Icosahedron

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

The icosahedron is related to various Johnson solids. Most obviously, it can be constructed by joining two pentagonal pyramids to a pentagonal antiprism. Joining a single pentagonal pyramid yields the gyroelongated pentagonal pyramid, and replacing the antiprism by a pentagonal prism yields the elongated pentagonal pyramid and the elongated pentagonal bipyramid. A much less obvious connection is with the hebesphenomegacorona, which may be derived from the icosahedron by expanding a single edge into a square, thus turning the two adjacent faces into squares as well. Similarly, if we take two opposite edges of the icosahedron and "stretch" them into squares via a partial Stott expansion, we obtain the bilunabirotunda.

An alternate, lower symmetry construction as a snub tetrahedron, also relates the icosahedron to the snub polytopes, most notably to the snub disicositetrachoron, of which it is a cell.

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.