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...

space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces...

group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types...

below § Classification). The set of Euclidean plane isometries forms a group under composition: the Euclidean group in two dimensions. It is generated...

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

ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle. Each triangle group is the symmetry group of a tiling of the Euclidean plane...

(also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between...

In 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...

non-Euclidean crystallographic group, NEC group or N.E.C. group is a discrete group of isometries of the hyperbolic plane. These symmetry groups correspond...

collection of them is then said to form an isometry group of the pseudo-Euclidean space. The isometry group of the subspace of a metric space consisting of...

mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that...

mathematical group, the symmetry group of the object. The most common group of transforms applied to objects are termed the Euclidean group of "isometries"...

In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers...

In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a...

has no analogue in higher dimensions. Euclidean group Galilean group Representation theory of the Poincaré group Wigner's classification Symmetry in quantum...

In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric...

representations: in the category of affine spaces. For example, the Euclidean group acts affinely upon Euclidean space. Irreducible representations Character table Character...

pieces or the edge pieces are normal. The translation group is a normal subgroup of the Euclidean group in any dimension. This means: applying a rigid transformation...

specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows...

Euclidean geometry was more restrictive than affine geometry, which in turn is more restrictive than projective geometry. Klein proposed that group theory...

centrally symmetric. A point group including a point reflection among its symmetries is called centrosymmetric. In Euclidean space, a point reflection is...

Bianchi V subalgebra, isomorphic to the Lie algebra of Hom(2), the group of euclidean homotheties, X 1 , X 2 , X 4 {\displaystyle X_{1},X_{2},X_{4}} generate...

translation and rotation is a global isometry on Euclidean spaces. See also Euclidean group and Euclidean space § Isometries. The map x ↦ | x | {\displaystyle...

four-dimensional Euclidean space insofar as it treats time differently than the three spatial dimensions. In 3-dimensional Euclidean space, the isometry group (maps...

embedded in Euclidean space. It is named after French mathematician Jean Gaston Darboux. Let S be an oriented surface in three-dimensional Euclidean space E3...

orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally...

the group of translations is a normal subgroup of the Euclidean group, that the Euclidean group is a semidirect product of the translation group and O(2)...

objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it; in particular...

within the Euclidean group E(n) of a translation is the set of all translations by the same distance. The smallest subgroup of the Euclidean group containing...

orientation. In the language of group theory the distinction is expressed as direct vs indirect isometries in the Euclidean group, where the former comprise...