Four-dimensional Space

In mathematics, four-dimensional space ("4D") is an abstract concept derived by generalizing the rules of three-dimensional space. It has been studied by mathematicians and philosophers for almost three hundred years, both for its own interest and for the insights it offered into mathematics and related fields.

Algebraically it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions. In particular a vector with four elements (a 4-tuple) can be used to represent a position in four-dimensional space. The space is a Euclidean space, so has a metric and norm, and so all directions are treated as the same: the additional dimension is indistinguishable from the other three.

In modern physics, space and time are unified in a four-dimensional Minkowski continuum called spacetime, whose metric treats the time dimension differently from the three spatial dimensions (see below for the definition of the Minkowski metric/pairing). Spacetime is thus not a Euclidean space.