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Tetrahedral symmetry

Index 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. [1]

48 relations: Alternating group, Archimedean solid, Binary tetrahedral group, Catalan solid, Conjugacy class, Coxeter notation, Cubic crystal system, Cuboid, Cycle graph (algebra), Cyclic group, Dihedral group, Dihedral symmetry in three dimensions, Dodecahedron, Fundamental domain, Harold Scott MacDonald Coxeter, Hermann–Mauguin notation, Icosahedral symmetry, Index of a subgroup, Isomorphism, Lagrange's theorem (group theory), List of finite spherical symmetry groups, List of small groups, Near-miss Johnson solid, Normal subgroup, Norman Johnson (mathematician), Octahedral symmetry, Orbifold notation, Orientation (vector space), Parity of a permutation, Permutation, Platonic solid, Point groups in three dimensions, Point reflection, Quotient group, Rotation, Schoenflies notation, Stereographic projection, Symmetric group, Symmetry number, Tetrahedron, Tetrahemihexahedron, Tetrakis hexahedron, Tetrated dodecahedron, Triakis tetrahedron, Triangle group, Truncated tetrahedron, Truncated triakis tetrahedron, Uniform star polyhedron.

Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set.

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Archimedean solid

In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes.

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Binary tetrahedral group

In mathematics, the binary tetrahedral group, denoted 2T or 2,3,3 is a certain nonabelian group of order 24.

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Catalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid.

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Conjugacy class

In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure.

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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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Cubic crystal system

In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube.

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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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Cycle graph (algebra)

In group theory, a sub-field of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups.

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Cyclic group

In algebra, a cyclic group or monogenous group is a group that is generated by a single element.

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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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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).

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Dodecahedron

In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces.

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Fundamental domain

Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action.

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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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Hermann–Mauguin notation

In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups.

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Icosahedral symmetry

A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation.

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Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H).

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Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

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Lagrange's theorem (group theory)

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.

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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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List of small groups

The following list in mathematics contains the finite groups of small order up to group isomorphism.

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Near-miss Johnson solid

In geometry, a near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular.

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Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.

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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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Octahedral symmetry

A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation.

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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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Orientation (vector space)

In mathematics, orientation is a geometric notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed.

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Parity of a permutation

In mathematics, when X is a finite set of at least two elements, the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations.

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Permutation

In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.

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Platonic solid

In three-dimensional space, a Platonic solid is a regular, convex polyhedron.

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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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Point reflection

In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space.

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Quotient group

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.

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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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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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Stereographic projection

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.

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Symmetric group

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.

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Symmetry number

The symmetry number or symmetry order of an object is the number of different but indistinguishable (or equivalent) arrangements (or views) of the object, i.e. the order of its symmetry group.

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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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Tetrahemihexahedron

In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4.

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Tetrakis hexahedron

In geometry, a tetrakis hexahedron (also known as a tetrahexahedron, hextetrahedron, tetrakis cube, and kiscube) is a Catalan solid.

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Tetrated dodecahedron

The tetrated dodecahedron is a near-miss Johnson solid.

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Triakis tetrahedron

In geometry, a triakis tetrahedron (or kistetrahedron) is an Archimedean dual solid, or a Catalan solid.

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Triangle group

In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle.

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Truncated tetrahedron

In geometry, the truncated tetrahedron is an Archimedean solid.

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Truncated triakis tetrahedron

The truncated triakis tetrahedron is a convex polyhedron with 16 faces: 4 sets of 3 pentagons arranged in a tetrahedral arrangement, with 4 hexagons in the gaps.

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Uniform star polyhedron

In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron.

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Redirects here:

332 symmetry, Achiral tetrahedral symmetry, Chiral tetrahedral symmetry, Full tetrahedral symmetry, Pyritohedral, Pyritohedral group, Pyritohedral symmetry, Tetrahedral group.

References

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

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