Icosahedral symmetry
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron.
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] = | |
Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120. The full symmetry group is the Coxeter group of type H3. It may be represented by Coxeter notation [5,3] and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group A5 on 5 letters.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.