Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
It is named after the Ancient Greek mathematician Euclid of Alexandria.
The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions.
Description
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. Geometric constructions are also used to define rational numbers.
When algebra and mathematical analysis became developed enough, this relation reversed and now it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry.
It means that points of the space are specified with collections of real numbers, and geometric shapes are defined as equations and inequalities. 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.
See also
- Euclidean geometry
- Function of several real variables, a coordinate presentation of a function on a Euclidean space
- Geometric algebra, an alternative algebraic formalism
- High-dimensional space
- Real coordinate space, a frequently used representation of Euclidean space
- Three-dimensional space
- Two-dimensional space
- Vector calculus, a standard algebraic formalism
- Vector space
External links
- Euclidean geometry @ Wikipedia
