Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional...

height (often labeled x, y, and z). This concept of ordinary space is called Euclidean space because it corresponds to Euclid's geometry, which was originally...

In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from...

three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called...

mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry...

the parent space which retains the same structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological...

In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted E 2 {\displaystyle {\textbf {E}}^{2}} or E 2 {\displaystyle \mathbb {E}...

that has magnitude (or length) and direction. Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical...

idealization of a flat surface in physical space such as a sheet of paper or a chalkboard. On the Euclidean plane, any two points can be joined by a unique...

a Euclidean space of dimension n, En (Euclidean line, E; Euclidean plane, E2; Euclidean three-dimensional space, E3) form a real coordinate space of...

topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications...

mathematics and theoretical physics, a pseudo-Euclidean space of signature (k, n-k) is a finite-dimensional real n-space together with a non-degenerate quadratic...

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements...

is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the n {\displaystyle...

a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere...

dimensions. In 3-dimensional Euclidean space, the isometry group (maps preserving the regular Euclidean distance) is the Euclidean group. It is generated by...

three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the whole space. Several...

of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently...

generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it...

n-dimensional space. Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in non-Euclidean spaces, such as...

number and zero. In vector spaces, the Euclidean norm is a measure of magnitude used to define a distance between two points in space. In physics, magnitude...

defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in n dimensions...

six-dimensional Euclidean space, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional elliptical space and hyperbolic spaces are also studied...

In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space E n {\displaystyle \mathbb {E} ^{n}} ; that is, the transformations...

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n {\displaystyle n} -dimensional...

analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples...

In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent...

is a fundamental result about convergence in a finite-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . The theorem states that each...

examine geometries that are non-Euclidean, in which space is conceived as curved, rather than flat, as in the Euclidean space. According to Albert Einstein's...

especially general topology, the Euclidean topology is the natural topology induced on n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb...