# Dimension

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Property of a mathematical space

This article is about the dimension of a space. For the dimension of an object, see [size](/source/Size). For the dimension of a quantity, see [Dimensional analysis](/source/Dimensional_analysis). For other uses, see [Dimension (disambiguation)](/source/Dimension_(disambiguation)).

From left to right: a [square](/source/Square_(geometry)), a [cube](/source/Cube) and a [tesseract](/source/Tesseract). The square is two-dimensional (2D) and bounded by one-dimensional [line segments](/source/Line_segment); 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.

1. Two points can be connected to create a [line segment](/source/Line_segment).
1. Two parallel line segments can be connected to form a [square](/source/Square).
1. Two parallel squares can be connected to form a [cube](/source/Cube).
1. Two parallel cubes can be connected to form a [tesseract](/source/Tesseract).

Geometry Projecting a sphere to a plane Branches Euclidean Non-Euclidean Elliptic Spherical Hyperbolic Non-Archimedean geometry Projective Affine Synthetic Analytic Algebraic Arithmetic Diophantine Differential Riemannian Symplectic Discrete differential Complex Finite Discrete/Combinatorial Digital Convex Computational Fractal Incidence Noncommutative geometry Noncommutative algebraic geometry Concepts Features Dimension Straightedge and compass constructions Angle Curve Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence Similarity Symmetry Zero-dimensional Point One-dimensional Line Line segment Ray Curve Length Two-dimensional Surface Plane Area Polygon Triangle Centers Altitude Hypotenuse Pythagorean theorem Circular Hyperbolic Spherical Quadrilateral Parallelogram Square Rectangle Rhombus Rhomboid Trapezoid Kite Circle Radius Diameter Circumference Disk Area Three-dimensional Surface area Volume Polyhedron Platonic Solid Tetrahedron cuboid Cube Octahedron Dodecahedron Icosahedron Pyramid Solid of revolution Sphere Great circle Cylinder Cone Four - other-dimensional 4-polytope Simplex 5-cell Hypercube Tesseract n-sphere Hypersphere Geometers by name Aida Aryabhata Ahmes Alhazen Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Chern Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Huygens Jyeṣṭhadeva Kātyāyana Khayyám Klein Lobachevsky Manava Minkowski Minggatu Pascal Pythagoras Parameshvara Poincaré Riemann Sakabe Sijzi al-Tusi Veblen Virasena Yang Hui al-Yasamin Zhang List of geometers by period BCE Ahmes Baudhayana Manava Pythagoras Euclid Archimedes Apollonius 1–1400s Zhang Kātyāyana Aryabhata Brahmagupta Virasena Alhazen Sijzi Khayyám al-Yasamin al-Tusi Yang Hui Parameshvara 1400s–1700s Jyeṣṭhadeva Descartes Pascal Huygens Minggatu Euler Sakabe Aida 1700s–1900s Gauss Lobachevsky Bolyai Riemann Klein Poincaré Hilbert Minkowski Cartan Veblen Coxeter Chern Present day Atiyah Gromov v t e

In [physics](/source/Physics) and [mathematics](/source/Mathematics), the **dimension** of a [mathematical space](/source/Space_(mathematics)) (or [object](/source/Mathematical_object)) is informally defined as the minimum number of [coordinates](/source/Coordinates) needed to specify any [point](/source/Point_(geometry)) within it.[1][2] Thus, a [line](/source/Line_(geometry)) has a [dimension of one](/source/One-dimensional_space) (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A [surface](/source/Surface_(mathematics)), such as the [boundary](/source/Boundary_(mathematics)) of a [cylinder](/source/Cylinder_(geometry)) or [sphere](/source/Sphere), has a [dimension of two](/source/Two-dimensional_space) (2D) because two coordinates are needed to specify a point on it – for example, both a [latitude](/source/Latitude) and [longitude](/source/Longitude) are required to locate a point on the surface of a sphere. A [two-dimensional Euclidean space](/source/Two-dimensional_Euclidean_space) is a two-dimensional space on the [plane](/source/Euclidean_plane). The inside of a [cube](/source/Cube), a cylinder or a sphere is [three-dimensional](/source/Three-dimensional) (3D) because three coordinates are needed to locate a point within these spaces.

In [classical mechanics](/source/Classical_mechanics), [space](/source/Space) and [time](/source/Time) are different categories and refer to [absolute space and time](/source/Absolute_space_and_time). That conception of the world is a [four-dimensional space](/source/Four-dimensional_space) but not the one that was found necessary to describe [electromagnetism](/source/Electromagnetism). The four dimensions (4D) of [spacetime](/source/Spacetime) consist of [events](/source/Event_(relativity)) that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an [observer](/source/Observer_(special_relativity)). [Minkowski space](/source/Minkowski_space) first approximates the universe without [gravity](/source/Gravity); the [pseudo-Riemannian manifolds](/source/Pseudo-Riemannian_manifold) of [general relativity](/source/General_relativity) describe spacetime with matter and gravity. 10 dimensions are used to describe [superstring theory](/source/Superstring_theory) (6D [hyperspace](/source/Hyperspace) + 4D), 11 dimensions can describe [supergravity](/source/Supergravity) and [M-theory](/source/M-theory) (7D hyperspace + 4D), and the state-space of [quantum mechanics](/source/Quantum_mechanics) is an infinite-dimensional [function space](/source/Function_space).

The concept of dimension is not restricted to physical objects. **High-dimensional spaces** frequently occur in mathematics and the [sciences](/source/Science). They may be [Euclidean spaces](/source/Euclidean_space) or more general [parameter spaces](/source/Parameter_space) or [configuration spaces](/source/Configuration_space_(mathematics)) such as in [Lagrangian](/source/Lagrangian_mechanics) or [Hamiltonian mechanics](/source/Hamiltonian_mechanics); these are [abstract spaces](/source/Space_(mathematics)), independent of the [physical space](/source/Physical_space).

## In mathematics

In [mathematics](/source/Mathematics), the dimension of an object is, roughly speaking, the number of [degrees of freedom](/source/Degrees_of_freedom) of a point that moves on this object. In other words, the dimension is the number of independent [parameters](/source/Parameter) or [coordinates](/source/Coordinates) 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 is [zero](/source/Zero-dimensional_space); the dimension of a [line](/source/Line_(geometry)) is [one](/source/One-dimensional_space), as a point can move on a line in only one direction (or its opposite); the dimension of a [plane](/source/Plane_(geometry)) is [two](/source/Two-dimensional_space), 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 be [embedded](/source/Embedding_(mathematics)). For example, a [curve](/source/Curve), such as a [circle](/source/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. This is independent from the fact that a curve cannot be embedded in a [Euclidean space](/source/Euclidean_space) of dimension lower than two, unless it is a line. Similarly, a [surface](/source/Surface_(mathematics)) is of dimension two, even if embedded in [three-dimensional space](/source/Three-dimensional_space).

The dimension of [Euclidean *n*-space](/source/Euclidean_space) **E***n* is *n*. When trying to generalize to other types of spaces, one is faced with the question "what makes **E***n* *n*-dimensional?" One answer is that to cover a fixed [ball](/source/Ball_(mathematics)) 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](/source/Minkowski_dimension) and its more sophisticated variant, the [Hausdorff dimension](/source/Hausdorff_dimension), but there are also other answers to that question. For example, the boundary of a ball in **E***n* looks locally like **E***n*-1 and this leads to the notion of the [inductive dimension](/source/Inductive_dimension). While these notions agree on **E***n*, they turn out to be different when one looks at more general spaces.

A [tesseract](/source/Tesseract) is an example of a four-dimensional object. 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".

Although the notion of higher dimensions goes back to [René Descartes](/source/Ren%C3%A9_Descartes), substantial development of a higher-dimensional geometry only began in the 19th century, via the work of [Arthur Cayley](/source/Arthur_Cayley), [William Rowan Hamilton](/source/William_Rowan_Hamilton), [Ludwig Schläfli](/source/Ludwig_Schl%C3%A4fli) and [Bernhard Riemann](/source/Bernhard_Riemann). Riemann's 1854 [Habilitationsschrift](/source/Habilitationsschrift), Schläfli's 1852 *[Theorie der vielfachen Kontinuität](/source/Theorie_der_vielfachen_Kontinuit%C3%A4t)*, and Hamilton's discovery of the [quaternions](/source/Quaternion) and [John T. Graves](/source/John_T._Graves)' discovery of the [octonions](/source/Octonion) in 1843 marked the beginning of higher-dimensional geometry.

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

### Vector spaces

Main article: [Dimension (vector space)](/source/Dimension_(vector_space))

The dimension of a [vector space](/source/Vector_space) is the number of vectors in any [basis](/source/Basis_(linear_algebra)) for the space, [i.e.](/source/Id_est) the number of coordinates necessary to specify any vector. This notion of dimension (the [cardinality](/source/Cardinality) of a basis) is often referred to as the *Hamel dimension* or *algebraic dimension* to distinguish it from other notions of dimension.

For the non-[free](/source/Free_module) case, this generalizes to the notion of the [length of a module](/source/Length_of_a_module).

### Manifolds

The uniquely defined dimension of every [connected](/source/Connectedness) topological [manifold](/source/Manifold) can be calculated. A connected topological manifold is [locally](/source/Local_property) [homeomorphic](/source/Homeomorphic) to Euclidean *n*-space, in which the number *n* is the manifold's dimension.

For connected [differentiable manifolds](/source/Differentiable_manifold), the dimension is also the dimension of the [tangent vector space](/source/Tangent_space) at any point.

In [geometric topology](/source/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. This state of affairs was highly marked in the various cases of the [Poincaré conjecture](/source/Poincar%C3%A9_conjecture), in which four different proof methods are applied.

#### Complex dimension

Main article: [Complex dimension](/source/Complex_dimension)

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 a [point at infinity](/source/Point_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 the [real numbers](/source/Real_numbers), it is sometimes useful in the study of [complex manifolds](/source/Complex_manifold) and [algebraic varieties](/source/Dimension_of_an_algebraic_variety) to work over the [complex numbers](/source/Complex_numbers) instead. A complex number ( x + i y {\displaystyle x+iy} ) has a [real part](/source/Real_part) x {\displaystyle x} and an [imaginary part](/source/Imaginary_part) y {\displaystyle y} , in which x {\displaystyle x} and y {\displaystyle 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-dimensional [spherical surface](/source/Sphere), when given a complex metric, becomes a [Riemann sphere](/source/Riemann_sphere) of one complex dimension.[3]

### Varieties

Main article: [Dimension of an algebraic variety](/source/Dimension_of_an_algebraic_variety)

The dimension of an [algebraic variety](/source/Algebraic_variety) may be defined in various equivalent ways. The most intuitive way is probably the dimension of the [tangent space](/source/Tangent_space) at any [Regular point of an algebraic variety](/source/Regular_point_of_an_algebraic_variety). Another intuitive way is to define the dimension as the number of [hyperplanes](/source/Hyperplane) 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.

An [algebraic set](/source/Algebraic_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 V 0 ⊊ V 1 ⊊ ⋯ ⊊ V d {\displaystyle V_{0}\subsetneq V_{1}\subsetneq \cdots \subsetneq V_{d}} of sub-varieties of the given algebraic set (the length of such a chain is the number of " ⊊ {\displaystyle \subsetneq } ").

Each variety can be considered as an [algebraic stack](/source/Stack_(mathematics)), 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, if V {\displaystyle V} is a variety of dimension m {\displaystyle m} and G {\displaystyle G} is an [algebraic group](/source/Algebraic_group) of dimension n {\displaystyle n} [acting on *V*](/source/Group_action), then the [quotient stack](/source/Quotient_stack) [ V / G ] {\displaystyle [V/G]} has dimension m − n {\displaystyle m-n} .[4]

### Krull dimension

The [Krull dimension](/source/Krull_dimension) of a [commutative ring](/source/Commutative_ring) is the maximal length of chains of [prime ideals](/source/Prime_ideal) in it, a chain of length *n* being a sequence P 0 ⊊ P 1 ⊊ ⋯ ⊊ P n {\displaystyle {\mathcal {P}}_{0}\subsetneq {\mathcal {P}}_{1}\subsetneq \cdots \subsetneq {\mathcal {P}}_{n}} 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 an [algebra over a field](/source/Algebra_over_a_field), the dimension as [vector space](/source/Vector_space) is finite if and only if its Krull dimension is 0.

### Topological spaces

For any [normal topological space](/source/Normal_topological_space) *X*, the [Lebesgue covering dimension](/source/Lebesgue_covering_dimension) of *X* is defined to be the smallest [integer](/source/Integer) *n* for which the following holds: any [open cover](/source/Open_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 than *n* + 1 elements. In this case 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 one writes dim *X* = ∞. Moreover, *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](/source/Tychonoff_space) merely by replacing the term "open" in the definition by the term "**functionally open**".

An [inductive dimension](/source/Inductive_dimension) may be defined [inductively](/source/Mathematical_induction) as follows. Consider a [discrete set](/source/Isolated_point) 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 a *new direction*, one obtains a 2-dimensional object. In general, one obtains an (*n* + 1)-dimensional 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, in the case of metric spaces, (*n* + 1)-dimensional balls have *n*-dimensional [boundaries](/source/Boundary_(topology)), 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 of [CW complexes](/source/CW_complexes), the dimension of an object is the largest n for which the [n-skeleton](/source/N-skeleton) is nontrivial. Intuitively, this can be described as follows: if the original space can be [continuously deformed](/source/Homotopy) into a collection of [higher-dimensional triangles](/source/Simplex) joined at their faces with a complicated surface, then the dimension of the object is the dimension of those triangles.[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

See also: [Dimension of a scheme](/source/Dimension_of_a_scheme)

### Hausdorff dimension

The [Hausdorff dimension](/source/Hausdorff_dimension) is useful for studying structurally complicated sets, especially [fractals](/source/Fractal). The Hausdorff dimension is defined for all [metric spaces](/source/Metric_space) and, unlike the dimensions considered above, can also have non-integer real values.[6] The [box dimension](/source/Box-counting_dimension) or [Minkowski dimension](/source/Minkowski_dimension) is a variant of the same idea. In general, there exist more definitions of [fractal dimensions](/source/Fractal_dimension) that work for highly irregular sets and attain non-integer positive real values.

### Hilbert spaces

Every [Hilbert space](/source/Hilbert_space) admits an [orthonormal basis](/source/Orthonormal_basis), and any two such bases for a particular space have the same [cardinality](/source/Cardinality). This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's [Hamel dimension](/source/Hamel_dimension) is finite, and in this case the two dimensions coincide.

## In physics

### Spatial dimensions

Classical physics theories describe three [physical dimensions](/source/Size): from a particular point in [space](/source/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](/source/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](/source/Space) and [Cartesian coordinate system](/source/Cartesian_coordinate_system).)

Number of dimensions Example co-ordinate systems 1 Number line Angle 2 Cartesian (two-dimensional) Polar Latitude and longitude 3 Cartesian (three-dimensional) Cylindrical Spherical

### Time

A **temporal dimension**, or **time dimension**, is a dimension of time. Time is often referred to as the "[fourth dimension](/source/Spacetime)" for this reason, but that is not to imply that it is a spatial dimension.[7] 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](/source/Arrow_of_time).

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](/source/Classical_mechanics) are [symmetric with respect to time](/source/T-symmetry), and equations of quantum mechanics are typically symmetric if both time and other quantities (such as [charge](/source/C-symmetry) and [parity](/source/Parity_(physics))) are reversed. In these models, the perception of time flowing in one direction is an artifact of the [laws of thermodynamics](/source/Laws_of_thermodynamics) (we perceive time as flowing in the direction of increasing [entropy](/source/Entropy)).

The best-known treatment of time as a dimension is [Poincaré](/source/Henri_Poincar%C3%A9) and [Einstein](/source/Albert_Einstein)'s [special relativity](/source/Special_relativity) (and extended to [general relativity](/source/General_relativity)), which treats perceived space and time as components of a four-dimensional [manifold](/source/Manifold), known as [spacetime](/source/Spacetime), and in the special, flat case as [Minkowski space](/source/Minkowski_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 a [fourth spatial dimension](/source/Four-dimensional_space). Time is not however present in a single point of absolute infinite [singularity](/source/Singularity_(mathematics)) as defined as a [geometric point](/source/Geometric_point), as an infinitely small point can have no change and therefore no time. Just as when an object moves through [positions](/source/Position_(geometry)) in space, it also moves through positions in time. In this sense the [force](/source/Force) moving any [object](/source/Physical_object) to change is *time*.[8][9][10]

### Additional dimensions

In physics, three dimensions of space and one of time is the accepted norm. However, there are theories that attempt to unify the four [fundamental forces](/source/Fundamental_forces) by introducing [extra dimensions](/source/Extra_dimensions)/[hyperspace](/source/Hyperspace). Most notably, [superstring theory](/source/Superstring_theory) requires 10 spacetime dimensions, and originates from a more fundamental 11-dimensional theory tentatively called [M-theory](/source/M-theory) which subsumes five previously distinct superstring theories. [Supergravity theory](/source/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" ([compactified](/source/Compactification_(physics))) at such tiny scales as to be effectively invisible to current experiments.

Illustration of a Calabi–Yau manifold

In 1921, [Kaluza–Klein theory](/source/Kaluza%E2%80%93Klein_theory) presented 5D including an extra dimension of space. At the level of [quantum field theory](/source/Quantum_field_theory), Kaluza–Klein theory unifies [gravity](/source/Gravity) with [gauge](/source/Gauge_theory) 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 reproduces [electromagnetism](/source/Electromagnetism). 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](/source/Quantum_gravity). Therefore, these models still require a [UV completion](/source/UV_completion), of the kind that string theory is intended to provide. In particular, superstring theory requires six compact dimensions (6D hyperspace) forming a [Calabi–Yau manifold](/source/Calabi%E2%80%93Yau_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 be [large extra dimensions](/source/Large_extra_dimensions). [D-branes](/source/D-brane) 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 the [brane](/source/Brane) 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 to [cosmology](/source/Brane_cosmology). For example, brane gas cosmology[11][12] 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 be [universal](/source/Universal_extra_dimension) if all fields are equally free to propagate within them.

## In computer graphics and spatial data

Main article: [Geometric primitive](/source/Geometric_primitive)

Several types of digital systems are based on the storage, analysis, and visualization of geometric shapes, including [illustration software](/source/Vector_graphics_editor), [computer-aided design](/source/Computer-aided_design), and [geographic information systems](/source/Geographic_information_systems). 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](/source/Geometric_primitive) corresponding to the spatial dimensions:[13]

- **Point** (0-dimensional), a single coordinate in a [Cartesian coordinate system](/source/Cartesian_coordinate_system).

- **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](/source/Interpolation) 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 a [polyhedron](/source/Polyhedron) 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 and [cartography](/source/Cartography), 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. This *dimensional 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).

## More dimensions

- Degrees of freedom - in [mechanics](/source/Degrees_of_freedom_(mechanics)) - in [physics and chemistry](/source/Degrees_of_freedom_(physics_and_chemistry)) - in [statistics](/source/Degrees_of_freedom_(statistics))

- [Exterior dimension](/source/Exterior_dimension)

- [Hurst exponent](/source/Hurst_exponent)

- [Isoperimetric dimension](/source/Isoperimetric_dimension)

- [Metric dimension](/source/Metric_dimension_(graph_theory))

- [Order dimension](/source/Order_dimension)

- [*q*-dimension](https://en.wikipedia.org/w/index.php?title=Q-dimension&action=edit&redlink=1) - [Fractal (*q* = 1)](/source/Fractal_dimension) - [Correlation (*q* = 2)](/source/Correlation_dimension)

## List of topics by dimension

- 0 dimension - [Point](/source/Point_(geometry)) - [Zero-dimensional space](/source/Zero-dimensional_space) - [Integer](/source/Integer)

- 1 dimension - [Line](/source/Line_(geometry)) - [Curve](/source/Curve) - [Graph (combinatorics)](/source/Graph_(discrete_mathematics)) - [Real number](/source/Real_number) - [Length](/source/Length)

- 2 dimensions - [Plane](/source/Plane_(geometry)) - [Surface](/source/Surface_(topology)) - [Polygon](/source/Polygon) - [Net](/source/Net_(polyhedron)) - [Complex number](/source/Complex_number) - [Cartesian coordinate system](/source/Cartesian_coordinate_system) - [List of uniform tilings](/source/List_of_uniform_tilings) - [Area](/source/Area)

- 3 dimensions - [Platonic solid](/source/Platonic_solid) - [Polyhedron](/source/Polyhedron) - [Stereoscopy (3-D imaging)](/source/Stereoscopy) - [3-manifold](/source/3-manifold) - [Axis of rotation](/source/Axis_of_rotation) - [Knots](/source/Knot_(mathematics)) - [Skew lines](/source/Skew_lines) - [Skew polygon](/source/Skew_polygon) - [Volume](/source/Volume)

- 4 dimensions - [Spacetime](/source/Spacetime) - [Fourth spatial dimension](/source/Four-dimensional_space) - [Convex regular 4-polytope](/source/Convex_regular_4-polytope) - [Quaternion](/source/Quaternion) - [4-manifold](/source/4-manifold) - [Polychoron](/source/Polychoron) - [Rotations in 4-dimensional Euclidean space](/source/Rotations_in_4-dimensional_Euclidean_space) - [Fourth dimension in art](/source/Fourth_dimension_in_art) - [Fourth dimension in literature](/source/Fourth_dimension_in_literature)

- 5 dimensions - [Kaluza–Klein theory](/source/Kaluza%E2%80%93Klein_theory)

- 8 dimensions - [Octonion](/source/Octonion)

- 10 dimensions - [Superstring theory](/source/Superstring_theory)

- 11 dimensions - [M-theory](/source/M-theory)

- 12 dimensions - [F-theory](/source/F-theory)

- 16 dimensions - [Sedenion](/source/Sedenion)

- 26 dimensions - [Bosonic string theory](/source/Bosonic_string_theory)

- 32 dimensions - [Trigintaduonion](/source/Trigintaduonion)

- Higher dimensions - [Vector space](/source/Vector_space) - [Plane of rotation](/source/Plane_of_rotation) - [Curse of dimensionality](/source/Curse_of_dimensionality) - [String theory](/source/String_theory)

- Infinite - [Hilbert space](/source/Hilbert_space) - [Function space](/source/Function_space)

## See also

- [Dimension (data warehouse)](/source/Dimension_(data_warehouse)) - [Dimension tables](/source/Dimension_table)

- [Dimensional analysis](/source/Dimensional_analysis)

- *[Flatland](/source/Flatland)*

- [Hyperspace (disambiguation)](/source/Hyperspace_(disambiguation))

- [Intrinsic dimension](/source/Intrinsic_dimension)

- [Mean dimension](/source/Mean_dimension)

- [Multidimensional analysis](/source/Multidimensional_analysis)

- [Space-filling curve](/source/Space-filling_curve)

- [Technical drawing](/source/Technical_drawing)

## References

1. **[^](#cite_ref-1)** Dave Kornreich (January 1999). ["Curious About Astronomy"](https://web.archive.org/web/20140111191053/http://curious.astro.cornell.edu/question.php?number=4). astro.[cornell.edu](/source/Cornell.edu). Archived from [the original](https://curious.astro.cornell.edu/question.php?number=4) on 2014-01-11. Retrieved 2014-03-03.

1. **[^](#cite_ref-2)** ["MathWorld: Dimension"](http://mathworld.wolfram.com/Dimension.html). [wolfram.com](/source/Wolfram.com). 2014-02-27. [Archived](https://web.archive.org/web/20140325220941/http://mathworld.wolfram.com/Dimension.html) from the original on 2014-03-25. Retrieved 2014-03-03.

1. **[^](#cite_ref-3)** [Yau, Shing-Tung](/source/Shing-Tung_Yau); Nadis, Steve (2010). ["4. Too Good to be True"](https://books.google.com/books?id=vlA4DgAAQBAJ&pg=PT60). *The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions*. Basic Books. pp. 60–. [ISBN](/source/ISBN_(identifier)) [978-0-465-02266-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-465-02266-3).

1. **[^](#cite_ref-4)** Fantechi, Barbara (2001), ["Stacks for everybody"](http://www.mathematik.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/fantechi.pdf) (PDF), *European Congress of Mathematics Volume I*, Progr. Math., vol. 201, Birkhäuser, pp. 349–359, [archived](https://web.archive.org/web/20060117052957/http://www.mathematik.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/fantechi.pdf) (PDF) from the original on 2006-01-17

1. **[^](#cite_ref-5)** [Hurewicz, Witold](/source/Witold_Hurewicz); [Wallman, Henry](/source/Henry_Wallman) (2015). [*Dimension Theory (PMS-4), Volume 4*](https://books.google.com/books?id=_xTWCgAAQBAJ). [Princeton University Press](/source/Princeton_University_Press). p. 24. [ISBN](/source/ISBN_(identifier)) [978-1-4008-7566-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4008-7566-5). [Extract of page 24](https://books.google.com/books?id=_xTWCgAAQBAJ&pg=PA24)

1. **[^](#cite_ref-Hausdorff_dimension_6-0)** [Fractal Dimension](http://math.bu.edu/DYSYS/chaos-game/node6.html) [Archived](https://web.archive.org/web/20061027003440/http://math.bu.edu/DYSYS/chaos-game/node6.html) 2006-10-27 at the [Wayback Machine](/source/Wayback_Machine), Boston University Department of Mathematics and Statistics

1. **[^](#cite_ref-7)** Lafleur, Laurence J. (1940). ["Time as a Fourth Dimension"](https://www.jstor.org/stable/2017930). *The Journal of Philosophy*. **37** (7): 169–178. [doi](/source/Doi_(identifier)):[10.2307/2017930](https://doi.org/10.2307%2F2017930). [ISSN](/source/ISSN_(identifier)) [0022-362X](https://search.worldcat.org/issn/0022-362X). [JSTOR](/source/JSTOR_(identifier)) [2017930](https://www.jstor.org/stable/2017930).

1. **[^](#cite_ref-8)** Rylov, Yuri A. (2007). "Non-Euclidean method of the generalized geometry construction and its application to space-time geometry". [arXiv](/source/ArXiv_(identifier)):[math/0702552](https://arxiv.org/abs/math/0702552).

1. **[^](#cite_ref-9)** Lane, Paul M.; Lindquist, Jay D. (May 22, 2015). ["Definitions for The Fourth Dimension: A Proposed Time Classification System1"](https://link.springer.com/chapter/10.1007/978-3-319-17046-6_8). In Bahn, Kenneth D. (ed.). *Proceedings of the 1988 Academy of Marketing Science (AMS) Annual Conference*. Developments in Marketing Science: Proceedings of the Academy of Marketing Science. Springer International Publishing. pp. 38–46. [doi](/source/Doi_(identifier)):[10.1007/978-3-319-17046-6_8](https://doi.org/10.1007%2F978-3-319-17046-6_8). [ISBN](/source/ISBN_(identifier)) [978-3-319-17045-9](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-17045-9) – via Springer Link.

1. **[^](#cite_ref-10)** Wilson, Edwin B.; Lewis, Gilbert N. (1912). ["The Space-Time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics"](http://www.jstor.org/stable/20022840). *Proceedings of the American Academy of Arts and Sciences*. **48** (11): 389–507. [doi](/source/Doi_(identifier)):[10.2307/20022840](https://doi.org/10.2307%2F20022840). [JSTOR](/source/JSTOR_(identifier)) [20022840](https://www.jstor.org/stable/20022840).

1. **[^](#cite_ref-11)** Brandenberger, R.; [Vafa, C.](/source/Cumrun_Vafa) (1989). "Superstrings in the early universe". *Nuclear Physics B*. **316** (2): 391–410. [Bibcode](/source/Bibcode_(identifier)):[1989NuPhB.316..391B](https://ui.adsabs.harvard.edu/abs/1989NuPhB.316..391B). [doi](/source/Doi_(identifier)):[10.1016/0550-3213(89)90037-0](https://doi.org/10.1016%2F0550-3213%2889%2990037-0).

1. **[^](#cite_ref-12)** Scott Watson, [Brane Gas Cosmology](http://www-astro-theory.fnal.gov/Conferences/cosmo02/poster/watson.pdf). [Archived](https://web.archive.org/web/20141027144123/http://www-astro-theory.fnal.gov/Conferences/cosmo02/poster/watson.pdf) 2014-10-27 at the [Wayback Machine](/source/Wayback_Machine) (pdf).

1. **[^](#cite_ref-13)** [Vector Data Models](https://saylordotorg.github.io/text_essentials-of-geographic-information-systems/s08-02-vector-data-models.html), *Essentials of Geographic Information Systems*, Saylor Academy, 2012

## Further reading

- Murty, Katta G. (2014). ["1. Systems of Simultaneous Linear Equations"](http://www.worldscientific.com/doi/suppl/10.1142/8261/suppl_file/8261_chap01.pdf) (PDF). *Computational and Algorithmic Linear Algebra and n-Dimensional Geometry*. World Scientific Publishing. [doi](/source/Doi_(identifier)):[10.1142/8261](https://doi.org/10.1142%2F8261). [ISBN](/source/ISBN_(identifier)) [978-981-4366-62-5](https://en.wikipedia.org/wiki/Special:BookSources/978-981-4366-62-5).

- [Abbott, Edwin A.](/source/Edwin_A._Abbott) (1884). [*Flatland: A Romance of Many Dimensions*](/source/Flatland). London: Seely & Co. - —. [*Flatland: ...*](https://www.gutenberg.org/ebooks/201) [Project Gutenberg](/source/Project_Gutenberg). - —; [Stewart, Ian](/source/Ian_Stewart_(mathematician)) (2008). [*The Annotated Flatland: A Romance of Many Dimensions*](https://books.google.com/books?id=mvE4DgAAQBAJ). Basic Books. [ISBN](/source/ISBN_(identifier)) [978-0-7867-2183-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7867-2183-2).

- [Banchoff, Thomas F.](/source/Thomas_Banchoff) (1996). [*Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions*](https://books.google.com/books?id=cstzQgAACAAJ). Scientific American Library. [ISBN](/source/ISBN_(identifier)) [978-0-7167-6015-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7167-6015-3).

- [Pickover, Clifford A.](/source/Clifford_A._Pickover) (2001). [*Surfing through Hyperspace: Understanding Higher Universes in Six Easy Lessons*](https://books.google.com/books?id=LOUiWjsgue8C). [Oxford University Press](/source/Oxford_University_Press). [ISBN](/source/ISBN_(identifier)) [978-0-19-992381-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-992381-6).

- [Rucker, Rudy](/source/Rudy_Rucker) (2014) [1984]. [*The Fourth Dimension: Toward a Geometry of Higher Reality*](/source/The_Fourth_Dimension_(book)). Courier Corporation. [ISBN](/source/ISBN_(identifier)) [978-0-486-77978-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-77978-2). [Google preview](https://books.google.com/books?id=Vgk7BAAAQBAJ)

- [Kaku, Michio](/source/Michio_Kaku) (1994). [*Hyperspace, a Scientific Odyssey Through the 10th Dimension*](/source/Hyperspace_(book)). Oxford University Press. [ISBN](/source/ISBN_(identifier)) [978-0-19-286189-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-286189-4).

- [Krauss, Lawrence M.](/source/Lawrence_M._Krauss) (2005). [*Hiding in the Mirror*](/source/Hiding_in_the_Mirror). Viking Press. [ISBN](/source/ISBN_(identifier)) [978-0-670-03395-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-670-03395-9).

## External links

Wikiquote has quotations related to ***[Dimension](https://en.wikiquote.org/wiki/Special:Search/Dimension)***.

Wikimedia Commons has media related to [Dimensions](https://commons.wikimedia.org/wiki/Category:Dimensions).

[Wikisource](/source/Wikisource) has the text of the [1911 *Encyclopædia Britannica*](/source/Encyclop%C3%A6dia_Britannica_Eleventh_Edition) article "[Dimension](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Dimension)".

- Copeland, Ed (2009). ["Extra Dimensions"](http://www.sixtysymbols.com/videos/dimensions.htm). *Sixty Symbols*. [Brady Haran](/source/Brady_Haran) for the [University of Nottingham](/source/University_of_Nottingham).

v t e Dimension Dimensional spaces Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime Other dimensions Krull Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category

v t e Tensors Glossary of tensor theory Scope Mathematics Coordinate system Differential geometry Dyadic algebra Euclidean geometry Exterior calculus Multilinear algebra Tensor algebra Tensor calculus Physics Engineering Computer vision Continuum mechanics Electromagnetism General relativity Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor field Tensor density Tensors in curvilinear coordinates Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative Exterior derivative Exterior product Hodge star operator Lie derivative Raising and lowering indices Symmetrization Tensor contraction Tensor product Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors Differential form One-form Dimension Exterior form Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker delta Levi-Civita symbol Metric tensor Nonmetricity tensor Ricci curvature Riemann curvature tensor Torsion tensor Weyl tensor Physics Moment of inertia Angular momentum tensor Spin tensor Cauchy stress tensor stress–energy tensor Einstein tensor EM tensor Gluon field strength tensor Metric tensor (GR) Mathematicians Élie Cartan Augustin-Louis Cauchy Elwin Bruno Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus Schouten Woldemar Voigt Hermann Weyl

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