From Wikipedia, the free encyclopedia
From left to right, the
square, the
cube, and the
tesseract. The square is bounded by 1dimensional lines, the cube by 2dimensional areas, and the tesseract by 3dimensional volumes. A
projection of the cube is given since it is viewed on a twodimensional screen. The same applies to the tesseract, which additionally can only be shown as a projection even in threedimensional space.
A diagram showing the first four spatial dimensions.
In
mathematics and
physics, the
dimension of a
space or
object is informally defined as the minimum number of
coordinates needed to specify each
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. 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, to locate a point on the surface of a sphere you need both its
latitude and its
longitude). The inside of a
cube, a cylinder or a sphere is threedimensional because three coordinates are needed to locate a point within these spaces.
In mathematics
In mathematics, the dimension of an object is an intrinsic property, independent of the space in which the object may happen to be embedded. For example: a point on the
unit circle in the plane can be specified by two
Cartesian coordinates but one can make do with a single coordinate (the
polar coordinate angle), so the circle is 1dimensional even though it exists in the 2dimensional plane. This
intrinsic notion of dimension is one of the chief ways in which the mathematical notion of dimension differs from its common usages.
The dimension of
Euclidean nspace E^{ n} is
n. When trying to generalize to other types of spaces, one is faced with the question “what makes
E^{ n} ndimensional?" One answer is that to cover a fixed ball in
E^{ n} 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, one may observe that the boundary of a ball in
E^{ n} looks locally like
E^{ n − 1} and this leads to the notion of the
inductive dimension. While these notions agree on
E^{ n}, they turn out to be different when one looks at more general spaces.
A
tesseract is an example of a fourdimensional object. Whereas outside of 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."
The rest of this section examines some of the more important mathematical definitions of the dimensions.
Dimension of a 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.
Manifolds
A
connected topological
manifold is
locally homeomorphic to Euclidean
nspace, 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.
The theory of manifolds, in the field of
geometric topology, is characterized by the way dimensions 1 and 2 are relatively elementary, the
highdimensional 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.
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 we write 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 we write dim
X = ∞. Note also that we say
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".
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 0dimensional. By dragging a 0dimensional object in some direction, one obtains a 1dimensional object. By dragging a 1dimensional object in a
new direction, one obtains a 2dimensional object. In general one obtains an
n+1dimensional 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.
Hausdorff dimension
For sets which are of a complicated structure, especially
fractals, the
Hausdorff dimension is useful. The Hausdorff dimension is defined for all
metric spaces and, unlike the Hamel dimension, can also attain noninteger real values.
^{[3]} 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 noninteger positive real values.
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.
In physics
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.)
Number of dimensions 
Example coordinate systems 
1 

2 

3 

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).
Additional dimensions
Theories such as
string theory and
Mtheory predict that physical space in general has in fact 10 and 11 dimensions, respectively. The extra dimensions are spatial. We perceive only three spatial dimensions, and no physical experiments have confirmed the reality of additional dimensions. A possible explanation that has been suggested is that space acts as if it were "curled up" in the extra dimensions on a subatomic scale, possibly at the quark/string level of scale or below. Another lessheld unsupported fringe view asserts that dimensions beyond the fourth progressively condense timelines and universes into single spatial points in the above dimension, until the tenth, where a 0dimensional point equates to all possible timelines in all possible universes.
^{[4]}
Literature
Perhaps the most basic way in which the word dimension is used in literature is as a hyperbolic synonym for feature, attribute, aspect, or magnitude. Frequently the hyperbole is quite literal as in he's so 2dimensional, meaning that one can see at a glance what he is. This contrasts with 3dimensional objects which have an interior that is hidden from view, and a back that can only be seen with further examination.
Science fiction texts often mention the concept of dimension, when really referring to
parallel universes, alternate universes, or other
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 spatial 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 dimension, not the standard ones.
One of the most heralded science fiction novellas regarding true geometric dimensionality, and often recommended as a starting point for those just starting to investigate such matters, is the 1884 novel
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 FifthDimension Catapult” (1931); and appeared irregularly in science fiction by the 1940s. Some of the classic stories involving other dimensions include
Robert A. Heinlein's 1941 '
"—And He Built a Crooked House—" ', in which a California architect designs a house based on a threedimensional projection of a tesseract, and
Alan E. Nourse's "Tiger by the Tail" and "The Universe Between," both 1951. Another reference would be
Madeleine L'Engle's novel "
A Wrinkle In Time" (1962) which uses the 5th Dimension as a way for "tesseracting the universe," or in a better sense, "folding" space in half to move across it quickly.
Philosophy
In 1783,
Kant 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."
^{[5]}
More dimensions
See also
A list of topics indexed by dimension
 Zero dimensions:
 One dimension:
 Two dimensions:
 Three dimensions
 Four dimensions:
 Highdimensional topics from mathematics:
 Highdimensional topics from physics:
 Infinitely many dimensions:
Notes
Further reading
 Edwin A. Abbott, (1884) Flatland: A Romance of Many Dimensions, Public Domain. Online version with ASCII approximation of illustrations at Project Gutenberg.
 Thomas Banchoff, (1996) Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Second Edition, Freeman.
 Clifford A. Pickover, (1999) Surfing through Hyperspace: Understanding Higher Universes in Six Easy Lessons, Oxford University Press.
 Rudy Rucker, (1984) The Fourth Dimension, HoughtonMifflin.
Dimension  (category) 

Dimensional spaces 



Polytopes and Shapes 


Concepts and mathematics 

