POINT (GEOMETRY)

5.8 Text and image sources, contributors, and licenses
5.8.1

Text

• Dimension Source: https://en.wikipedia.org/wiki/Dimension?oldid=675219095 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, 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, LariAnn, Gbaor, Andyo2000,
Jmath666, Wenli, HX Aeternus, Wolfrock, 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, Inc ru, 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, Jay-Sebastos, L Kensington, Bomazi, BioPupil, RockMagnetist, 28bot,
Isocliff, Khestwol, ClueBot NG, Wcherowi, Thekk2007, Lanthanum-138, 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,
I3roly, Anonymous-232, Brandon Ernst, Kylejaylee, Chuluojun, Prachi.apomr, Loraof, BakedLikaBiscuit, Inkanyamba, Knight victor,
Srednuas Lenoroc and Anonymous: 481
• Extended real number line Source: https://en.wikipedia.org/wiki/Extended_real_number_line?oldid=671639548 Contributors: AxelBoldt, B4hand, Patrick, Michael Hardy, TakuyaMurata, Loren Rosen, Revolver, Rbraunwa, Dysprosia, Fibonacci, Robbot, Tobias Bergemann, Giftlite, Dbenbenn, Inter, Stevenmattern, Paul August, RoyBoy, Kevin Lamoreau, Eric Kvaalen, Keenan Pepper, ABCD, Sligocki,
Schapel, David W. Cantrell, Linas, Georgia guy, Isnow, DVdm, RussBot, Trovatore, Saric, Cojoco, SmackBot, Eskimbot, Skylarkk,
Ser Amantio di Nicolao, Mr Stephen, EdC~enwiki, Wjejskenewr, Zero sharp, CRGreathouse, FilipeS, Xantharius, Epbr123, Dugwiki,
P64, David Eppstein, STBotD, TXiKiBoT, A4bot, DumZiBoT, SilvonenBot, Addbot, Some jerk on the Internet, LaaknorBot, Yobot,
Boleyn3, HRoestBot, Thinking of England, Schubi87, Distortiondude, Sheeana, Rnddim, ZéroBot, SporkBot, Brad7777, Limit-theorem,
Connorpark and Anonymous: 44
• Interval (mathematics) Source: https://en.wikipedia.org/wiki/Interval_(mathematics)?oldid=674865208 Contributors: Zundark, Ed
Poor, JeLuF, Patrick, Michael Hardy, Andres, Bjcairns, Charles Matthews, Dcoetzee, Dysprosia, Jitse Niesen, OlivierM, McKay, Robbot,
Ruinia, Ojigiri~enwiki, Ambarish, Tobias Bergemann, Alan Liefting, Tosha, Giftlite, Markus Krötzsch, MSGJ, Markus Kuhn, Gareth
Wyn, Jorge Stolfi, Gazibara, Sam Hocevar, PhotoBox, Mormegil, Paul August, Rgdboer, Liberatus, EmilJ, Jpgordon, Teorth, Jsoulie,
Jumbuck, Burn, Krellion, Oleg Alexandrov, Simetrical, Mindmatrix, Ikescs, OdedSchramm, Rejnal, Theboywonder, Dpr, Dpv, Salix
alba, Mathbot, Greg321, AttishOculus, Gurch, Dmitry-kazakov, Chobot, YurikBot, Hairy Dude, RussBot, Michael Slone, Stephenb,
Gaius Cornelius, PaulGarner, Scilicet, GrinBot~enwiki, JJL, SmackBot, Incnis Mrsi, Tomáš Petříček, SmartGuy Old, Silly rabbit, Nbarth,
DHN-bot~enwiki, Cybercobra, SashatoBot, Cronholm144, Amine Brikci N, Pet-ro, Eassin, Ylloh, CRGreathouse, Woudloper, CBM, He
Who Is, Thijs!bot, Epbr123, Jojan, Pjvpjv, Marek69, NERIUM, Opelio, JAnDbot, Martinkunev, Nyq, Albmont, Email4mobile, David
Eppstein, Anaxial, J.delanoy, Pharaoh of the Wizards, Extransit, Redrad, NewEnglandYankee, Cometstyles, DavidCBryant, Hulten, TheNewPhobia, Pleasantville, Philip Trueman, TXiKiBoT, Enviroboy, Insanity Incarnate, Dmcq, Monty845, Cowlinator, Quietbritishjim,
SieBot, ToePeu.bot, Nathan B. Kitchen, Ezh, Lightmouse, Skeptical scientist, Msrasnw, Anchor Link Bot, ClueBot, Rumping, Marino-slo,
The Thing That Should Not Be, Idleloop~enwiki, CptCutLess, Otolemur crassicaudatus, Excirial, Bremerenator, Hans Adler, SchreiberBike, Ottawa4ever, Kikos, SoxBot III, Knopfkind, SilvonenBot, Addbot, Mammothx, LaaknorBot, Glane23, AndersBot, DavidBParker,

5.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

31

Jasper Deng, 5 albert square, Tide rolls, Zorrobot, Luckas-bot, Yobot, Vs64vs, JorgeFierro, Allent511, AnomieBOT, Qdinar, AdjustShift,
RandomAct, Flewis, Xelnx, ArthurBot, Pownuk, LilHelpa, Obersachsebot, Capricorn42, SteveWoolf, TonyHagale, Charvest, Aghajanpour, Shadowjams, LucienBOT, Calmer Waters, ItsZippy, Sumone10154, DARTH SIDIOUS 2, Woogee, Hyarmendacil, Noommos,
Jowa fan, EmausBot, K6ka, John Cline, Quondum, Wayne Slam, Sassospicco, Mayur, Wikiloop, Bean49Bot, DASHBotAV, 28bot, ClueBot NG, Jack Greenmaven, Wcherowi, Tideflat, Widr, Vibhijain, Helpful Pixie Bot, ‫اقرأ‬, Webclient101, Stephan Kulla, I am One of
Many, Theopolito, Ginsuloft and Anonymous: 204
• Line (geometry) Source: https://en.wikipedia.org/wiki/Line_(geometry)?oldid=670299951 Contributors: Zundark, XJaM, William Avery, Stevertigo, Patrick, Liftarn, Tango, Dcljr, AugPi, Dcoetzee, Furrykef, SEWilco, Mazin07, Fredrik, Altenmann, Mikalaari, Stewartadcock, Carnildo, Tobias Bergemann, Giftlite, BenFrantzDale, Tom harrison, No Guru, Cantus, Python eggs, Lakefall~enwiki, H
Padleckas, Tomruen, Zfr, Neutrality, Zeman, PhotoBox, Ta bu shi da yu, Discospinster, Rich Farmbrough, Paul August, El C, Edwinstearns, Ascorbic, Rgdboer, Longhair, La goutte de pluie, James Foster, Obradovic Goran, Zaraki~enwiki, Tsirel, Alansohn, Anthony
Appleyard, Free Bear, Qrc, Andrewpmk, Burn, Velella, Frankman, HenkvD, Max Naylor, Oleg Alexandrov, Gatewaycat, Linas, Btyner,
Enzo Aquarius, Juan Marquez, Ravidreams, PlatypeanArchcow, Mathbot, Wars, Fresheneesz, Srleffler, CiaPan, Rija, DVdm, Algebraist,
Wavelength, Splash, David R. Ingham, Leutha, Dbfirs, Cheeser1, Botteville, Ms2ger, Closedmouth, ArielGold, Brentt, Minnesota1, Sardanaphalus, Tttrung, SmackBot, RDBury, Adam majewski, Diggers2004, Incnis Mrsi, Melchoir, WookieInHeat, Yamaguchi , Gilliam,
JAn Dudík, Octahedron80, Nbarth, Maxstokols, Gypsyheart, Can't sleep, clown will eat me, Tamfang, Ioscius, Talmage, Midnightcomm, Cybercobra, Jiddisch~enwiki, Nplus~enwiki, Ybact, Bjankuloski06en~enwiki, CredoFromStart, IronGargoyle, Timmeh, Makyen,
Iridescent, Newone, Tpl, Courcelles, Blahstickman, Baqu11, Vaughan Pratt, CRGreathouse, Jackzhp, Nczempin, Black and White,
MarsRover, Gregbard, Equendil, WillowW, Mato, Gogo Dodo, Dr.enh, Manfroze, Xantharius, AbcXyz, Icep, Escarbot, AntiVandalBot,
John.d.page, Salgueiro~enwiki, JAnDbot, Sangwinc, Instinct, GurchBot, Snowolfd4, Tarif Ezaz, Bongwarrior, VoABot II, JamesBWatson,
28421u2232nfenfcenc, David Eppstein, DerHexer, MartinBot, BetBot~enwiki, Treyd500, Owsteele, Tgeairn, J.delanoy, Mike.lifeguard,
Jaxha, Policron, VolkovBot, Orphic, Oktalist, Am Fiosaigear~enwiki, KevinTR, Anonymous Dissident, Javed666, LeaveSleaves, Terwilleger, Temporaluser, Nssbm117, Symane, SieBot, Tresiden, Gerakibot, Lucasbfrbot, Arthur81~enwiki, Hxhbot, Paolo.dL, ScAvenger lv,
Oxymoron83, Ubermammal, BenoniBot~enwiki, MegaBrutal, JohnnyMrNinja, Anchor Link Bot, Gon56, Hariva, DEMcAdams, ClueBot, Justin W Smith, The Thing That Should Not Be, General Epitaph, Farras Octara, CounterVandalismBot, MARKELLOS, Iohannes
Animosus, Hans Adler, Smarkflea, Egmontaz, Kiensvay, Mitch Ames, WikHead, WikiDao, Torchflame, Sparky568, Hunter Kahn, D.M.
from Ukraine, Thebestofall007, Addbot, Landon1980, Kongr43gpen, Fgnievinski, Ronhjones, CanadianLinuxUser, MrOllie, Download,
Glane23, Favonian, Tide rolls, Վազգեն, Bananas321, Luckas-bot, Yobot, Fraggle81, Ciphers, Götz, Piano non troppo, Roux-HG, Xqbot,
Anschmid13, Amaury, BoomerAB, Gamewinner375, FrescoBot, StaticVision, Sae1962, DrilBot, AwesomeHersh, Achim1999, MastiBot, SpaceFlight89, Jujutacular, Rausch, Lapasotka, Vrenator, Duoduoduo, Distortiondude, Onel5969, TjBot, EmausBot, Zerkroz, Rapture1420, JSquish, AmigoDoPaulo, ZéroBot, Josve05a, Dffgd, Alpha Quadrant (alt), Quondum, D.Lazard, L Kensington, Peter Karlsen,
Kizzerdalz, GrayFullbuster, 28bot, Nirakka, ClueBot NG, Wcherowi, Satellizer, Taty711, Wikiinvictoriabc, Kelgog1, MEPRIYANSHU,
Kangaroopower, Allenjambalaya, Joydeep, Ummsomeone, MannertB, GoShow, JYBot, Dexbot, Lugia2453, JPaestpreornJeolhlna, Tentinator, Bg9989, Tarpuq, Rabin.jcet13, WillemienH, Loraof, Dtaylor369, Warepig, Tommygeiss and Anonymous: 305
• Point (geometry) Source: https://en.wikipedia.org/wiki/Point_(geometry)?oldid=670007931 Contributors: The Anome, William Avery,
Michael Hardy, Fred Bauder, Aarchiba, Andres, Revolver, Charles Matthews, Dysprosia, Maximus Rex, Furrykef, Robbot, Kuszi, Giftlite,
BenFrantzDale, MrMambo, Knutux, LiDaobing, Antandrus, Joseph Myers, Pmanderson, Neutrality, Freakofnurture, Discospinster, Paul
August, Elwikipedista~enwiki, Rgdboer, Bobo192, Smalljim, PWilkinson, Sebastian Goll, Alansohn, Oleg Alexandrov, Linas, Madmardigan53, Bkkbrad, Unixer, Hdante, MassGalactusUniversum, Magister Mathematicae, Jobnikon, Tlroche, Salix alba, MikeJ9919,
Sango123, Ravidreams, FlaBot, Chobot, Roboto de Ajvol, YurikBot, Wavelength, Trovatore, Bota47, Lt-wiki-bot, Arthur Rubin, Profero, Sardanaphalus, SmackBot, RDBury, Benjaminb, Incnis Mrsi, Bggoldie~enwiki, Gilliam, Anwar saadat, TimBentley, Octahedron80,
Baronnet, Darth Panda, JustUser, Demoeconomist, Cybercobra, SashatoBot, Vriullop, JoshuaZ, Muadd, Laurens-af, Newone, Vanisaac,
CmdrObot, MarsRover, TheTito, Gregbard, Grandexandi, Doctormatt, Thijs!bot, Futurebird, Porqin, Seaphoto, John.d.page, Salgueiro~enwiki,
JAnDbot, Hut 8.5, Moralist, Magioladitis, VoABot II, Twsx, R'n'B, Pbroks13, J.delanoy, Kimse, SHAN3, Maurice Carbonaro, Drahgo,
Bobianite, Barneca, Philip Trueman, TXiKiBoT, Olly150, Siddthekidd, Falcon8765, Symane, Dogah, SieBot, Tiddly Tom, ToePeu.bot,
Flyer22, Martarius, Dom96, ClueBot, WDavis1911, Liempt, Alexbot, IJA, Addbot, Some jerk on the Internet, Fgnievinski, LaaknorBot, Numbo3-bot, Romanskolduns, Angrysockhop, Math Champion, Luckas-bot, Yobot, Fraggle81, TaBOT-zerem, Reindra, EricWester, AnomieBOT, Rubinbot, Galoubet, Gowr, ArthurBot, Xqbot, Aiuw, RibotBOT, Brayan Jaimes, Aaron Kauppi, MacMan4891,
BoomerAB, Pinethicket, Piandcompany, Distortiondude, EmausBot, JSquish, AmigoDoPaulo, ZéroBot, D.Lazard, Card Zero, Paulmiko, Chewings72, ClueBot NG, Wcherowi, SusikMkr, Braincricket, Usageunit, Rm1271, David.karpay, Pbierre, Sriharsh1234, Brirush,
Bg9989, Wingconeel, Muhamad ittal, Akwillis and Anonymous: 123

5.8.2

Images

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• Spherical_Coordinates_(Colatitude,_Longitude).svg Original artist: Spherical_Coordinates_(Colatitude,_Longitude).svg: Inductiveload

32

CHAPTER 5. POINT (GEOMETRY)

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domain Contributors: Own work Original artist: Andeggs
• File:Coord_XY.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/49/Coord_XY.svg License: Public domain Contributors: Own work Original artist: Andeggs
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• File:Squarecubetesseract.png Source: https://upload.wikimedia.org/wikipedia/commons/2/25/Squarecubetesseract.png License: CC
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providing “attribution is given to Robert Webb’s Great Stella software as the creator of this image along with a link to the website:
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• File:Tesseract.gif Source: https://upload.wikimedia.org/wikipedia/commons/5/55/Tesseract.gif License: Public domain Contributors: ?
Original artist: ?
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5.8.3

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