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|Faces per vertex||5|
|Vertices per face||3|
|Symmetry group||icosahedral (Ih)|
In geometry, the regular icosahedron is one of the five Platonic solids. It is a convex regular polyhedron composed of twenty triangular faces, with five meeting at each of the twelve vertices. It has 30 edges. Its dual polyhedron is the dodecahedron.
The area A and the volume V of a regular icosahedron of edge length a are:
There are distortions of the icosahedron that, while no longer regular, are nevertheless vertex-uniform. These are invariant under the same rotations as the tetrahedron, and are somewhat analogous to the snub cube and snub dodecahedron, including some forms which are chiral and some with Th-symmetry, i.e. have different planes of symmetry than the tetrahedron. The icosahedron has a large number of stellations, including one of the Kepler-Poinsot solids and some of the regular compounds, which could be discussed here.
Many viruses, including HIV, have the shape of an icosahedron. Viruses need to be small, and the icosahedron is the regular polyhedron with largest volume per diameter. A regular polyhedron is used because it can be built from a single basic unit protein that's reused over and over again; that saves space in the viral genome.