Dimension
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Me, Myself, and I
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[[WP:UNDO|Undid]] revision 226064026 by [[Special:Contributions/78.61.224.240|78.61.224.240]] ([[User talk:78.61.224.240|talk]])
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[[Image:Squarecubetesseract.png|right|thumb|400px|From left to right, the [[square (geometry)|square]] has two dimensions, the [[cube]] has three and the [[tesseract]] has four.]]
In mathematics the '''dimension''' of a [[space]] is roughly defined as the minimum number of [[coordinates]] needed to specify every [[point (geometry)|point]] within it<ref>[http://curious.astro.cornell.edu/question.php?number=4 Curious About Astronomy]</ref><ref>[http://mathworld.wolfram.com/Dimension.html MathWorld: Dimension]]</ref>. Dimensions can be thought of as the [[axes]] in a [[Cartesian coordinate system]], which in a three-dimensional system run left-right, up-down and forward-backward. A set of three [[coordinate]]s on these axes, or any other three-dimensional coordinate system, specifies the position of a particular point in [[space]]<ref name="Oxford Encyclopedia">Oxford Illustrated Encyclopedia: The Physical World</ref>. In the physical world, according to the [[theory of relativity]] the [[fourth dimension]] is [[time]], which runs before-after. An event’s position in space and time is therefore specified if [[Four-vector|four coordinates]] are given.
On [[surface]]s such as a [[plane (mathematics)|plane]] or the surface of a [[sphere]], a point can be specified using just two numbers and so this space is said to be two-dimensional. Similarly a [[line]] is one-dimensional because only one coordinate is needed, whereas a point has zero dimensions. In [[mathematics]], spaces with [[higher dimensions|more than three dimensions]] are used to describe other manifolds. In these [[n-dimensional space]]s a point is located by n coordinates (x<sub>1</sub>, x<sub>2</sub>, … x<sub>n</sub>). Some theories, such as those used in [[fractal geometry]], make use of [[integer|non-integer]] and [[Negative and non-negative numbers|negative]] dimensions.
Another meaning of the term "dimension" in physics relates to the ''nature'' of a measurable quantity. In general, physical measurements that must be expressed in [[unit of measurement|units of measurement]], and quantities obtained by such measurements are [[dimensionful]]. An example of a dimension is [[length]], abbreviated L, which is the dimension for measurements expressed in units of length, be they [[meters]], [[nautical mile]]s, or [[lightyear]]s. Another example is [[time]], abbreviated T, whether the measurement is expressed in [[second]]s or in [[hour]]s. Speed, which is the distance (length) travelled in a certain amount of time, is a dimensionful quantity that has the dimension LT<sup> −1</sup> (meaning L/T). Acceleration, the change in speed per time unit, has dimension LT<sup> −2</sup>.
In mathematics, the dimension of [[Euclidean space|Euclidean ''n''-space]] ''E''<sup> ''n''</sup> is ''n''. When trying to generalize to other types of spaces, one is faced with the question “what makes ''E''<sup> ''n''</sup> ''n'' dimensional?" One answer is that in order to cover a fixed ball in ''E''<sup> ''n''</sup> by small balls of raius <math>\epsilon</math>, one needs on the order of <math>\epsilon^{-n}</math> 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 <math>E^n</math> looks localy like <math>E^{n-1}</math> and this leads to the notion of the [[inductive dimension]]. While these notions agree on <math>E^n</math>, 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 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."
Historically, the notion of [[higher dimensions]] in mathematics was introduced by [[Bernhard Riemann]], in his 1854 [[Habilitationsschrift]], where he considered a point to be any ''n'' numbers <math>(x_1,\dots,x_n)</math>, abstractly, without any geometric picture needed nor implied.
The rest of this section examines some of the more important mathematical definitions of dimension.
=== Hamel dimension ===
{{main|Hamel dimension}}
For [[vector space]]s, there is a natural concept of dimension, namely the [[cardinality]] of a [[Basis_(linear_algebra)|basis]].
=== Manifolds ===<!-- This section is linked from [[Ball (mathematics)]] -->
A [[connectedness|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.
The theory of manifolds, in the field of [[geometric topology]], 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.
=== 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 thaft 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===
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 [[boundary (topology)|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 [[fractal]]s, the [[Hausdorff dimension]] is useful. The Hausdorff dimension is defined for all [[metric space]]s and, unlike the Hamel dimension, can also attain non-integer real values.<ref name="Hausdorff dimension">[http://math.bu.edu/DYSYS/chaos-game/node6.html Fractal Dimension], Boston University Department of Mathematics and Statistics</ref> The [[box-counting dimension|box dimension]] or [[Minkowski dimension]] is a variant of the same idea. In general, there exist more definitions of [[fractal dimension]]s that work for highly irregular sets and attain non-integer 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 two dimensions coincide.
=== Krull dimension of commutative rings ===
The [[Krull dimension]] of a commutative [[ring (algebra)|ring]], named after [[Wolfgang Krull]] (1899–1971), is defined to be the maximal number of strict inclusions in an increasing chain of [[prime ideal]]s in the ring.
=== Negative dimension ===
The negative (fractal) dimension is introduced by [[Benoit Mandelbrot]], in which, when it is positive gives the known definition, and when it is negative measures the degree of "emptiness" of empty sets.<ref name="Negative dimension">Benoit B. Mandelbrot, [http://www.math.yale.edu/mandelbrot/webbooks/wb_neg.html Negative Fractal Dimension], Yale Mathematics Department</ref>
== In physics ==
=== Spatial dimensions ===
[[Image:Coord planes color.svg|324px|thumb|A three dimensional Cartesian coordinate system.]]
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 amount. 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]].)
=== Time ===
Time is often referred to as the "[[Spacetime|fourth dimension]]". It is one way to measure physical change. It is perceived differently from the three spatial dimensions in that there is only one of it, that movement in time occurs at the fixed rate of one second per second, and that we cannot move freely in time but subjectively move [[arrow of time|in one direction]].
The equations used in physics to model reality do not treat time in the same way that humans perceive it. The equations of [[classical mechanics]] are [[T-symmetry|symmetric with respect to time]], and equations of quantum mechanics are typically symmetric if both time and other quantities (such as [[C-symmetry|charge]] and [[Parity (physics)|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 [[Henri Poincaré|Poincaré]] and [[Albert Einstein|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]].
=== Additional dimensions ===
Theories such as [[string theory]] and [[M-theory]] predict that physical space in general has in fact 10 and 11 dimensions, respectively. The extra dimensions are spacelike. 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 is as it were "curled up" in the extra dimensions on a very small, subatomic scale, possibly at the quark/string level of scale or below.
===Penrose's singularity theorem===
In his book ''[[The Road to Reality: A Complete Guide to the Laws of the Universe]]'', [[scientist]] Sir [[Roger Penrose]] explained his [[Penrose-Hawking singularity theorems|singularity theorem]]. It asserts that all theories that attribute more than three spatial dimensions and one [[Time|temporal]] dimension to the world of experience are unstable. The instabilities that exist in systems of such extra dimensions would result in their rapid collapse into a singularity. For that reason, Penrose wrote, the unification of [[gravitation]] with other [[force]]s through extra dimensions cannot occur.
=== Dimensionful quantities ===
{{main|Dimensional analysis}}
In the physical sciences and in engineering, the ''dimension'' of a physical quantity is the expression of the class of [[Units of measurement|physical unit]] that such a quantity is measured against. The dimension of speed, for example, is LT<sup>−1</sup>, that is, length divided by time. The units in which the quantity is expressed, such as ms<sup>−1</sup> (meters per second) or [[mph]] (miles per hour), has to conform to the dimension.
== Science fiction ==
[[Science fiction]] texts often mention the concept of dimension, when really referring to [[parallel universe (fiction)|parallel universe]]s, alternate universes, or other [[planes of existence]]. This usage is derived from the idea that in order 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."
== More dimensions ==
<div style="-moz-column-count:2; column-count:2;">
* [[Dimension of an algebraic variety]]
* [[Lebesgue covering dimension]]
* [[Isoperimetric dimension]]
* [[Poset dimension]]
* [[Metric dimension (graph theory)|Metric dimension]]
* [[Pointwise dimension]]
* [[Lyapunov dimension]]
* [[Kaplan-Yorke dimension]]
* [[Exterior dimension]]
* [[Hurst exponent]]
* [[q-dimension]]; especially:
** [[Information dimension]] (corresponding to q = 1)
** [[Correlation dimension]] (corresponding to q = 2)
</div>
== See also ==
=== By number of dimensions ===
* Zero dimensions:
**[[Point (geometry)|Point]]
**[[Zero-dimensional space]]
* One dimension:
**[[Line (mathematics)|Line]]
* Two dimensions:
**[[2D geometric model]]s
**[[2D computer graphics]]
* Three dimensions
**[[3D computer graphics]]
**[[3-D film]]s and video
**[[Stereoscopy]] (3-D imaging)
* Four dimensions:
**[[Time]] (4th dimension)
**[[Fourth dimension|Fourth spatial dimension]]
**[[Tesseract]] (four dimensional shapes)
* Five dimensions:
**[[Kaluza-Klein theory]]
**[[Fifth dimension]]
**[[Seventh dimension]]
* Ten, eleven or twenty-six dimensions:
**[[String theory]]
**[[M-theory]]
**[[Calabi-Yau spaces]]
* Infinitely many dimensions:
**[[Hilbert space]]
=== Other ===
* [[Degrees of freedom]]
* [[Dimension (data warehouse)]] and [[dimension table]]s
* [[Hyperspace]] (disambiguation page)
== Further reading ==
* [[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'', Houghton-Mifflin.
* [[Edwin A. Abbott]], (1884) ''[[Flatland|Flatland: A Romance of Many Dimensions]]'', Public Domain. [http://www.gutenberg.org/etext/201 Online version with ASCII approximation of illustrations] at [[Project Gutenberg]].
==References==
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