Involutional symmetry Cs, (*) [ ] = | Cyclic symmetry Cnv, (*nn) [n] = | Dihedral symmetry Dnh, (*n22) [n,2] = | |
Polyhedral group, [n,3], (*n32) | |||
---|---|---|---|
Tetrahedral symmetry Td, (*332) [3,3] = | Octahedral symmetry Oh, (*432) [4,3] = | Icosahedral symmetry Ih, (*532) [5,3] = |
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry.
This article lists the groups by Schoenflies notation, Coxeter notation, [1] orbifold notation, [2] and order. John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion. [3]
Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6. [4]
There are four involutional groups: no symmetry (C1), reflection symmetry (Cs), 2-fold rotational symmetry (C2), and central point symmetry (Ci).
Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract | Fund. domain | |
---|---|---|---|---|---|---|---|---|---|
1 | 1 | 11 | C1 | C1 | ][ [ ]+ | 1 | Z1 | ||
2 | 2 | 22 | D1 = C2 | D2 = C2 | [2]+ | 2 | Z2 | ||
1 | 22 | × | Ci = S2 | CC2 | [2+,2+] | 2 | Z2 | ||
2 = m | 1 | * | Cs = C1v = C1h | ±C1 = CD2 | [ ] | 2 | Z2 |
There are four infinite cyclic symmetry families, with n = 2 or higher. (n may be 1 as a special case as no symmetry)
Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract | Fund. domain | |
---|---|---|---|---|---|---|---|---|---|
4 | 42 | 2× | S4 | CC4 | [2+,4+] | 4 | Z4 | ||
2/m | 22 | 2* | C2h = D1d | ±C2 = ±D2 | [2,2+] [2+,2] | 4 | Z4 |
Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract | Fund. domain | |
---|---|---|---|---|---|---|---|---|---|
2 3 4 5 6 n | 2 3 4 5 6 n | 22 33 44 55 66 nn | C2 C3 C4 C5 C6 Cn | C2 C3 C4 C5 C6 Cn | [2]+ [3]+ [4]+ [5]+ [6]+ [n]+ | 2 3 4 5 6 n | Z2 Z3 Z4 Z5 Z6 Zn | ||
2mm 3m 4mm 5m 6mm nm (n is odd) nmm (n is even) | 2 3 4 5 6 n | *22 *33 *44 *55 *66 *nn | C2v C3v C4v C5v C6v Cnv | CD4 CD6 CD8 CD10 CD12 CD2n | [2] [3] [4] [5] [6] [n] | 4 6 8 10 12 2n | D4 D6 D8 D10 D12 D2n | ||
3 8 5 12 - | 62 82 10.2 12.2 2n.2 | 3× 4× 5× 6× n× | S6 S8 S10 S12 S2n | ±C3 CC8 ±C5 CC12 CC2n / ±Cn | [2+,6+] [2+,8+] [2+,10+] [2+,12+] [2+,2n+] | 6 8 10 12 2n | Z6 Z8 Z10 Z12 Z2n | ||
3/m=6 4/m 5/m=10 6/m n/m | 32 42 52 62 n2 | 3* 4* 5* 6* n* | C3h C4h C5h C6h Cnh | CC6 ±C4 CC10 ±C6 ±Cn / CC2n | [2,3+] [2,4+] [2,5+] [2,6+] [2,n+] | 6 8 10 12 2n | Z6 Z2×Z4 Z10 Z2×Z6 Z2×Zn ≅Z2n (odd n) |
There are three infinite dihedral symmetry families, with n = 2 or higher (n may be 1 as a special case).
Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract | Fund. domain |
---|---|---|---|---|---|---|---|---|
222 | 2.2 | 222 | D2 | D4 | [2,2]+ | 4 | D4 | |
42m | 42 | 2*2 | D2d | DD8 | [2+,4] | 8 | D4 | |
mmm | 22 | *222 | D2h | ±D4 | [2,2] | 8 | Z2×D4 |
Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract | Fund. domain | |
---|---|---|---|---|---|---|---|---|---|
32 422 52 622 | 3.2 4.2 5.2 6.2 n.2 | 223 224 225 226 22n | D3 D4 D5 D6 Dn | D6 D8 D10 D12 D2n | [2,3]+ [2,4]+ [2,5]+ [2,6]+ [2,n]+ | 6 8 10 12 2n | D6 D8 D10 D12 D2n | ||
3m 82m 5m 12.2m | 62 82 10.2 12.2 n2 | 2*3 2*4 2*5 2*6 2*n | D3d D4d D5d D6d Dnd | ±D6 DD16 ±D10 DD24 DD4n / ±D2n | [2+,6] [2+,8] [2+,10] [2+,12] [2+,2n] | 12 16 20 24 4n | D12 D16 D20 D24 D4n | ||
6m2 4/mmm 10m2 6/mmm | 32 42 52 62 n2 | *223 *224 *225 *226 *22n | D3h D4h D5h D6h Dnh | DD12 ±D8 DD20 ±D12 ±D2n / DD4n | [2,3] [2,4] [2,5] [2,6] [2,n] | 12 16 20 24 4n | D12 Z2×D8 D20 Z2×D12 Z2×D2n ≅D4n (odd n) |
There are three types of polyhedral symmetry: tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, named after the triangle-faced regular polyhedra with these symmetries.
Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract | Fund. domain |
---|---|---|---|---|---|---|---|---|
23 | 3.3 | 332 | T | T | [3,3]+ | 12 | A4 | |
m3 | 43 | 3*2 | Th | ±T | [4,3+] | 24 | 2×A4 | |
43m | 33 | *332 | Td | TO | [3,3] | 24 | S4 |
Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract | Fund. domain |
---|---|---|---|---|---|---|---|---|
432 | 4.3 | 432 | O | O | [4,3]+ | 24 | S4 | |
m3m | 43 | *432 | Oh | ±O | [4,3] | 48 | 2×S4 |
Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract | Fund. domain |
---|---|---|---|---|---|---|---|---|
532 | 5.3 | 532 | I | I | [5,3]+ | 60 | A5 | |
532/m | 53 | *532 | Ih | ±I | [5,3] | 120 | 2×A5 |
All of the discrete point symmetries are subgroups of certain continuous symmetries. They can be classified as products of orthogonal groups O(n) or special orthogonal groups SO(n). O(1) is a single orthogonal reflection, dihedral symmetry order 2, Dih1. SO(1) is just the identity. Half turns, C2, are needed to complete.
Rank 3 groups | Other names | Example geometry | Example finite subgroups | |
---|---|---|---|---|
O(3) | Full symmetry of the sphere | [3,3] = , [4,3] = , [5,3] = [4,3+] = | ||
SO(3) | Sphere group | Rotational symmetry | [3,3]+ = , [4,3]+ = , [5,3]+ = | |
O(2)×O(1) O(2)⋊C2 | Dih∞×Dih1 Dih∞⋊C2 | Full symmetry of a spheroid, torus, cylinder, bicone or hyperboloid Full circular symmetry with half turn | [p,2] = [p]×[ ] = [2p,2+] = , [2p+,2+] = | |
SO(2)×O(1) | C∞×Dih1 | Rotational symmetry with reflection | [p+,2] = [p]+×[ ] = | |
SO(2)⋊C2 | C∞⋊C2 | Rotational symmetry with half turn | [p,2]+ = | |
O(2)×SO(1) | Dih∞ Circular symmetry | Full symmetry of a hemisphere, cone, paraboloid or any surface of revolution | [p,1] = [p] = | |
SO(2)×SO(1) | C∞ Circle group | Rotational symmetry | [p,1]+ = [p]+ = |
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules.
In geometry, a uniform 4-polytope is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
In mathematics, the Coxeter numberh is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.
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. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.
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) that does not change the object.
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(1/5), or approximately 78.46°.
In geometry, a snub is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube and snub dodecahedron. In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron: moving the faces apart, twisting them about their centers, adding new polygons centered on the original vertices, and adding pairs of triangles fitting between the original edges.
In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a parameter n, the dimension of the hypercube.
In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 8, the set of all such locations is called 8-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance. Eight-dimensional Euclidean space is eight-dimensional space equipped with the Euclidean metric.
In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.
In geometry, a point group in four dimensions is an isometry group in four dimensions that leaves the origin fixed, or correspondingly, an isometry group of a 3-sphere.
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In geometry, a regular skew apeirohedron is an infinite regular skew polyhedron, with either skew regular faces or skew regular vertex figures.