Euclidean group

Types of subgroups of E(n): Finite groups. They always have a fixed point. In 3D, for every point there are for every orientation two which are maximal (with respect to inclusion) among the finite groups: Oh and Ih. The groups Ih are even maximal among the groups including the next category. Countably infinite groups without arbitrarily small translations, rotations, or combinations, i.e., for every point the set of images under the isometries is topologically ...

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Euclidean group, Euclidean group - Subgroup structure matrix and vector representation, Euclidean group - Subgroups, Euclidean group - Relation to the affine group, Euclidean group - Rigid body motions, Euclidean group - Overview of isometries in up to three dimensions, Euclidean group - Commuting isometries, Euclidean group - Conjugacy classes

Read more here: » Euclidean group: Encyclopedia II - Euclidean group - Subgroups

The notion of isometry comes in two main flavors: global isometry and a weaker notion path isometry or arcwise isometry. Both are often called just isometry and one should guess from context which one is intended. Let X and Y be metric spaces with metrics dX and dY. A map is called distance preserving if for any one has < ...

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Isometry, Isometry - Definitions, Isometry - Examples, Isometry - Generalizations

Read more here: » Isometry: Encyclopedia II - Isometry - Definitions

A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete symmetry group. A major application is in crystallography, to categorize crystals, but by itself the topic is one of 3D Euclidean geometry. There are 7 crystal systems: Triclinic, all cases not satisfying the requirements of any other system; thus there is no other symmetry than translational symmetry, or the only extra kind is inversion.Including:

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

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In geometry, two shapes are called congruent if one can be transformed into the other by an isometry, i.e. a combination of translations, rotations and reflections. Note: This article is about congruences in geometry. For notions of congruence in algebra, see congruence relation. Congruence geometry - Definition of congruence in analytic geometry. In a Euclidean system, congruence is fundamental; it's the counterpart of an equals sign in numerical analysis. In analytic geometry, congru ...

Including:

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

Read more here: » Congruence geometry: Encyclopedia - Congruence geometry

Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by one of the operations. In 2D geometry the main kinds of symmetry of interest are with respect to the basic Euclidean plane isometries: translations, rotations, reflections, ...

Including:

• 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

Read more here: » Symmetry: Encyclopedia - Symmetry

An example of abstract group Dihn, and a common way to visualize it, is the group Dn of Euclidean plane isometries which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions. Dn consists of n rotations of multiples of 360°/n about the origin, and reflections across n lines through the origin, making angles of multiples of 180°/n with each other. This is the symmetry group of a regular polygon with n sides ( ...

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Dihedral group, Dihedral group - The dihedral group as symmetry group in 2D and rotation group in 3D, Dihedral group - Equivalent definitions and properties, Dihedral group - Examples of automorphism groups, Dihedral group - Infinite dihedral group, Dihedral group - Generalized dihedral group, Dihedral group - Topology

Read more here: » Dihedral group: Encyclopedia II - Dihedral group - The dihedral group as symmetry group in 2D and rotation group in 3D

The infinite series have an index n, which can be any integer; in each series, the nth symmetry group contains n-fold rotational symmetry about an axis, i.e. symmetry with respect to a rotation by an angle 360°/n. n=1 covers the cases of no rotational symmetry at all. There are four series with no other axes of rotational symmetry, see cyclic symmetries, an ...

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Point groups in three dimensions, Point groups in three dimensions - Group structure, Point groups in three dimensions - 3D isometries which leave the origin fixed, Point groups in three dimensions - Conjugacy, Point groups in three dimensions - Infinite isometry groups, Point groups in three dimensions - Finite isometry groups, Point groups in three dimensions - The seven infinite series, Point groups in three dimensions - The seven remaining point groups, Point groups in three dimensions - Relation between orbifold notation and order, Point groups in three dimensions - Rotation groups, Point groups in three dimensions - Correspondence between rotation groups and other groups, Point groups in three dimensions - Maximal symmetries, Point groups in three dimensions - The groups arranged by abstract group type, Point groups in three dimensions - Symmetry groups in 3D which are cyclic as abstract group, Point groups in three dimensions - Symmetry groups in 3D which are dihedral as abstract group, Point groups in three dimensions - Other, Point groups in three dimensions - Impossible discrete symmetries, Point groups in three dimensions - Examples, Point groups in three dimensions - Fundamental domain

Read more here: » Point groups in three dimensions: Encyclopedia II - Point groups in three dimensions - The seven infinite series

If G is a semidirect product of N and H, then the map φ : H → Aut(N) (where Aut(N) denotes the group of all automorphisms of N) defined by φ(h)(n) = hnh–1 for all h in H and n in N is a group homomorphism. Together N, H and φ determine G up to isomorphism, as we now show. Given any two groups N and H (not necessarily subgroups of a given group) and a group homomorphism φ : See also:

Semidirect product, Semidirect product - Some equivalent definitions, Semidirect product - Elementary facts and caveats, Semidirect product - Outer semidirect products, Semidirect product - Examples, Semidirect product - Relation to direct products, Semidirect product - Generalizations, Semidirect product - Notation

Read more here: » Semidirect product: Encyclopedia II - Semidirect product - Outer semidirect products

Normal subgroups are of relevance because if N is normal, then the quotient group G/N may be formed: if N is normal, we can define a multiplication on cosets by (a1N)(a2N) := (a1a2)N This turns the set of cosets into a group called the quotient group G/N. There is a natural homomorphism f : G → G/N given by f(a) = aN. The image f(N) consists only ...

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Normal subgroup, Normal subgroup - Example, Normal subgroup - Normal subgroups and homomorphisms, Normal subgroup - Attributes of normality

Read more here: » Normal subgroup: Encyclopedia II - Normal subgroup - Normal subgroups and homomorphisms

If h is a translation, then its conjugate by an isometry can be described as applying the isometry to the translation: the conjugate of a translation by a translation is the first translation the conjugate of a translation by a rotation is a translation by a rotated translation vector the conjugate of a translation by a reflection is a translation by a reflected translation vector Thus the conjugacy class within the Euclidean group E( ...

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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 - Translation

E(2) is a semidirect product of O(2) and the translation group T. In other words O(2) is a subgroup of E(2) isomorphic to the quotient group of E(2) by T: O(2) E(2) / T There is a "natural" surjective group homomorphism p : E(2) → E(2)/ T, sending each element g of E(2) to the coset of T to which g belongs, that is: p (g) = gT, sometimes called the canonical projec ...

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Point groups in two dimensions, Point groups in two dimensions - Symmetry groups, Point groups in two dimensions - Combinations with translational symmetry

Read more here: » Point groups in two dimensions: Encyclopedia II - Point groups in two dimensions - Combinations with translational symmetry

Rotational symmetry of order n, also called n-fold rotational symmetry, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of 360°/n (180°, 120°, 90°, 72°, 60°, 51 3/7 °, etc.) does not change the object. Note that "1-fold" symmetry is no symmetry, and "2-fold" is the simplest symmetry, so it does mean "more than basic". The notation for n-fold symmetry is Cn or simply "n". The actual symmetry group is specified by ...

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Rotational symmetry, Rotational symmetry - n-fold rotational symmetry, Rotational symmetry - Examples, Rotational symmetry - C2, Rotational symmetry - C3, Rotational symmetry - C4, Rotational symmetry - Mixed, Rotational symmetry - Multiple symmetry axes through the same point, Rotational symmetry - Rotational symmetry with respect to any angle, Rotational symmetry - Rotational symmetry together with translational symmetry

Read more here: » Rotational symmetry: Encyclopedia II - Rotational symmetry - n-fold rotational symmetry

Was there one 'geometry' or many? Since Euclid, geometry had meant the geometry of Euclidean space of two dimensions (plane geometry) or of three dimensions (solid geometry). In the first half of the nineteenth century there had been several developments complicating the picture. Mathematical applications required geometry of four or more dimensions; the close scrutiny of the foundations of the traditional Euclidean geometry had revealed the independence of the Parallel Axiom from the others, and non-Euclidean geometry had been born; and in projective geometry new 'points' (p ...

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Erlangen program, Erlangen program - The problems of nineteenth century geometry, Erlangen program - Homogeneous spaces, Erlangen program - Examples: affine geometry, Erlangen program - Influence on later work, Erlangen program - Abstract returns from the Erlangen program

Read more here: » Erlangen program: Encyclopedia II - Erlangen program - The problems of nineteenth century geometry

Mathematically, a space group is a symmetry group or symmetry group type of n-dimensional structures with translational symmetry in n independent directions, such as, for n = 3, a crystal. Only discrete symmetry groups are included in the categorization; i.e., infinitely fine structure or homogeneity in one or more directions is excluded. Two symmetry groups are of the same crystallographic space group type if they are the same up to an affine transformation of space that preserves orientation. Thus e.g. a change of angle between translation vectors does not affect the space group typ ...

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Space group, Space group - Space groups in crystallography, Space group - Glide planes and screw axes, Space group - Notation, Space group - Group theory, Space group - Space groups in various dimensions, Space group - Grouping space groups by point group, Space group - Further categorizing of space groups

Read more here: » Space group: Encyclopedia II - Space group - Group theory

Since a translation is an affine transformation but not a linear transformation, homogeneous coordinates are normally used to represent the translation operator by a matrix. Thus we write the 3-dimensional vector w = (wx, wy, wz) using 4 homogeneous coordinates as w = (wx, wy, wz, 1). To translate an object by a vector v, each homogeneous vector p (written in homogeneous coordinates) would need to be mult ...

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Translation geometry, Translation geometry - Matrix representation, Translation geometry - External link

Read more here: » Translation geometry: Encyclopedia II - Translation geometry - Matrix representation

More generally, we can have spontaneous symmetry breaking in nonvacuum situations and for systems not described by actions. The crucial concept here is the order parameter. If there is a field (often a background field) which acquires an expectation value (not necessarily a vacuum expectation value) which is not invariant under the symmetry in question, we say that the system is in the ordered phase and the symmetry is spontaneously broken. This is because other subsystems interact with the order parameter which ...

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Spontaneous symmetry breaking, Spontaneous symmetry breaking - Mathematical example: the Mexican hat potential, Spontaneous symmetry breaking - Broader concept, Spontaneous symmetry breaking - Examples

Read more here: » Spontaneous symmetry breaking: Encyclopedia II - Spontaneous symmetry breaking - Broader concept

Reflections, or mirror isometries, can be combined to produce any isometry. Thus isometries are an example of a reflection group. Euclidean plane isometry - Mirror combinations. In the Euclidean plane, we have the following possibilities. [d  ] Identity Two reflections in the same mirror restore each point to its original position. All points are left fixed. Any pair of identical mirrors has the same effect. See also:

Euclidean plane isometry, Euclidean plane isometry - Informal discussion, Euclidean plane isometry - Formal definition, Euclidean plane isometry - Classification of Euclidean plane isometries, Euclidean plane isometry - Isometries as reflection group, Euclidean plane isometry - Mirror combinations, Euclidean plane isometry - Three mirrors suffice, Euclidean plane isometry - Recognition, Euclidean plane isometry - Group structure, Euclidean plane isometry - Composition, Euclidean plane isometry - Translation rotation and orthogonal subgroups, Euclidean plane isometry - Nested group construction, Euclidean plane isometry - Discrete subgroups, Euclidean plane isometry - Isometries in the complex plane

Read more here: » Euclidean plane isometry: Encyclopedia II - Euclidean plane isometry - Isometries as reflection group

Over the field R of real numbers, the orthogonal group O(n,R) and the special orthogonal group SO(n,R) are often simply denoted by O(n) and SO(n) if no confusion is possible. They form real compact Lie groups of dimension n(n-1)/2. O(n,R) has two connected components, with SO(n,R) being the identity component, i.e., the connected component containing the identity matrix. The real orthogonal and real special ...

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Orthogonal group, Orthogonal group - Over the real number field, Orthogonal group - 3D isometries which leave the origin fixed, Orthogonal group - Over the complex number field, Orthogonal group - The Dickson invariant, Orthogonal group - Orthogonal groups of characteristic 2, Orthogonal group - The spinor norm, Orthogonal group - Galois cohomology and orthogonal groups

Read more here: » Orthogonal group: Encyclopedia II - Orthogonal group - Over the real number field

The basic mathematics underlying spacetime physics is affine differential geometry, in which we endow an n dimensional differentiable manifold M with a law of parallel translation of vectors along paths in M. (At each point of a differentiable manifold, we have a linear space of tangent vectors, but we have no way to transport vectors to another point, or to compare vectors at two points in M.) The parallel translation preserves linear relationships between vectors; that is, if two vectors u and v at the same point of M parallel translate along a curve to vectors u' and v ...

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Einstein-Cartan theory, Einstein-Cartan theory - Introduction, Einstein-Cartan theory - Derivation of field equations of Einstein-Cartan theory, Einstein-Cartan theory - Geometric insights from Einstein-Cartan theory, Einstein-Cartan theory - First geometric insight, Einstein-Cartan theory - Second geometric insight, Einstein-Cartan theory - Third geometric insight, Einstein-Cartan theory - Fourth geometric insight, Einstein-Cartan theory - General relativity plus matter with spin implies Einstein-Cartan theory

Read more here: » Einstein-Cartan theory: Encyclopedia II - Einstein-Cartan theory - Introduction

List of Lie group topics - Physical theories. Pauli matrices Gell-Mann matrices Poisson bracket Noether's theorem Wigner's classification Gauge theory Grand unification theory Supergroup Lie superalgebra Twistor Anyon Witt algebra Virasoro algebra List of Lie group topics - Geometry. Erlangen programme Homogeneous space Principal homogeneous space Invariant theory Lie deriv ...

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List of Lie group topics, List of Lie group topics - Examples, List of Lie group topics - Lie algebras, List of Lie group topics - Foundational results, List of Lie group topics - Semisimple theory, List of Lie group topics - Representation theory, List of Lie group topics - Applications, List of Lie group topics - Physical theories, List of Lie group topics - Geometry, List of Lie group topics - Discrete groups, List of Lie group topics - Algebraic groups, List of Lie group topics - Special functions, List of Lie group topics - Automorphic forms, List of Lie group topics - People

Read more here: » List of Lie group topics: Encyclopedia II - List of Lie group topics - Applications