From left to right: asquare, acube and atesseract. The square is two-dimensional (2D) and bounded by one-dimensionalline segments; the cube is three-dimensional (3D) and bounded by two-dimensional squares; the tesseract is four-dimensional (4D) and bounded by three-dimensional cubes. The first four spatial dimensions, represented in a two-dimensional picture.
Two points can be connected to create aline segment.
Two parallel line segments can be connected to form asquare.
Two parallel squares can be connected to form acube.
Two parallel cubes can be connected to form atesseract.
Inmathematics, the dimension of an object is, roughly speaking, the number ofdegrees of freedom of a point that moves on this object. In other words, the dimension is the number of independentparameters orcoordinates that are needed for defining the position of a point that is constrained to be on the object. For example, the dimension of a point iszero; the dimension of aline isone, as a point can move on a line in only one direction (or its opposite); the dimension of aplane istwo, etc.
The dimension is an intrinsic property of an object, in the sense that it is independent of the dimension of the space in which the object is or can beembedded. For example, acurve, such as acircle, is of dimension one, because the position of a point on a curve is determined by its signed distance along the curve to a fixed point on the curve. This is independent from the fact that a curve cannot be embedded in aEuclidean space of dimension lower than two, unless it is a line. Similarly, asurface is of dimension two, even if embedded inthree-dimensional space.
The dimension ofEuclideann-spaceEnisn. When trying to generalize to other types of spaces, one is faced with the question "what makesEnn-dimensional?" One answer is that to cover a fixedball inEn by small balls of radiusε, one needs on the order ofε−n such small balls. This observation leads to the definition of theMinkowski dimension and its more sophisticated variant, theHausdorff dimension, but there are also other answers to that question. For example, the boundary of a ball inEn looks locally likeEn-1 and this leads to the notion of theinductive dimension. While these notions agree onEn, they turn out to be different when one looks at more general spaces.
Atesseract is an example of a four-dimensional object. Whereas outside mathematics the use of the term "dimension" is as in: "A tesseracthas four dimensions", mathematicians usually express this as: "The tesseracthas dimension 4", or: "The dimension of the tesseractis 4".
The dimension of avector space is the number of vectors in anybasis for the space,i.e. the number of coordinates necessary to specify any vector. This notion of dimension (thecardinality of a basis) is often referred to as theHamel dimension oralgebraic dimension to distinguish it from other notions of dimension.
The uniquely defined dimension of everyconnected topologicalmanifold can be calculated. A connected topological manifold islocallyhomeomorphic to Euclideann-space, in which the numbern is the manifold's dimension.
Ingeometric topology, the theory of manifolds is characterized by the way dimensions 1 and 2 are relatively elementary, thehigh-dimensional casesn > 4 are simplified by having extra space in which to "work"; and the casesn = 3 and4 are in some senses the most difficult. This state of affairs was highly marked in the various cases of thePoincaré conjecture, in which four different proof methods are applied.
The complex plane can be mapped to the surface of a sphere, called the Riemann sphere, with the complex number 0 mapped to one pole, the unit circle mapped to the equator, and apoint at infinity mapped to the other pole.
The dimension of a manifold depends on the base field with respect to which Euclidean space is defined. While analysis usually assumes a manifold to be over thereal numbers, it is sometimes useful in the study ofcomplex manifolds andalgebraic varieties to work over thecomplex numbers instead. A complex number (x +iy) has areal partx and animaginary party, in which x and y are both real numbers; hence, the complex dimension is half the real dimension.
Conversely, in algebraically unconstrained contexts, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensionalspherical surface, when given a complex metric, becomes aRiemann sphere of one complex dimension.[3]
The dimension of analgebraic variety may be defined in various equivalent ways. The most intuitive way is probably the dimension of thetangent space at anyRegular point of an algebraic variety. Another intuitive way is to define the dimension as the number ofhyperplanes 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.
Analgebraic 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 of sub-varieties of the given algebraic set (the length of such a chain is the number of "").
Each variety can be considered as analgebraic 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, ifV is a variety of dimensionm andG is analgebraic group of dimensionnacting onV, then thequotient stack [V/G] has dimensionm − n.[4]
TheKrull dimension of acommutative ring is the maximal length of chains ofprime ideals in it, a chain of lengthn being a sequence 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 anynormal topological spaceX, theLebesgue covering dimension ofX is defined to be the smallestintegern for which the following holds: anyopen cover has an open refinement (a second open cover in which each element is a subset of an element in the first cover) such that no point is included in more thann + 1 elements. In this case dimX =n. ForX a manifold, this coincides with the dimension mentioned above. If no such integern exists, then the dimension ofX is said to be infinite, and one writes dimX = ∞. Moreover,X has dimension −1, i.e. dimX = −1 if and only ifX is empty. This definition of covering dimension can be extended from the class of normal spaces to allTychonoff spaces merely by replacing the term "open" in the definition by the term "functionally open".
Aninductive dimension may be definedinductively as follows. Consider adiscrete 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 1-dimensional object. By dragging a 1-dimensional object in anew direction, one obtains a 2-dimensional object. In general, one obtains an (n + 1)-dimensional object by dragging ann-dimensional object in anew direction. The inductive dimension of a topological space may refer to thesmall inductive dimension or thelarge inductive dimension, and is based on the analogy that, in the case of metric spaces,(n + 1)-dimensional balls haven-dimensionalboundaries, permitting an inductive definition based on the dimension of the boundaries of open sets. Moreover, the boundary of a discrete set of points is the empty set, and therefore the empty set can be taken to have dimension -1.[5]
Similarly, for the class ofCW complexes, the dimension of an object is the largestn for which then-skeleton is nontrivial. Intuitively, this can be described as follows: if the original space can becontinuously deformed into a collection ofhigher-dimensional triangles joined at their faces with a complicated surface, then the dimension of the object is the dimension of those triangles.[citation needed]
TheHausdorff dimension is useful for studying structurally complicated sets, especiallyfractals. The Hausdorff dimension is defined for allmetric spaces and, unlike the dimensions considered above, can also have non-integer real values.[6] Thebox dimension orMinkowski dimension is a variant of the same idea. In general, there exist more definitions offractal dimensions that work for highly irregular sets and attain non-integer positive real values.
EveryHilbert space admits anorthonormal basis, and any two such bases for a particular space have the samecardinality. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space'sHamel dimension is finite, and in this case the two dimensions coincide.
Classical physics theories describe threephysical dimensions: from a particular point inspace, 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 impliesi.e., moving in alinear 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. (SeeSpace andCartesian coordinate system.)
Atemporal dimension, ortime 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[citation needed]. 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 movein 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 ofclassical mechanics aresymmetric with respect to time, and equations of quantum mechanics are typically symmetric if both time and other quantities (such ascharge andparity) are reversed. In these models, the perception of time flowing in one direction is an artifact of thelaws of thermodynamics (we perceive time as flowing in the direction of increasingentropy).
The best-known treatment of time as a dimension isPoincaré andEinstein'sspecial relativity (and extended togeneral relativity), which treats perceived space and time as components of a four-dimensionalmanifold, known asspacetime, and in the special, flat case asMinkowski space. Time is different from other spatial dimensions as time operates in all spatial dimensions. Time operates in the first, second and third as well as theoretical spatial dimensions such as afourth spatial dimension. Time is not however present in a single point of absolute infinitesingularity as defined as ageometric point, as an infinitely small point can have no change and therefore no time. Just as when an object moves throughpositions in space, it also moves through positions in time. In this sense theforce moving anyobject to change istime.[7][8][9]
In physics, three dimensions of space and one of time is the accepted norm. However, there are theories that attempt to unify the fourfundamental forces by introducingextra dimensions/hyperspace. Most notably,superstring theory requires 10 spacetime dimensions, and originates from a more fundamental 11-dimensional theory tentatively calledM-theory which subsumes five previously distinct superstring theories.Supergravity theory also promotes 11D spacetime = 7D hyperspace + 4 common dimensions. To date, no direct experimental or observational evidence is available to support the existence of these extra dimensions. If hyperspace exists, it 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.
Illustration of a Calabi–Yau manifold
In 1921,Kaluza–Klein theory presented 5D including an extra dimension of space. At the level ofquantum field theory, Kaluza–Klein theory unifiesgravity withgauge 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 reproduceselectromagnetism. However, at sufficiently high energies or short distances, this setup still suffers from the same pathologies that famously obstruct direct attempts to describequantum gravity. Therefore, these models still require aUV completion, of the kind that string theory is intended to provide. In particular, superstring theory requires six compact dimensions (6D hyperspace) forming aCalabi–Yau manifold. 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 are not 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 belarge 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 thebrane 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 tocosmology. For example, brane gas cosmology[10][11] attempts to explain why there are three dimensions of space using topological and thermodynamic considerations. According to this idea it would be since three is the largest number of spatial dimensions in which strings can generically intersect. If initially there are many 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 beuniversal if all fields are equally free to propagate within them.
Several types of digital systems are based on the storage, analysis, and visualization of geometric shapes, includingillustration software,Computer-aided design, andGeographic information systems. Different vector systems use a wide variety of data structures to represent shapes, but almost all are fundamentally based on a set ofgeometric primitives corresponding to the spatial dimensions:[12]
Line orPolyline (1-dimensional) usually represented as an ordered list of points sampled from a continuous line, whereupon the software is expected tointerpolate the intervening shape of the line as straight- or curved-line segments.
Polygon (2-dimensional) usually represented as a line that closes at its endpoints, representing the boundary of a two-dimensional region. The software is expected to use this boundary to partition 2-dimensional space into an interior and exterior.
Surface (3-dimensional) represented using a variety of strategies, such as apolyhedron consisting of connected polygon faces. The software is expected to use this surface to partition 3-dimensional space into an interior and exterior.
Frequently in these systems, especially GIS andCartography, a representation of a real-world phenomenon may have a different (usually lower) dimension than the phenomenon being represented. For example, a city (a two-dimensional region) may be represented as a point, or a road (a three-dimensional volume of material) may be represented as a line. Thisdimensional generalization correlates with tendencies in spatial cognition. For example, asking the distance between two cities presumes a conceptual model of the cities as points, while giving directions involving travel "up," "down," or "along" a road imply a one-dimensional conceptual model. This is frequently done for purposes of data efficiency, visual simplicity, or cognitive efficiency, and is acceptable if the distinction between the representation and the represented is understood but can cause confusion if information users assume that the digital shape is a perfect representation of reality (i.e., believing that roads really are lines).
^Yau, Shing-Tung; Nadis, Steve (2010)."4. Too Good to be True".The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. Basic Books. pp. 60–.ISBN978-0-465-02266-3.
^Fantechi, Barbara (2001),"Stacks for everybody"(PDF),European Congress of Mathematics Volume I, Progr. Math., vol. 201, Birkhäuser, pp. 349–359,archived(PDF) from the original on 2006-01-17
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