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Dihedral symmetry in three dimensions

Index Dihedral symmetry in three dimensions

In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn (n ≥ 2). [1]

34 relations: Antiprism, Arthur Moritz Schoenflies, Baseball, Basketball, Beach ball, Bipyramid, Coxeter notation, Cuboid, Cyclic symmetry in three dimensions, Dihedral group, Frieze group, Geometry, Harold Scott MacDonald Coxeter, Improper rotation, Klein four-group, List of finite spherical symmetry groups, Norman Johnson (mathematician), Orbifold notation, Pentagonal antiprism, Pentagonal trapezohedron, Pentagrammic antiprism, Pentagrammic crossed-antiprism, Pentagrammic prism, Point groups in three dimensions, Prism (geometry), Reflection symmetry, Rotation, Rotational symmetry, Schoenflies notation, Snub square antiprism, Symmetry group, Tetrahedral symmetry, Tetrahedron, Trapezohedron.

Antiprism

In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles.

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Arthur Moritz Schoenflies

Arthur Moritz Schoenflies (17 April 1853 – 27 May 1928), sometimes written as Schönflies, was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology.

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Baseball

Baseball is a bat-and-ball game played between two opposing teams who take turns batting and fielding.

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Basketball

Basketball is a team sport played on a rectangular court.

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Beach ball

A beach ball is an inflatable ball for beach and water games.

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Bipyramid

An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.

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Coxeter notation

In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups.

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Cuboid

In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube.

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Cyclic symmetry in three dimensions

In three dimensional geometry, there are four infinite series of point groups in three dimensions (n≥1) with n-fold rotational or reflectional symmetry about one axis (by an angle of 360°/n) does not change the object.

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Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.

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Frieze group

In mathematics, a frieze or frieze pattern is a design on a two-dimensional surface that is repetitive in one direction.

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Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

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Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.

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Improper rotation

In geometry, an improper rotation,.

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Klein four-group

In mathematics, the Klein four-group (or just Klein group or Vierergruppe, four-group, often symbolized by the letter V or as K4) is the group, the direct product of two copies of the cyclic group of order 2.

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List of finite spherical symmetry groups

Finite spherical symmetry groups are also called point groups in three dimensions.

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Norman Johnson (mathematician)

Norman Woodason Johnson (November 12, 1930 – July 13, 2017) was a mathematician, previously at Wheaton College, Norton, Massachusetts.

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Orbifold notation

In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature.

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Pentagonal antiprism

In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.

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Pentagonal trapezohedron

The pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms.

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Pentagrammic antiprism

In geometry, the pentagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams.

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Pentagrammic crossed-antiprism

In geometry, the pentagrammic crossed-antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams.

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Pentagrammic prism

In geometry, the pentagrammic prism is one in an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams.

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Point groups in three dimensions

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere.

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Prism (geometry)

In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases.

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Reflection symmetry

Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection.

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Rotation

A rotation is a circular movement of an object around a center (or point) of rotation.

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Rotational symmetry

Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn.

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Schoenflies notation

The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is one of two conventions commonly used to describe point groups.

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Snub square antiprism

In geometry, the snub square antiprism is one of the Johnson solids (J85).

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Symmetry group

In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.

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Tetrahedral symmetry

A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.

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Tetrahedron

In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.

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Trapezohedron

The n-gonal trapezohedron, antidipyramid, antibipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism.

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Redirects here:

Antiprismatic symmetry, Prismatic symmetry.

References

[1] https://en.wikipedia.org/wiki/Dihedral_symmetry_in_three_dimensions

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