Dimension

dimensionsdimensionalone-dimensionalthree-dimensional3Ddimensionalitydimension theorytwo-dimensionalhigh-dimensionalhigher dimensions
[[File:Dimension levels.svg|thumb | 236px | The first four spatial dimensions, represented in a two-dimensional picture.wikipedia
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Two-dimensional space

Euclidean planetwo-dimensional2D
A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere.
For a generalization of the concept, see dimension.

Space

spatialphysical spacereal space
In classical mechanics, space and time are different categories and refer to absolute space and time.
Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime.

Four-dimensional space

four-dimensionalfourth dimensionfour dimensions
That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism.
Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world.

Time

temporaldurationsequence of events
In classical mechanics, space and time are different categories and refer to absolute space and time.
Time is often referred to as a fourth dimension, along with three spatial dimensions.

Spacetime

space-timespace-time continuumspace and time
The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer.
In physics, spacetime is any mathematical model which fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.

Line (geometry)

linestraight linelines
Thus a line has a dimension of one because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line.
In two dimensions, i.e., the Euclidean plane, two lines which do not intersect are called parallel.

Point (geometry)

pointpointslocation
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.
In particular, the geometric points do not have any length, area, volume or any other dimensional attribute.

Plane (geometry)

planeplanarplanes
A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere.
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.

Euclidean space

EuclideanspaceEuclidean vector space
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.
Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the Euclidean plane (dimension two).

M-theory

M theoryMysterious dualityE 8 gauge theory approach to M-theory
Ten dimensions are used to describe superstring theory, eleven dimensions can describe supergravity and M-theory, and the state-space of quantum mechanics is an infinite-dimensional function space.
In string theory, the point-like particles of particle physics are replaced by one-dimensional objects called strings.

Ludwig Schläfli

SchläfliSchläfli, Ludwig
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.
Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional spaces.

Supergravity

supergravity theorySUGRAmSUGRA
Ten dimensions are used to describe superstring theory, eleven dimensions can describe supergravity and M-theory, and the state-space of quantum mechanics is an infinite-dimensional function space.
It was quickly generalized to many different theories in various numbers of dimensions and involving additional (N) supersymmetries.

Sphere

sphericalhemisphereglobose
A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere.
Spheres can be generalized to spaces of any number of dimensions.

Coordinate system

coordinatescoordinateaxis
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.
For example, in 1D, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to −3, so that the coordinate of each point becomes 3 more.

Physics

physicistphysicalphysicists
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.
Beyond the known universe, the field of theoretical physics also deals with hypothetical issues, such as parallel universes, a multiverse, and higher dimensions.

Hyperplane

hyperplanesaffine hyperplanehyper-plane
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).
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.

Degrees of freedom

degree of freedomdegrees of freedom (DoF)N-dof
In mathematics, the dimension of an object is, roughly speaking, the number of degrees of freedom of a point that moves on this object.

Superstring theory

superstringsuperstring theoriessuperstrings
Ten dimensions are used to describe superstring theory, eleven dimensions can describe supergravity and M-theory, and the state-space of quantum mechanics is an infinite-dimensional function space.
Our physical space is observed to have three large spatial dimensions and, along with time, is a boundless 4-dimensional continuum known as spacetime.

Vector space

vectorvector spacesvectors
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.
In his work, the concepts of linear independence and dimension, as well as scalar products are present.

Manifold

manifoldsboundarymanifold with boundary
The uniquely defined dimension of every connected topological manifold can be calculated.
Generally manifolds are taken to have a fixed dimension (the space must be locally homeomorphic to a fixed n-ball), and such a space is called an n-manifold; however, some authors admit manifolds where different points can have different dimensions.

Bernhard Riemann

RiemannGeorg Friedrich Bernhard RiemannRiemann, Bernhard
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.
He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality.

Lebesgue covering dimension

topological dimensioncovering dimensiondimension
, the Lebesgue covering dimension of
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a

Simplex

simplicesstandard simplex4-simplex
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.
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

Length of a module

lengthfinite lengthfinite length module
For the non-free case, this generalizes to the notion of the length of a module.
There are also various ideas of dimension that are useful.

Hilbert space

Hilbert spacesHilbertseparable Hilbert space
Every Hilbert space admits an orthonormal basis, and any two such bases for a particular space have the same cardinality.
It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions.