Wikipedia
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.
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. With Cartesian coordinates it is modelled by the real coordinate space of the same dimension. In one dimension, this is the real line; in two dimensions, it is the Cartesian plane; and in higher dimensions it is a coordinate space with three or more real number coordinates. Mathematicians denote the -dimensional Euclidean space by if they wish to emphasize its Euclidean nature, but is used as well since the latter is assumed to have the standard Euclidean structure, and these two structures are not always distinguished. Euclidean spaces have finite dimension.
Usage examples of "euclidean space".
N-sphere: An N-dimensional space without boundaries which can be embedded in (N+1)-dimensional Euclidean space as the surface (or hypersurface) equidistant from some point.
Because one is using Euclidean space-times, in which the time direction is on the same footing as directions in space, it is possible for space-time to be finite in extent and yet to have no singularities that formed a boundary or edge.
She swung her sword at the kitchen walls and sliced them into great sheets curving away into a gray Euclidean space, reality curling like orange peel.
Ginny and I are pretty good shots in the nearly Euclidean space of this plenum.
It's true that as far as Euclidean space is concerned, mazes ramble all over the place, with some planets as much as a thousand light-years away from the home system.
C60 is the most symmetric molecule possible in three-dimensional Euclidean space.
Here was topiary on a grand scale, arranged across Hilbert rather than Euclidean space.