Icosahedron

The icosahedron, or ike, is one of the five Platonic solids. It has 20 triangles as faces, joining 5 to a vertex.

It is the only Platonic solid that does not appear as a cell in one of the convex regular polychora. It does, however, appear as the vertex figure of the hexacosichoron.

Vertex coordinates
The vertices of an icosahedron of edge length 1, centered at the origin, are all cyclic permutations of


 * (0, ±1/2, ±($\sqrt{10+2√5}$+1)/4).

Snub tetrahedron
The icosahedron can also be considered to be a kind of snub tetrahedron, by analogy with the snub cube and snub dodecahedron. It is the result of alternating the vertices of a truncated octahedron and then adjusting edge lengths to be equal. It can be represented as s3s3s.