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.
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] = | |
The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.