Dimension

This article is about dimensions of space. For the dimen-sion of a quantity, see Dimensional analysis. For otheruses, see Dimension (disambiguation).

In physics and mathematics, the dimension of a

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-dimensionalvolumes. For display on a two-dimensional surface such as ascreen, the 3d cube and 4d tesseract require projection.

The first four spatial dimensions.

mathematical space (or object) is informally defined asthe minimum number of coordinates needed to specifyany point within it.[1][2] Thus a line has a dimension ofone because only one coordinate is needed to specify apoint on it – for example, the point at 5 on a number line.A surface such as a plane or the surface of a cylinder orsphere has a dimension of two because two coordinatesare needed to specify a point on it – for example, both alatitude and longitude are required to locate a point on thesurface of a sphere. The inside of a cube, a cylinder ora sphere is three-dimensional because three coordinatesare needed to locate a point within these spaces.In classical mechanics, space and time are different cat-egories and refer to absolute space and time. Thatconception of the world is a four-dimensional spacebut not the one that was found necessary to describeelectromagnetism. The four dimensions of spacetimeconsist of events that are not absolutely defined spatiallyand temporally, but rather are known relative to the mo-tion of an observer. Minkowski space first approximatesthe universe without gravity; the pseudo-Riemannianmanifolds of general relativity describe spacetime with

matter and gravity. Ten dimensions are used to describestring theory, and the state-space of quantum mechanicsis an infinite-dimensional function space.The concept of dimension is not restricted to physicalobjects. High-dimensional spaces frequently occur inmathematics and the sciences. They may be parameterspaces or configuration spaces such as in Lagrangian orHamiltonian mechanics; these are abstract spaces, inde-pendent of the physical space we live in.

1 In mathematics

In mathematics, the dimension of an object is an intrinsicproperty independent of the space in which the object isembedded. For example, a point on the unit circle in theplane can be specified by two Cartesian coordinates, buta single polar coordinate (the angle) would be sufficient,so the circle is 1-dimensional even though it exists in the2-dimensional plane. This intrinsic notion of dimensionis one of the chief ways the mathematical notion of di-mension differs from its common usages.The dimension of Euclidean n-spaceEn is n. When tryingto generalize to other types of spaces, one is faced withthe question “what makes En n-dimensional?" One an-swer is that to cover a fixed ball in En by small balls of ra-dius ε, one needs on the order of ε−n such small balls. Thisobservation leads to the definition of the Minkowski di-mension and its more sophisticated variant, the Hausdorffdimension, but there are also other answers to that ques-tion. For example, the boundary of a ball in En looks lo-cally likeEn−1 and this leads to the notion of the inductivedimension. While these notions agree on En , they turnout 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 “di-mension” is as in: “A tesseract has four dimensions",mathematicians usually express this as: “The tesseracthas dimension 4", or: “The dimension of the tesseract is4”.Although the notion of higher dimensions goes back toRené Descartes, substantial development of a higher-dimensional geometry only began in the 19th century,via the work of Arthur Cayley, William Rowan Hamil-ton, Ludwig Schläfli and Bernhard Riemann. Riemann’s1854 Habilitationsschrift, Schläfli’s 1852 Theorie dervielfachen Kontinuität, Hamilton’s 1843 discovery of thequaternions and the construction of the Cayley algebra

1

2 1 IN MATHEMATICS

marked the beginning of higher-dimensional geometry.The rest of this section examines some of the more im-portant mathematical definitions of the dimensions.

1.1 Dimension of a vector space

Main article: Dimension (vector space)

The dimension of a vector space is the number of vectorsin any basis for the space, i.e. the number of coordinatesnecessary to specify any vector. This notion of dimen-sion (the cardinality of a basis) is often referred to as theHamel dimension or algebraic dimension to distinguish itfrom other notions of dimension.

1.2 Manifolds

A connected topological manifold is locallyhomeomorphic to Euclidean n-space, and the num-ber n is called the manifold’s dimension. One can showthat this yields a uniquely defined dimension for everyconnected topological manifold.For connected differentiable manifolds, the dimension isalso the dimension of the tangent vector space at anypoint.In geometric topology, the theory of manifolds is charac-terized by the way dimensions 1 and 2 are relatively ele-mentary, the high-dimensional cases n > 4 are simplifiedby having extra space in which to “work"; and the cases n= 3 and 4 are in some senses the most difficult. This stateof affairs was highly marked in the various cases of thePoincaré conjecture, where four different proof methodsare applied.

1.3 Varieties

Main article: Dimension of an algebraic variety

The dimension of an algebraic variety may be defined invarious equivalent ways. The most intuitive way is prob-ably the dimension of the tangent space at any regularpoint. Another intuitive way is to define the dimensionas the number of hyperplanes that are needed in order tohave an intersection with the variety that is reduced to afinite number of points (dimension zero). This definitionis based on the fact that the intersection of a variety witha hyperplane reduces the dimension by one unless if thehyperplane contains the variety.An algebraic set being a finite union of algebraic vari-eties, its dimension is the maximum of the dimensions ofits components. It is equal to the maximal length of thechains V0 ⊊ V1 ⊊ . . . ⊊ Vd of sub-varieties of the givenalgebraic set (the length of such a chain is the number of

" ⊊ ").Each variety can be considered as an algebraic stack, andits dimension as variety agrees with its dimension as stack.There are however many stacks which do not correspondto varieties, and some of these have negative dimension.Specifically, if V is a variety of dimension m and G isan algebraic group of dimension n acting on V, then thequotient stack [V/G] has dimension m−n.[3]

1.4 Krull dimension

Main article: Krull dimension

The Krull dimension of a commutative ring is the maxi-mal length of chains of prime ideals in it, a chain of lengthn being a sequenceP0 ⊊ P1 ⊊ . . . ⊊ Pn of prime idealsrelated by inclusion. It is strongly related to the dimensionof an algebraic variety, because of the natural correspon-dence between sub-varieties and prime ideals of the ringof the polynomials on the variety.For an algebra over a field, the dimension as vector spaceis finite if and only if its Krull dimension is 0.

1.5 Lebesgue covering dimension

Main article: Lebesgue covering dimension

For any normal topological space X, the Lebesgue cover-ing dimension of X is defined to be n if n is the smallestinteger for which the following holds: any open cover hasan open refinement (a second open cover where each el-ement is a subset of an element in the first cover) suchthat no point is included in more than n + 1 elements. Inthis case dim X = n. For X a manifold, this coincideswith the dimension mentioned above. If no such integern 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 def-inition of covering dimension can be extended from theclass of normal spaces to all Tychonoff spaces merely byreplacing the term “open” in the definition by the term"functionally open".

1.6 Inductive dimension

Main article: Inductive dimension

An inductive definition of dimension can be created asfollows. Consider a discrete set of points (such as a finitecollection of points) to be 0-dimensional. By dragging a0-dimensional object in some direction, one obtains a 1-dimensional object. By dragging a 1-dimensional objectin a new direction, one obtains a 2-dimensional object.

2.3 Additional dimensions 3

In general one obtains an (n + 1)-dimensional object bydragging an n-dimensional object in a new direction.The inductive dimension of a topological space may re-fer to the small inductive dimension or the large induc-tive dimension, and is based on the analogy that (n +1)-dimensional balls have n-dimensional boundaries, per-mitting an inductive definition based on the dimension ofthe boundaries of open sets.

1.7 Hausdorff dimension

Main article: Hausdorff dimension

For structurally complicated sets, especially fractals, theHausdorff dimension is useful. The Hausdorff dimensionis defined for all metric spaces and, unlike the dimen-sions considered above, can also attain non-integer realvalues.[4] The box dimension or Minkowski dimension isa variant of the same idea. In general, there exist moredefinitions of fractal dimensions that work for highly ir-regular sets and attain non-integer positive real values.Fractals have been found useful to describe many natu-ral objects and phenomena.[5][6]

1.8 Hilbert spaces

Every Hilbert space admits an orthonormal basis, andany two such bases for a particular space have the samecardinality. This cardinality is called the dimension ofthe Hilbert space. This dimension is finite if and only ifthe space’s Hamel dimension is finite, and in this case theabove dimensions coincide.

2 In physics

2.1 Spatial dimensions

Classical physics theories describe three physical dimen-sions: from a particular point in space, the basic direc-tions in which we can move are up/down, left/right, andforward/backward. Movement in any other direction canbe expressed in terms of just these three. Moving downis the same as moving up a negative distance. Moving di-agonally upward and forward is just as the name of thedirection implies; i.e., moving in a linear combination ofup and forward. In its simplest form: a line describes onedimension, a plane describes two dimensions, and a cubedescribes three dimensions. (See Space and Cartesian co-ordinate system.)

2.2 Time

A temporal dimension is a dimension of time. Time isoften referred to as the "fourth dimension" for this rea-

son, but that is not to imply that it is a spatial dimension.A temporal dimension is one way to measure physicalchange. It is perceived differently from the three spa-tial dimensions in that there is only one of it, and that wecannot move freely in time but subjectively move in onedirection.The equations used in physics to model reality do not treattime in the same way that humans commonly perceiveit. The equations of classical mechanics are symmetricwith respect to time, and equations of quantum mechan-ics are typically symmetric if both time and other quan-tities (such as charge and parity) are reversed. In thesemodels, the perception of time flowing in one directionis an artifact of the laws of thermodynamics (we perceivetime as flowing in the direction of increasing entropy).The best-known treatment of time as a dimension isPoincaré and Einstein's special relativity (and extended togeneral relativity), which treats perceived space and timeas components of a four-dimensional manifold, knownas spacetime, and in the special, flat case as Minkowskispace.

2.3 Additional dimensions

In physics, three dimensions of space and one of time isthe accepted norm. However, there are theories that at-tempt to unify the four fundamental forces by introduc-ing more dimensions. Most notably, superstring theoryrequires 10 spacetime dimensions, and originates froma more fundamental 11-dimensional theory tentativelycalled M-theory which subsumes five previously distinctsuperstring theories. To date, no experimental or obser-vational evidence is available to confirm the existence ofthese extra dimensions. If extra dimensions exist, theymust be hidden from us by some physical mechanism.One well-studied possibility is that the extra dimensionsmay be “curled up” at such tiny scales as to be effectivelyinvisible to current experiments. Limits on the size andother properties of extra dimensions are set by particle ex-periments such as those at the Large Hadron Collider.[7]

At the level of quantum field theory, Kaluza–Klein the-ory unifies gravity with gauge interactions, based on therealization that gravity propagating in small, compact ex-tra dimensions is equivalent to gauge interactions at longdistances. In particular when the geometry of the ex-tra dimensions is trivial, it reproduces electromagnetism.However at sufficiently high energies or short distances,this setup still suffers from the same pathologies thatfamously obstruct direct attempts to describe quantumgravity. Therefore, these models still require a UV com-pletion, of the kind that string theory is intended to pro-vide. Thus Kaluza-Klein theory may be considered eitheras an incomplete description on its own, or as a subset ofstring theory model building.In addition to small and curled up extra dimensions, theremay be extra dimensions that instead aren't apparent be-

4 5 IN PHILOSOPHY

cause the matter associated with our visible universe islocalized on a (3 + 1)-dimensional subspace. Thus theextra dimensions need not be small and compact but maybe large extra dimensions. D-branes are dynamical ex-tended objects of various dimensionalities predicted bystring theory that could play this role. They have theproperty that open string excitations, which are associ-ated with gauge interactions, are confined to the brane bytheir endpoints, whereas the closed strings that mediatethe gravitational interaction are free to propagate into thewhole spacetime, or “the bulk”. This could be related towhy 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 tocosmology. For example, brane gas cosmology[8][9] at-tempts to explain why there are three dimensions ofspace using topological and thermodynamic considera-tions. According to this idea it would be because threeis the largest number of spatial dimensions where stringscan generically intersect. If initially there are lots ofwindings of strings around compact dimensions, spacecould only expand to macroscopic sizes once these wind-ings are eliminated, which requires oppositely woundstrings to find each other and annihilate. But strings canonly find each other to annihilate at a meaningful rate inthree dimensions, so it follows that only three dimensionsof space are allowed to grow large given this kind of ini-tial configuration.Extra dimensions are said to be universal if all fields areequally free to propagate within them.

3 Networks and dimension

Some complex networks are characterized by fractal di-mensions.[10] The concept of dimension can be general-ized to include networks embedded in space.[11] The di-mension characterize their spatial constraints.

4 In literature

Main article: Fourth dimension in literature

Science fiction texts often mention the concept of “di-mension” when referring to parallel or alternate universesor other imagined planes of existence. This usage isderived from the idea that to travel to parallel/alternateuniverses/planes of existence one must travel in a direc-tion/dimension besides the standard ones. In effect, theother universes/planes are just a small distance away fromour 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 regardingtrue geometric dimensionality, and often recommended

as a starting point for those just starting to investigate suchmatters, is the 1884 novella Flatland by Edwin A. Abbott.Isaac Asimov, in his foreword to the Signet Classics 1984edition, described Flatland as “The best introduction onecan find into the manner of perceiving dimensions.”The idea of other dimensions was incorporated into manyearly science fiction stories, appearing prominently, forexample, in Miles J. Breuer's The Appendix and the Spec-tacles (1928) and Murray Leinster's The Fifth-DimensionCatapult (1931); and appeared irregularly in science fic-tion by the 1940s. Classic stories involving other di-mensions include Robert A. Heinlein's —And He Builta Crooked House (1941), in which a California architectdesigns a house based on a three-dimensional projectionof a tesseract; and Alan E. Nourse's Tiger by the Tail andThe Universe Between (both 1951). Another referenceis Madeleine L'Engle's novel A Wrinkle In Time (1962),which uses the fifth dimension as a way for “tesseractingthe universe” or “folding” space in order to move across itquickly. The fourth and fifth dimensions were also a keycomponent of the book The Boy Who Reversed Himselfby William Sleator.

5 In philosophy

Immanuel Kant, in 1783, wrote: “That everywhere space(which is not itself the boundary of another space) hasthree dimensions and that space in general cannot havemore dimensions is based on the proposition that notmore than three lines can intersect at right angles in onepoint. This proposition cannot at all be shown from con-cepts, but rests immediately on intuition and indeed onpure intuition a priori because it is apodictically (demon-strably) certain.”[12]

“Space has Four Dimensions” is a short story published in1846 by German philosopher and experimental psychol-ogist Gustav Fechner under the pseudonym “Dr. Mises”.The protagonist in the tale is a shadow who is aware ofand able to communicate with other shadows, but whois trapped on a two-dimensional surface. According toFechner, this “shadow-man” would conceive of the thirddimension as being one of time.[13] The story bears astrong similarity to the "Allegory of the Cave" presentedin Plato's The Republic (c. 380 BC).Simon Newcomb wrote an article for the Bulletin of theAmerican Mathematical Society in 1898 entitled “ThePhilosophy of Hyperspace”.[14] Linda Dalrymple Hen-derson coined the term “hyperspace philosophy”, usedto describe writing that uses higher dimensions to ex-plore metaphysical themes, in her 1983 thesis about thefourth dimension in early-twentieth-century art.[15] Ex-amples of “hyperspace philosophers” include CharlesHoward Hinton, the first writer, in 1888, to use the word“tesseract";[16] and the Russian esotericist P. D. Ouspen-sky.

5

6 More dimensions• Degrees of freedom in mechanics / physics andchemistry / statistics

7 See also

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)

• 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 dimensionsin 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

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 ofMathematics and Statistics

[5] Bunde, Armin; Havlin, Shlomo, eds. (1991). Fractals andDisordered Systems. Springer.

[6] Bunde, Armin; Havlin, Shlomo, eds. (1994). Fractals inScience. Springer.

[7] CMS Collaboration, “Search for Microscopic Black HoleSignatures at the Large Hadron Collider” (arxiv.org)

[8] Brandenberger, R., Vafa, C., Superstrings in the early uni-verse

[9] Scott Watson, Brane Gas Cosmology (pdf).

[10] Song, Chaoming; Havlin, Shlomo; Makse,Hernán A. (2005). “Self-similarity of complexnetworks”. 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 spa-tially embedded networks”. Nature Physics 7 (6).Bibcode:2011NatPh...7..481D. doi:10.1038/nphys1932.

6 10 EXTERNAL LINKS

[12] Prolegomena, § 12

[13] Banchoff, Thomas F. (1990). “From Flatland toHypergraphics: Interacting with Higher Dimen-sions”. Interdisciplinary Science Reviews 15 (4): 364.doi:10.1179/030801890789797239.

[14] Newcomb, Simon (1898). “The Philosophy of Hyper-space”. 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 PietMondrian” (PDF). Spaces of Utopia: an Electronic Jour-nal (5): 11.

[16] Pickover, CliffordA. (2009), “Tesseract”, TheMath Book:From Pythagoras to the 57th Dimension, 250 Milestones inthe History of Mathematics, Sterling Publishing Company,Inc., p. 282, ISBN 9781402757969.

9 Further reading• Katta G Murty, “Systems of Simultaneous LinearEquations” (Chapter 1 of Computational and Algo-rithmic Linear Algebra and n-Dimensional Geome-try, World Scientific Publishing: 2014 (ISBN 978-981-4366-62-5).

• Edwin A. Abbott, Flatland: A Romance of ManyDimensions (1884) (Public domain: Online versionwith ASCII approximation of illustrations at ProjectGutenberg).

• Thomas Banchoff, Beyond the Third Dimension:Geometry, Computer Graphics, and Higher Dimen-sions, Second Edition, W. H. Freeman and Com-pany: 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 OdysseyThrough the 10th Dimension, Oxford UniversityPress: 1994.

10 External links• Copeland, Ed (2009). “Extra Dimensions”. SixtySymbols. Brady Haran for the University of Not-tingham.

7

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Own work Original artist: Andeggs• File:Coord_XYZ.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/64/Coord_XYZ.svg License: Public domain Contrib-utors: Own work Original artist: Andeggs

8 11 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

• File:Cylindrical_Coordinates.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b7/Cylindrical_Coordinates.svg Li-cense: Public domain Contributors: Own work Original artist: Inductiveload

• File:Dimension_levels.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/45/Dimension_levels.svg License: CC BY-SA3.0 Contributors: Own work Original artist: NerdBoy1392

• File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0Contributors:Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist:Tkgd2007

• File:Spherical_Coordinates_(Colatitude,_Longitude).svg Source: https://upload.wikimedia.org/wikipedia/commons/5/51/Spherical_Coordinates_%28Colatitude%2C_Longitude%29.svg License: Public domain Contributors: Own work Original artist: Inductiveload

• File:Squarecubetesseract.png Source: https://upload.wikimedia.org/wikipedia/commons/2/25/Squarecubetesseract.png License: CCBY-SA 3.0 Contributors: self-made based on this (public domain), this (GFDL / CC-BY-SA-3.0) and this (use allowed for any pur-pose providing “attribution is given to Robert Webb’s Great Stella software as the creator of this image along with a link to the website:http://www.software3d.com/Stella.php"). Original artist: Andeggs

• File:Tesseract.gif Source: https://upload.wikimedia.org/wikipedia/commons/5/55/Tesseract.gif License: Public domain Contributors: ?Original artist: ?

• File:Wikiquote-logo.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikiquote-logo.svg License: Public domainContributors: ? Original artist: ?

11.3 Content license• Creative Commons Attribution-Share Alike 3.0