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...
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space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces...
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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...
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below § Classification). The set of Euclidean plane isometries forms a group under composition: the Euclidean group in two dimensions. It is generated...
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Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements...
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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...
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Rigid transformation (redirect from Euclidean transformation)
(also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between...
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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...
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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...
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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...
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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...
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has no analogue in higher dimensions. Euclidean group Galilean group Representation theory of the Poincaré group Wigner's classification Symmetry in quantum...
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Symmetry (geometry) (redirect from Euclidean symmetries)
mathematical group, the symmetry group of the object. The most common group of transforms applied to objects are termed the Euclidean group of "isometries"...
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In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers...
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In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a...
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In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric...
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representations: in the category of affine spaces. For example, the Euclidean group acts affinely upon Euclidean space. Irreducible representations Character table Character...
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Normal subgroup (redirect from Normal group)
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...
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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...
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Erlangen program (category Group theory)
Euclidean geometry was more restrictive than affine geometry, which in turn is more restrictive than projective geometry. Klein proposed that group theory...
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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...
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Point reflection (redirect from Point reflection group)
centrally symmetric. A point group including a point reflection among its symmetries is called centrosymmetric. In Euclidean space, a point reflection is...
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Isometry (redirect from Group of isometries)
translation and rotation is a global isometry on Euclidean spaces. See also Euclidean group and Euclidean space § Isometries. The map x ↦ | x | {\displaystyle...
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Minkowski space (section Pseudo-Euclidean metrics)
four-dimensional Euclidean space insofar as it treats time differently than the three spatial dimensions. In 3-dimensional Euclidean space, the isometry group (maps...
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Semidirect product (category Group products)
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)...
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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...
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Darboux frame (section Frames on Euclidean space)
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...
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orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally...
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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...
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Rotation (mathematics) (category Euclidean symmetries)
orientation. In the language of group theory the distinction is expressed as direct vs indirect isometries in the Euclidean group, where the former comprise...
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