Dimension

From Wikipedia for FEVERv2
Jump to navigation Jump to search

This article is about the dimension of a space. Dimension_sentence_0

For the dimension of an object, see size. Dimension_sentence_1

For the dimension of a quantity, see Dimensional analysis. Dimension_sentence_2

For other uses, see Dimension (disambiguation). Dimension_sentence_3

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Dimension_sentence_4

Thus a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. Dimension_sentence_5

A surface such as a plane or the surface of a cylinder or sphere has a dimension of two (2D) 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. Dimension_sentence_6

The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. Dimension_sentence_7

In classical mechanics, space and time are different categories and refer to absolute space and time. Dimension_sentence_8

That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. Dimension_sentence_9

The four dimensions (4D) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Dimension_sentence_10

Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Dimension_sentence_11

10 dimensions are used to describe superstring theory (6D hyperspace + 4D), 11 dimensions can describe supergravity and M-theory (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space. Dimension_sentence_12

The concept of dimension is not restricted to physical objects. Dimension_sentence_13

High-dimensional spaces frequently occur in mathematics and the sciences. Dimension_sentence_14

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. Dimension_sentence_15

In mathematics Dimension_section_0

In mathematics, the dimension of an object is, roughly speaking, the number of degrees of freedom of a point that moves on this object. Dimension_sentence_16

In other words, the dimension is the number of independent parameters or coordinates that are needed for defining the position of a point that is constrained to be on the object. Dimension_sentence_17

For example, the dimension of a point is zero; the dimension of a line is one, as a point can move on a line in only one direction (or its opposite); the dimension of a plane is two, etc. Dimension_sentence_18

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 be embedded. Dimension_sentence_19

For example, a curve, such as a circle 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. Dimension_sentence_20

This is independent from the fact that a curve cannot be embedded in a Euclidean space of dimension lower than two, unless it is a line. Dimension_sentence_21

The dimension of Euclidean n-space E is n. When trying to generalize to other types of spaces, one is faced with the question "what makes E n-dimensional?" Dimension_sentence_22

One answer is that to cover a fixed ball in E by small balls of radius ε, one needs on the order of ε such small balls. Dimension_sentence_23

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. Dimension_sentence_24

For example, the boundary of a ball in E looks locally like E and this leads to the notion of the inductive dimension. Dimension_sentence_25

While these notions agree on E, they turn out to be different when one looks at more general spaces. Dimension_sentence_26

A tesseract is an example of a four-dimensional object. Dimension_sentence_27

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" or: 4D. Dimension_sentence_28

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. Dimension_sentence_29

Riemann's 1854 Habilitationsschrift, Schläfli's 1852 Theorie der vielfachen Kontinuität, and Hamilton's discovery of the quaternions and John T. Graves' discovery of the octonions in 1843 marked the beginning of higher-dimensional geometry. Dimension_sentence_30

The rest of this section examines some of the more important mathematical definitions of dimension. Dimension_sentence_31

Vector spaces Dimension_section_1

Main article: Dimension (vector space) Dimension_sentence_32

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. Dimension_sentence_33

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. Dimension_sentence_34

For the non-free case, this generalizes to the notion of the length of a module. Dimension_sentence_35

Manifolds Dimension_section_2

The uniquely defined dimension of every connected topological manifold can be calculated. Dimension_sentence_36

A connected topological manifold is locally homeomorphic to Euclidean n-space, in which the number n is the manifold's dimension. Dimension_sentence_37

For connected differentiable manifolds, the dimension is also the dimension of the tangent vector space at any point. Dimension_sentence_38

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. Dimension_sentence_39

This state of affairs was highly marked in the various cases of the Poincaré conjecture, where four different proof methods are applied. Dimension_sentence_40

Complex dimension Dimension_section_3

The dimension of a manifold depends on the base field with respect to which Euclidean space is defined. Dimension_sentence_41

While analysis usually assumes a manifold to be over the real numbers, it is sometimes useful in the study of complex manifolds and algebraic varieties to work over the complex numbers instead. Dimension_sentence_42

A complex number (x + iy) has a real part x and an imaginary part y, where x and y are both real numbers; hence, the complex dimension is half the real dimension. Dimension_sentence_43

Conversely, in algebraically unconstrained contexts, a single complex coordinate system may be applied to an object having two real dimensions. Dimension_sentence_44

For example, an ordinary two-dimensional spherical surface, when given a complex metric, becomes a Riemann sphere of one complex dimension. Dimension_sentence_45

Varieties Dimension_section_4

Main article: Dimension of an algebraic variety Dimension_sentence_46

The dimension of an algebraic variety may be defined in various equivalent ways. Dimension_sentence_47

The most intuitive way is probably the dimension of the tangent space at any Regular point of an algebraic variety. Dimension_sentence_48

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). Dimension_sentence_49

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. Dimension_sentence_50

Each variety can be considered as an algebraic stack, and its dimension as variety agrees with its dimension as stack. Dimension_sentence_51

There are however many stacks which do not correspond to varieties, and some of these have negative dimension. Dimension_sentence_52

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. Dimension_sentence_53

Krull dimension Dimension_section_5

For an algebra over a field, the dimension as vector space is finite if and only if its Krull dimension is 0. Dimension_sentence_54

Topological spaces Dimension_section_6

For any normal topological space X, the Lebesgue covering dimension of X is defined to be the smallest integer n 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. Dimension_sentence_55

In this case dim X = n. For X a manifold, this coincides with the dimension mentioned above. Dimension_sentence_56

If no such integer n exists, then the dimension of X is said to be infinite, and one writes dim X = ∞. Dimension_sentence_57

Moreover, X has dimension −1, i.e. dim X = −1 if and only if X is empty. Dimension_sentence_58

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". Dimension_sentence_59

An inductive dimension may be defined inductively as follows. Dimension_sentence_60

Consider a discrete set of points (such as a finite collection of points) to be 0-dimensional. Dimension_sentence_61

By dragging a 0-dimensional object in some direction, one obtains a 1-dimensional object. Dimension_sentence_62

By dragging a 1-dimensional object in a new direction, one obtains a 2-dimensional object. Dimension_sentence_63

In general one obtains an (n + 1)-dimensional object by dragging an n-dimensional object in a new direction. Dimension_sentence_64

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, in the case of metric spaces, (n + 1)-dimensional balls have n-dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets. Dimension_sentence_65

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. Dimension_sentence_66

Similarly, for the class of CW complexes, the dimension of an object is the largest n for which the n-skeleton is nontrivial. Dimension_sentence_67

Intuitively, this can be described as follows: if the original space can be continuously deformed into a collection of higher-dimensional triangles joined at their faces with a complicated surface, then the dimension of the object is the dimension of those triangles. Dimension_sentence_68

See also: dimension of a scheme Dimension_sentence_69

Hausdorff dimension Dimension_section_7

The Hausdorff dimension is useful for studying structurally complicated sets, especially fractals. Dimension_sentence_70

The Hausdorff dimension is defined for all metric spaces and, unlike the dimensions considered above, can also have non-integer real values. Dimension_sentence_71

The box dimension or Minkowski dimension is a variant of the same idea. Dimension_sentence_72

In general, there exist more definitions of fractal dimensions that work for highly irregular sets and attain non-integer positive real values. Dimension_sentence_73

Fractals have been found useful to describe many natural objects and phenomena. Dimension_sentence_74

Hilbert spaces Dimension_section_8

Every Hilbert space admits an orthonormal basis, and any two such bases for a particular space have the same cardinality. Dimension_sentence_75

This cardinality is called the dimension of the Hilbert space. Dimension_sentence_76

This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the two dimensions coincide. Dimension_sentence_77

In physics Dimension_section_9

Spatial dimensions Dimension_section_10

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. Dimension_sentence_78

Movement in any other direction can be expressed in terms of just these three. Dimension_sentence_79

Moving down is the same as moving up a negative distance. Dimension_sentence_80

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. Dimension_sentence_81

In its simplest form: a line describes one dimension, a plane describes two dimensions, and a cube describes three dimensions. Dimension_sentence_82

(See Space and Cartesian coordinate system.) Dimension_sentence_83

Time Dimension_section_11

A temporal dimension, or time dimension, is a dimension of time. Dimension_sentence_84

Time is often referred to as the "fourth dimension" for this reason, but that is not to imply that it is a spatial dimension. Dimension_sentence_85

A temporal dimension is one way to measure physical change. Dimension_sentence_86

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. Dimension_sentence_87

The equations used in physics to model reality do not treat time in the same way that humans commonly perceive it. Dimension_sentence_88

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. Dimension_sentence_89

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). Dimension_sentence_90

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. Dimension_sentence_91

Additional dimensions Dimension_section_12

In physics, three dimensions of space and one of time is the accepted norm. Dimension_sentence_92

However, there are theories that attempt to unify the four fundamental forces by introducing extra dimensions/hyperspace. Dimension_sentence_93

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. Dimension_sentence_94

Supergravity theory also promotes 11D spacetime = 7D hyperspace + 4 common dimensions. Dimension_sentence_95

To date, no direct experimental or observational evidence is available to support the existence of these extra dimensions. Dimension_sentence_96

If hyperspace exists, it must be hidden from us by some physical mechanism. Dimension_sentence_97

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. Dimension_sentence_98

Limits on the size and other properties of extra dimensions are set by particle experiments such as those at the Large Hadron Collider. Dimension_sentence_99

In 1921, Kaluza-Klein theory presented 5D including an extra dimension of space. Dimension_sentence_100

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. Dimension_sentence_101

In particular when the geometry of the extra dimensions is trivial, it reproduces electromagnetism. Dimension_sentence_102

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. Dimension_sentence_103

Therefore, these models still require a UV completion, of the kind that string theory is intended to provide. Dimension_sentence_104

In particular, superstring theory requires six compact dimensions (6D hyperspace) forming a Calabi–Yau manifold. Dimension_sentence_105

Thus Kaluza-Klein theory may be considered either as an incomplete description on its own, or as a subset of string theory model building. Dimension_sentence_106

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. Dimension_sentence_107

Thus the extra dimensions need not be small and compact but may be large extra dimensions. Dimension_sentence_108

D-branes are dynamical extended objects of various dimensionalities predicted by string theory that could play this role. Dimension_sentence_109

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". Dimension_sentence_110

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. Dimension_sentence_111

Some aspects of brane physics have been applied to cosmology. Dimension_sentence_112

For example, brane gas cosmology attempts to explain why there are three dimensions of space using topological and thermodynamic considerations. Dimension_sentence_113

According to this idea it would be because three is the largest number of spatial dimensions where strings can generically intersect. Dimension_sentence_114

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. Dimension_sentence_115

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. Dimension_sentence_116

Extra dimensions are said to be universal if all fields are equally free to propagate within them. Dimension_sentence_117

In computer graphics and spatial data Dimension_section_13

Main article: Geometric primitive Dimension_sentence_118

Several types of digital systems are based on the storage, analysis, and visualization of geometric shapes, including illustration software, Computer-aided design, and Geographic information systems. Dimension_sentence_119

Different vector systems use a wide variety of data structures to represent shapes, but almost all are fundamentally based on a set of geometric primitives corresponding to the spatial dimensions: Dimension_sentence_120

Dimension_unordered_list_0

  • Point (0-dimensional), a single coordinate in a Cartesian coordinate system.Dimension_item_0_0
  • Line or Polyline (1-dimensional), usually represented as an ordered list of points sampled from a continuous line, whereupon the software is expected to interpolate the intervening shape of the line as straight or curved line segments.Dimension_item_0_1
  • 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.Dimension_item_0_2
  • Surface (3-dimensional), represented using a variety of strategies, such as a polyhedron consisting of connected polygon faces. The software is expected to use this surface to partition 3-dimensional space into an interior and exterior.Dimension_item_0_3

Frequently in these systems, especially GIS and Cartography, a representation of a real-world phenomena may have a different (usually lower) dimension than the phenomenon being represented. Dimension_sentence_121

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. Dimension_sentence_122

This dimensional generalization correlates with tendencies in spatial cognition. Dimension_sentence_123

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. Dimension_sentence_124

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). Dimension_sentence_125

Networks and dimension Dimension_section_14

Some complex networks are characterized by fractal dimensions. Dimension_sentence_126

The concept of dimension can be generalized to include networks embedded in space. Dimension_sentence_127

The dimension characterize their spatial constraints. Dimension_sentence_128

In literature Dimension_section_15

Main article: Fourth dimension in literature Dimension_sentence_129

Science fiction texts often mention the concept of "dimension" when referring to parallel or alternate universes or other imagined planes of existence. Dimension_sentence_130

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. Dimension_sentence_131

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. Dimension_sentence_132

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. Dimension_sentence_133

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." Dimension_sentence_134

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. Dimension_sentence_135

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). Dimension_sentence_136

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. Dimension_sentence_137

The fourth and fifth dimensions are also a key component of the book The Boy Who Reversed Himself by William Sleator. Dimension_sentence_138

In philosophy Dimension_section_16

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. Dimension_sentence_139

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." Dimension_sentence_140

"Space has Four Dimensions" is a short story published in 1846 by German philosopher and experimental psychologist Gustav Fechner under the pseudonym "Dr. Mises". Dimension_sentence_141

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. Dimension_sentence_142

According to Fechner, this "shadow-man" would conceive of the third dimension as being one of time. Dimension_sentence_143

The story bears a strong similarity to the "Allegory of the Cave" presented in Plato's The Republic (c. Dimension_sentence_144

380 BC). Dimension_sentence_145

Simon Newcomb wrote an article for the Bulletin of the American Mathematical Society in 1898 entitled "The Philosophy of Hyperspace". Dimension_sentence_146

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. Dimension_sentence_147

Examples of "hyperspace philosophers" include Charles Howard Hinton, the first writer, in 1888, to use the word "tesseract"; and the Russian esotericist P. Dimension_sentence_148 D. Ouspensky. Dimension_sentence_149

More dimensions Dimension_section_17

See also Dimension_section_18

Topics by dimension Dimension_section_19

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Dimension.