this article is about the dimension of a space. for the dimension of an object, see
size. for the dimension of a quantity, see
dimensional analysis. for other uses, see
dimension (disambiguation).
maximum number of independent directions within a mathematical space
from left to right: the
square, the
cube and the
tesseract. the
two-dimensional (2d) square is bounded by
one-dimensional (1d) lines; the
three-dimensional (3d) cube by two-dimensional areas; and the
four-dimensional (4d) tesseract by three-dimensional volumes. for display on a two-dimensional surface such as a screen, the 3d cube and 4d tesseract require
projection.
the first four spatial dimensions, represented in a two-dimensional picture.
- two points can be connected to create a line segment.
- two parallel line segments can be connected to form a square.
- two parallel squares can be connected to form a cube.
- two parallel cubes can be connected to form a tesseract.
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.^{[1]}^{[2]} 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. 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. 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.
in classical mechanics, space and time are different categories and refer to absolute space and time. that conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. 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. minkowski space first approximates the universe without gravity; the pseudo-riemannian manifolds of general relativity describe spacetime with matter and gravity. 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.
the concept of dimension is not restricted to physical objects. high-dimensional spaces frequently occur in mathematics and the sciences. 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.