Icosahedron

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.