Euclidean group

Representation theory divides into subtheories depending on the kind of group being represented. The various theories are quite different in detail, though some basic definitions and concepts are similar. The most important divisions are: Finite groups — Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to crystallography and to geometry. If the field of scalars of the vector space has characteristic p, and if p divides ...

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Group representation, Group representation - Branches of representation theory, Group representation - Basic definitions, Group representation - Simple example, Group representation - Equivalence of representations, Group representation - Reducibility, Group representation - Character theory, Group representation - Generalizations, Group representation - Set-theoretical representations, Group representation - Representations in other categories

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Consider the solid ball in R3 of radius π (that is, all points of R3 of distance π or less from the origin). Given the above, for every point in this ball there is a rotation, with axis through the point and rotation angle equal to the distance of the point from the origin. The identity rotation corresponds to the point at the center of the ball. Rotation through angles between 0 and -π correspond to the point on the same axis and distance from the origin but on the opposite side of the origin. The one ...

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Rotation group, Rotation group - Properties, Rotation group - Orthogonal matrices, Rotation group - Axis of rotation, Rotation group - Topology, Rotation group - Representations of rotations, Rotation group - Generalizations

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The set of all symmetry operations considered, on all objects in a set X, can be modelled as a group action g : G × X → X, where the image of g in G and x in X is written as g·x. If, for some g, g·x = y then x and y are said to be symmetric to each other. For each object x, operations g for which g·x = x form a group, the symmetry group of the object, a subgroup of GSee also:

Symmetry, Symmetry - Mathematical model for symmetry, Symmetry - Non-isometric symmetry, Symmetry - Reflection symmetry, Symmetry - Rotational symmetry, Symmetry - Translational symmetry, Symmetry - Glide reflection symmetry, Symmetry - Rotoreflection symmetry, Symmetry - Screw axis symmetry, Symmetry - Symmetry combinations, Symmetry - Color, Symmetry - Similarity vs. sameness, Symmetry - More on symmetry in geometry, Symmetry - Symmetry in mathematics, Symmetry - Symmetry in logic, Symmetry - Generalization of symmetry, Symmetry - Symmetry in physics, Symmetry - Symmetry in biology, Symmetry - Symmetry in chemistry, Symmetry - Symmetry in the arts and crafts, Symmetry - Architecture, Symmetry - Pottery, Symmetry - Quilts, Symmetry - Carpets rugs, Symmetry - Music, Symmetry - Other arts and crafts, Symmetry - Aesthetics, Symmetry - Symmetry in games and puzzles, Symmetry - Symmetry in literature, Symmetry - Symmetry in telecommunications, Symmetry - Moral symmetry

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The set of all symmetry operations considered, on all objects in a set X, can be modelled as a group action g : G × X → X, where the image of g in G and x in X is written as g·x. If, for some g, g·x = y then x and y are said to be symmetric to each other. For each object x, operations g for which g·x = x form a group, the symmetry group of the object, a subgroup of GSee also:

Symmetry, Symmetry - Mathematical model for symmetry, Symmetry - Non-isometric symmetry, Symmetry - Reflection symmetry, Symmetry - Rotational symmetry, Symmetry - Translational symmetry, Symmetry - Glide reflection symmetry, Symmetry - Rotoreflection symmetry, Symmetry - Screw axis symmetry, Symmetry - Symmetry combinations, Symmetry - Color, Symmetry - Similarity vs. sameness, Symmetry - More on symmetry in geometry, Symmetry - Symmetry in mathematics, Symmetry - Symmetry in logic, Symmetry - Generalization of symmetry, Symmetry - Symmetry in physics, Symmetry - Symmetry in biology, Symmetry - Symmetry in chemistry, Symmetry - Symmetry in the arts and crafts, Symmetry - Architecture, Symmetry - Pottery, Symmetry - Quilts, Symmetry - Carpets rugs, Symmetry - Music, Symmetry - Other arts and crafts, Symmetry - Aesthetics, Symmetry - Symmetry in games and puzzles, Symmetry - Symmetry in literature, Symmetry - Symmetry in telecommunications, Symmetry - Moral symmetry, Symmetry - Related topics

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Two triangles are congruent if their corresponding sides and angles are equal. Usually it is sufficient to establish the equality of three corresponding parts and use one of the following results to conclude the congruence of the two triangles: SAS (Side-Angle-Side): Two triangles are congruent if a pair of corresponding sides and the included angle are equal. SSS (Side-Side-Side): Two triangles are congruent if their corresponding sides are equal. ASA (Angle-Side-Angle): Two triangles are congruent if ...

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Congruence geometry, Congruence geometry - Definition of congruence in analytic geometry, Congruence geometry - Congruence of triangles

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The first three Frenet vectors and generalized curvatures can be visualized in three-dimensional space. They have additional names and more semantic information attached to them. Differential geometry of curves - Tangent vector. At every point of a C1 curve we can define a tangent vector. If γ is interpreted as the path of a particle then the tangent vector can be visualized as the path that the particle takes when free from outer force. The tangent ve ...

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Differential geometry of curves, Differential geometry of curves - Definitions, Differential geometry of curves - Examples, Differential geometry of curves - Reparametrization and equivalence relation, Differential geometry of curves - Length and natural parametrization, Differential geometry of curves - Frenet frame, Differential geometry of curves - Special Frenet vectors and generalized curvatures, Differential geometry of curves - Tangent vector, Differential geometry of curves - Normal or curvature vector, Differential geometry of curves - Curvature, Differential geometry of curves - Binormal vector, Differential geometry of curves - Torsion, Differential geometry of curves - Main theorem of curve theory, Differential geometry of curves - Frenet-Serret formulas, Differential geometry of curves - 2-dimensions, Differential geometry of curves - 3-dimensions, Differential geometry of curves - n dimensions general formula

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A symmetry group consists of isometric affine transformations; each is given by an orthogonal matrix and a translation vector (which may be the zero vector). Space groups can be grouped by the matrices involved, i.e. ignoring the translation vectors (see also Euclidean group). This corresponds to discrete symmetry groups with a fixed point. There are infinitely many of these point groups in three dimensions. However, only part of these are compatible with translational symmetry: the crystallographic point groups. This is expressed in the crystallographic restriction theorem. (In spite of these n ...

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Crystal system, Crystal system - Crystallographic point group, Crystal system - Overview of point groups by crystal system, Crystal system - Classification of lattices

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Up to conjugacy the discrete point groups in 2 dimensional space are the following classes: cyclic groups C1, C2, C3, C4,... where Cn consists of all rotations about a fixed point by multiples of the angle 360°/n dihedral groups D1, D2, D3, D4,... where Dn (of order 2n) consists of the rotations in Cn together with reflections in nSee also:

Symmetry group, Symmetry group - One dimension, Symmetry group - Two dimensions, Symmetry group - Three dimensions, Symmetry group - Symmetry groups in general, Symmetry group - External link

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In a Euclidean system, congruence is fundamental; it's the counterpart of an equals sign in numerical analysis. In analytic geometry, congruence may be defined intuitively thus: two mappings of figures onto one Cartesian coordinate system are congruent if and only if, for any two points in the first mapping, the Euclidean distance between them is equal to the Euclidean distance between the corresponding points in the second mapping. A more formal definition: two subsets A and B of Euclidean space Rn ...

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Congruence geometry, Congruence geometry - Definition of congruence in analytic geometry, Congruence geometry - Congruence of triangles

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The length l of a smooth curve γ : [a, b] → Rn can be defined as The length of a curve is invariant under reparametrization and therefore a differential geometric property of the curve. For each regular Cr-curve γ: [a, b] → Rn we can define a function Writing we get a reparametrization of γ which is called natural, arc-length or unit speed pa ...

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Differential geometry of curves, Differential geometry of curves - Definitions, Differential geometry of curves - Examples, Differential geometry of curves - Reparametrization and equivalence relation, Differential geometry of curves - Length and natural parametrization, Differential geometry of curves - Frenet frame, Differential geometry of curves - Special Frenet vectors and generalized curvatures, Differential geometry of curves - Tangent vector, Differential geometry of curves - Normal or curvature vector, Differential geometry of curves - Curvature, Differential geometry of curves - Binormal vector, Differential geometry of curves - Torsion, Differential geometry of curves - Main theorem of curve theory, Differential geometry of curves - Frenet-Serret formulas, Differential geometry of curves - 2-dimensions, Differential geometry of curves - 3-dimensions, Differential geometry of curves - n dimensions general formula

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Let n be a natural number, r an natural number or ∞, I be a non-empty interval of real numbers and t in I. A vector valued function of class Cr (i.e. γ is r times continuously differentiable) is called a parametric curve of class Cr or a Cr parametrization of the curve γ. t is called the parameter of the ...

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Differential geometry of curves, Differential geometry of curves - Definitions, Differential geometry of curves - Examples, Differential geometry of curves - Reparametrization and equivalence relation, Differential geometry of curves - Length and natural parametrization, Differential geometry of curves - Frenet frame, Differential geometry of curves - Special Frenet vectors and generalized curvatures, Differential geometry of curves - Tangent vector, Differential geometry of curves - Normal or curvature vector, Differential geometry of curves - Curvature, Differential geometry of curves - Binormal vector, Differential geometry of curves - Torsion, Differential geometry of curves - Main theorem of curve theory, Differential geometry of curves - Frenet-Serret formulas, Differential geometry of curves - 2-dimensions, Differential geometry of curves - 3-dimensions, Differential geometry of curves - n dimensions general formula

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Given the image of a curve one can define several different parametrizations of the curve. Differential geometry aims to describe properties of curves invariant under certain reparametrizations. So we have to define a suitable equivalence relation on the set of all parametric curves. The differential geometric properties of a curve (length, frenet frame and generalized curvature) are invariant under reparametrization and therefore properties of the equivalence class.The equivalence classes are called C< ...

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Differential geometry of curves, Differential geometry of curves - Definitions, Differential geometry of curves - Examples, Differential geometry of curves - Reparametrization and equivalence relation, Differential geometry of curves - Length and natural parametrization, Differential geometry of curves - Frenet frame, Differential geometry of curves - Special Frenet vectors and generalized curvatures, Differential geometry of curves - Tangent vector, Differential geometry of curves - Normal or curvature vector, Differential geometry of curves - Curvature, Differential geometry of curves - Binormal vector, Differential geometry of curves - Torsion, Differential geometry of curves - Main theorem of curve theory, Differential geometry of curves - Frenet-Serret formulas, Differential geometry of curves - 2-dimensions, Differential geometry of curves - 3-dimensions, Differential geometry of curves - n dimensions general formula

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You can find the use of symmetry across a wide variety of arts and crafts. Symmetry - Architecture. Symmetry has long been a predominant design element in architecture; prominent examples include the Leaning Tower of Pisa, Monticello, the Astrodome, the Sydney Opera House, Gothic church windows, and the Pantheon. Symmetry is used in the design of the overall floor plan of buildings as well as the design of individual building elements such as doors, windows, floors, frieze work, a ...

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Symmetry, Symmetry - Mathematical model for symmetry, Symmetry - Non-isometric symmetry, Symmetry - Reflection symmetry, Symmetry - Rotational symmetry, Symmetry - Translational symmetry, Symmetry - Glide reflection symmetry, Symmetry - Rotoreflection symmetry, Symmetry - Screw axis symmetry, Symmetry - Symmetry combinations, Symmetry - Color, Symmetry - Similarity vs. sameness, Symmetry - More on symmetry in geometry, Symmetry - Symmetry in mathematics, Symmetry - Symmetry in logic, Symmetry - Generalization of symmetry, Symmetry - Symmetry in physics, Symmetry - Symmetry in biology, Symmetry - Symmetry in chemistry, Symmetry - Symmetry in the arts and crafts, Symmetry - Architecture, Symmetry - Pottery, Symmetry - Quilts, Symmetry - Carpets rugs, Symmetry - Music, Symmetry - Other arts and crafts, Symmetry - Aesthetics, Symmetry - Symmetry in games and puzzles, Symmetry - Symmetry in literature, Symmetry - Symmetry in telecommunications, Symmetry - Moral symmetry

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A dyadic relation R is symmetric if and only if, whenever it's true that Rab, it's true that Rba. Thus, “is the same age as” is symmetrical, for if Paul is the same age as Mary, then Mary is the same age as Paul. Symmetric binary logical connectives are "and" (∧, , or &), "or" (∨), "biconditional" (iff) (↔), NAND ("not-and"), XOR ("not-biconditional"), and NOR ("not-or"). ...

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Symmetry, Symmetry - Mathematical model for symmetry, Symmetry - Non-isometric symmetry, Symmetry - Reflection symmetry, Symmetry - Rotational symmetry, Symmetry - Translational symmetry, Symmetry - Glide reflection symmetry, Symmetry - Rotoreflection symmetry, Symmetry - Screw axis symmetry, Symmetry - Symmetry combinations, Symmetry - Color, Symmetry - Similarity vs. sameness, Symmetry - More on symmetry in geometry, Symmetry - Symmetry in mathematics, Symmetry - Symmetry in logic, Symmetry - Generalization of symmetry, Symmetry - Symmetry in physics, Symmetry - Symmetry in biology, Symmetry - Symmetry in chemistry, Symmetry - Symmetry in the arts and crafts, Symmetry - Architecture, Symmetry - Pottery, Symmetry - Quilts, Symmetry - Carpets rugs, Symmetry - Music, Symmetry - Other arts and crafts, Symmetry - Aesthetics, Symmetry - Symmetry in games and puzzles, Symmetry - Symmetry in literature, Symmetry - Symmetry in telecommunications, Symmetry - Moral symmetry

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An example of a mathematical expression exhibiting symmetry is a2c + 3ab + b2c. If a and b are exchanged, the expression remains unchanged due to the commutativity of addition and multiplication. Like in geometry, for the terms there are two possibilities: it is itself symmetric it has one or more other t ...

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Symmetry, Symmetry - Mathematical model for symmetry, Symmetry - Non-isometric symmetry, Symmetry - Reflection symmetry, Symmetry - Rotational symmetry, Symmetry - Translational symmetry, Symmetry - Glide reflection symmetry, Symmetry - Rotoreflection symmetry, Symmetry - Screw axis symmetry, Symmetry - Symmetry combinations, Symmetry - Color, Symmetry - Similarity vs. sameness, Symmetry - More on symmetry in geometry, Symmetry - Symmetry in mathematics, Symmetry - Symmetry in logic, Symmetry - Generalization of symmetry, Symmetry - Symmetry in physics, Symmetry - Symmetry in biology, Symmetry - Symmetry in chemistry, Symmetry - Symmetry in the arts and crafts, Symmetry - Architecture, Symmetry - Pottery, Symmetry - Quilts, Symmetry - Carpets rugs, Symmetry - Music, Symmetry - Other arts and crafts, Symmetry - Aesthetics, Symmetry - Symmetry in games and puzzles, Symmetry - Symmetry in literature, Symmetry - Symmetry in telecommunications, Symmetry - Moral symmetry

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The German geometer Felix Klein enunciated a very influential Erlangen programme in 1872, suggesting symmetry as unifying and organising principle in geometry (at a time when that was read 'geometries'). This is a broad rather than deep principle. Initially it led to interest in the groups attached to geometries, and the slogan transformation geometry (an aspect of the New Math, but hardly controversial in modern mathematical practice). By now it has been applied i ...

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Symmetry, Symmetry - Mathematical model for symmetry, Symmetry - Non-isometric symmetry, Symmetry - Reflection symmetry, Symmetry - Rotational symmetry, Symmetry - Translational symmetry, Symmetry - Glide reflection symmetry, Symmetry - Rotoreflection symmetry, Symmetry - Screw axis symmetry, Symmetry - Symmetry combinations, Symmetry - Color, Symmetry - Similarity vs. sameness, Symmetry - More on symmetry in geometry, Symmetry - Symmetry in mathematics, Symmetry - Symmetry in logic, Symmetry - Generalization of symmetry, Symmetry - Symmetry in physics, Symmetry - Symmetry in biology, Symmetry - Symmetry in chemistry, Symmetry - Symmetry in the arts and crafts, Symmetry - Architecture, Symmetry - Pottery, Symmetry - Quilts, Symmetry - Carpets rugs, Symmetry - Music, Symmetry - Other arts and crafts, Symmetry - Aesthetics, Symmetry - Symmetry in games and puzzles, Symmetry - Symmetry in literature, Symmetry - Symmetry in telecommunications, Symmetry - Moral symmetry

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With a color image one can associate a greyshade or black-and-white image. One way is to associate with each color a greyshade or either black or white. Alternatively, boundaries may be represented in black, and interior areas in white. When considering symmetry "ignoring colors" this tends to mean that dark colors become black and light colors white, or that boundaries become black. Sometimes there is only one meaningful conversion, in other cases the conversion has to be specified to avoid ambiguity (see e.g. the tetrakis square tiling). T ...

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Symmetry, Symmetry - Mathematical model for symmetry, Symmetry - Non-isometric symmetry, Symmetry - Reflection symmetry, Symmetry - Rotational symmetry, Symmetry - Translational symmetry, Symmetry - Glide reflection symmetry, Symmetry - Rotoreflection symmetry, Symmetry - Screw axis symmetry, Symmetry - Symmetry combinations, Symmetry - Color, Symmetry - Similarity vs. sameness, Symmetry - More on symmetry in geometry, Symmetry - Symmetry in mathematics, Symmetry - Symmetry in logic, Symmetry - Generalization of symmetry, Symmetry - Symmetry in physics, Symmetry - Symmetry in biology, Symmetry - Symmetry in chemistry, Symmetry - Symmetry in the arts and crafts, Symmetry - Architecture, Symmetry - Pottery, Symmetry - Quilts, Symmetry - Carpets rugs, Symmetry - Music, Symmetry - Other arts and crafts, Symmetry - Aesthetics, Symmetry - Symmetry in games and puzzles, Symmetry - Symmetry in literature, Symmetry - Symmetry in telecommunications, Symmetry - Moral symmetry

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If we have a given set of objects with some structure, then it is possible for a symmetry to merely convert only one object into another, instead of acting upon all possible objects simultaneously. This requires a generalization from the concept of symmetry group to that of a groupoid. Physicists have come up with other directions of generalization, such as supersymmetry and quantum groups. ...

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Symmetry, Symmetry - Mathematical model for symmetry, Symmetry - Non-isometric symmetry, Symmetry - Reflection symmetry, Symmetry - Rotational symmetry, Symmetry - Translational symmetry, Symmetry - Glide reflection symmetry, Symmetry - Rotoreflection symmetry, Symmetry - Screw axis symmetry, Symmetry - Symmetry combinations, Symmetry - Color, Symmetry - Similarity vs. sameness, Symmetry - More on symmetry in geometry, Symmetry - Symmetry in mathematics, Symmetry - Symmetry in logic, Symmetry - Generalization of symmetry, Symmetry - Symmetry in physics, Symmetry - Symmetry in biology, Symmetry - Symmetry in chemistry, Symmetry - Symmetry in the arts and crafts, Symmetry - Architecture, Symmetry - Pottery, Symmetry - Quilts, Symmetry - Carpets rugs, Symmetry - Music, Symmetry - Other arts and crafts, Symmetry - Aesthetics, Symmetry - Symmetry in games and puzzles, Symmetry - Symmetry in literature, Symmetry - Symmetry in telecommunications, Symmetry - Moral symmetry

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(see main article: symmetry in physics) Symmetry in physics has been generalized to mean invariance under any kind of transformation. This has become one of the most powerful tools of theoretical physics. See Noether's theorem (which, as a gross oversimplification, states that for every symmetry law, there is a conservation law); and also, Wigner's theorem, which says that the symmetries of the laws of physics determine ...

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Symmetry, Symmetry - Mathematical model for symmetry, Symmetry - Non-isometric symmetry, Symmetry - Reflection symmetry, Symmetry - Rotational symmetry, Symmetry - Translational symmetry, Symmetry - Glide reflection symmetry, Symmetry - Rotoreflection symmetry, Symmetry - Screw axis symmetry, Symmetry - Symmetry combinations, Symmetry - Color, Symmetry - Similarity vs. sameness, Symmetry - More on symmetry in geometry, Symmetry - Symmetry in mathematics, Symmetry - Symmetry in logic, Symmetry - Generalization of symmetry, Symmetry - Symmetry in physics, Symmetry - Symmetry in biology, Symmetry - Symmetry in chemistry, Symmetry - Symmetry in the arts and crafts, Symmetry - Architecture, Symmetry - Pottery, Symmetry - Quilts, Symmetry - Carpets rugs, Symmetry - Music, Symmetry - Other arts and crafts, Symmetry - Aesthetics, Symmetry - Symmetry in games and puzzles, Symmetry - Symmetry in literature, Symmetry - Symmetry in telecommunications, Symmetry - Moral symmetry

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In 3D, the conjugate by a translation of a rotation about an axis is the corresponding rotation about the translated axis, etc. Thus the conjugacy class within the Euclidean group E(3) of a rotation about an axis is a rotation by the same angle about any axis. The conjugate closure of a singleton containing a rotation in 3D is E+(3). In 2D it is different in the case of a k-fold rotation: the conjugate closure contains See also:

Conjugation of isometries in Euclidean space, Conjugation of isometries in Euclidean space - Translation, Conjugation of isometries in Euclidean space - Inversion, Conjugation of isometries in Euclidean space - Rotation, Conjugation of isometries in Euclidean space - Reflection, Conjugation of isometries in Euclidean space - Rotoreflection, Conjugation of isometries in Euclidean space - Isometry groups, Conjugation of isometries in Euclidean space - Cyclic groups, Conjugation of isometries in Euclidean space - Dihedral groups

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The conjugate of the inversion in a point by a translation is the inversion in the translated point, etc. Thus the conjugacy class within the Euclidean group E(n) of inversion in a point is the set of inversions in all points. Since a combination of two inversions is a translation, the conjugate closure of a singleton containing inversion in a point is the set of all translations and the inversions in all points. ...

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Conjugation of isometries in Euclidean space, Conjugation of isometries in Euclidean space - Translation, Conjugation of isometries in Euclidean space - Inversion, Conjugation of isometries in Euclidean space - Rotation, Conjugation of isometries in Euclidean space - Reflection, Conjugation of isometries in Euclidean space - Rotoreflection, Conjugation of isometries in Euclidean space - Isometry groups, Conjugation of isometries in Euclidean space - Cyclic groups, Conjugation of isometries in Euclidean space - Dihedral groups

Read more here: » Conjugation of isometries in Euclidean space: Encyclopedia II - Conjugation of isometries in Euclidean space - Inversion