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...

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

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

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...

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

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...

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

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...

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

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...

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...

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...

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...

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...

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

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...

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

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

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

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

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...

group of A. An example is the action of the Euclidean group E(n) on the Euclidean space En. Since the affine group in dimension n is a matrix group in...

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...

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

point group, Schoenflies notation Discrete group Euclidean group Even and odd permutations Frieze group Frobenius group Fuchsian group Geometric group theory...

In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely...