Euclidean Space

In mathematics, Euclidean space is the Euclidean plane, the three-dimensional space of Euclidean geometry, and generalizations of these ideas to higher dimensions. The term “Euclidean” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity. It is named for the Greek mathematician Euclid of Alexandria.

Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. In modern mathematics, it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. This approach brings the tools of algebra and calculus to bear on questions of geometry, and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions.

From the modern viewpoint, there is essentially only one Euclidean space of each dimension. In dimension one this is the real line; in dimension two it is the Cartesian plane; and in higher dimensions it is a coordinate space with three or more real number coordinates – in short, an n-dimensional real coordinate space. A point in Euclidean space may be identified by a tuple of real numbers, and distances are defined using the Euclidean distance formula. Mathematicians often denote the n-dimensional Euclidean space by, or sometimes if they wish to emphasize its Euclidean nature. Euclidean spaces have finite dimension.