Icosahedron

The icosahedron is one of the five Platonic solids. It is a convex regular polyhedron composed of twenty triangular faces, with five meeting at each vertex. Its dual is the dodecahedron.

Canonical coordinates for the vertices of an icosahedron centered at the origin are (0,±1,±τ), (±1,±τ,0), (±τ,0,±1), where τ = (1+√5)/2 is the golden mean - note these form three mutually orthogonal golden rectangles. The edges of an octahedron can be partitioned in the golden mean so that the resulting vertices define a regular icosahedron, with the five octahedra defining any given icosahedron forming a regular compound.

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 T_h-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.

See also Truncated icosahedron.