Icosahedron

An icosahedron is a Platonic solid 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 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 of tetrahedral symmetry that, while no longer regular, are nevertheless vertex-uniform; these are chiral and somewhat analagous to the [snub cube]? and [snub dodecahedron]?. The icosahedron has a large number of stellation?s, including one of the Kepler-poinsot solids and some of the regular compounds, which could be discussed here.