Top Qs
Timeline
Chat
Perspective

List of spherical symmetry groups

From Wikipedia, the free encyclopedia

Remove ads

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.

More information Polyhedral group, [n,3], (*n32) ...

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]

Remove ads

Involutional symmetry

There are four involutional groups: no symmetry (C1), reflection symmetry (Cs), 2-fold rotational symmetry (C2), and central point symmetry (Ci).

More information Intl, Geo ...
Remove ads

Cyclic symmetry

There are four infinite cyclic symmetry families, with n = 2 or higher. (n may be 1 as a special case as no symmetry)

More information Intl, Geo ...
More information Intl, Geo ...
Remove ads

Dihedral symmetry

There are three infinite dihedral symmetry families, with n = 2 or higher (n may be 1 as a special case).

More information Intl, Geo ...
More information Intl, Geo ...

Polyhedral symmetry

There are three types of polyhedral symmetry: tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, named after the triangle-faced regular polyhedra with these symmetries.

More information Intl, Geo ...
More information Intl, Geo ...
More information Intl, Geo ...
Remove ads

Continuous symmetries

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.

More information Rank 3 groups, Other names ...
Remove ads

See also

References

Further reading

Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads