A point in three-dimensional Euclidean space can be located by three coordinates.
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension.[1] For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.
Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his Elements,[2] with the great innovation of proving all properties of the space as theorems, by starting from a few fundamental properties, called postulates, which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (parallel postulate).
After the introduction at the end of the 19th century of non-Euclidean geometries, the old postulates were re-formalized to define Euclidean spaces through axiomatic theory. Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article.[3] In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space.
There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are isomorphic. Therefore it is usually possible to work with a specific Euclidean space, denoted or , which can be represented using Cartesian coordinates as the real n-space equipped with the standard dot product.
Euclideanspace is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional...
In mathematics, the Euclidean distance between two points in Euclideanspace is the length of the line segment between them. It can be calculated from...
In mathematics, a Euclidean plane is a Euclideanspace of dimension two, denoted E 2 {\displaystyle {\textbf {E}}^{2}} or E 2 {\displaystyle \mathbb {E}...
examine geometries that are non-Euclidean, in which space is conceived as curved, rather than flat, as in the Euclideanspace. According to Albert Einstein's...
mathematician. It is the name of: Euclideanspace, the two-dimensional plane and three-dimensional space of Euclidean geometry as well as their higher...
generalize the notion of a closed and bounded subset of Euclideanspace. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it...
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements...
a point on the surface of a sphere. A two-dimensional Euclideanspace is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere...
product) of Euclideanspace, even though it is not the only inner product that can be defined on Euclideanspace (see Inner product space for more). Algebraically...
dimensions. In 3-dimensional Euclideanspace, the isometry group (maps preserving the regular Euclidean distance) is the Euclidean group. It is generated by...
calculus on Euclideanspace is a generalization of calculus of functions in one or several variables to calculus of functions on Euclideanspace R n {\displaystyle...
analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclideanspace with its usual notion of distance. Other well-known examples...
topological space that locally resembles real n-dimensional Euclideanspace. Topological manifolds are an important class of topological spaces, with applications...
In mathematics, a manifold is a topological space that locally resembles Euclideanspace near each point. More precisely, an n {\displaystyle n} -dimensional...
magnitude (or length) and direction. Euclidean vectors can be added and scaled to form a vector space. A Euclidean vector is frequently represented by...
rotations of three-dimensional space and the hyperbolic space, of which any representation as a submanifold of Euclideanspace will fail to represent their...
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclideanspaces in such a way that these are independent...
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclideanspace E n {\displaystyle \mathbb {E} ^{n}} ; that is, the transformations...
Curved space often refers to a spatial geometry which is not "flat", where a flat space has zero curvature, as described by Euclidean geometry. Curved...
Global Information
