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Icosahedral symmetry
3D symmetry group / From Wikipedia, the free encyclopedia
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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.
![]() | This article includes a list of general references, but it lacks sufficient corresponding inline citations. (May 2020) |
![]() Involutional symmetry Cs, (*) [ ] = ![]() |
![]() Cyclic symmetry Cnv, (*nn) [n] = ![]() ![]() ![]() |
![]() Dihedral symmetry Dnh, (*n22) [n,2] = ![]() ![]() ![]() ![]() ![]() | |
Polyhedral group, [n,3], (*n32) | |||
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![]() Tetrahedral symmetry Td, (*332) [3,3] = ![]() ![]() ![]() ![]() ![]() |
![]() Octahedral symmetry Oh, (*432) [4,3] = ![]() ![]() ![]() ![]() ![]() |
![]() Icosahedral symmetry Ih, (*532) [5,3] = ![]() ![]() ![]() ![]() ![]() |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Icosahedral_reflection_domains.png/640px-Icosahedral_reflection_domains.png)
![](http://upload.wikimedia.org/wikipedia/en/thumb/e/ec/Soccer_ball.svg/640px-Soccer_ball.svg.png)
![A great icosahedron](http://upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Sixteenth_stellation_of_icosahedron.png/640px-Sixteenth_stellation_of_icosahedron.png)
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.