Icosahedron

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

An alternate, lower symmetry construction as a snub tetrahedron, furthermore 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, and as the cell of the non-convex faceted hexacosichoron.

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


 * (0, ±1/2, ±($\sqrt{(5+√5)/8}$+1)/4).

In vertex figures
The icosahedron is seen as a vertex figure in two regular polychora: the hexacosichoron, where it has an edge length of 1, and the great stellated hecatonicosachoron, where it has an edge length of ($\sqrt{5}$–1)/2.

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.

Related polyhedra
The icosahedron is the colonel of a two-member regiment that also includes the great dodecahedron.

The icosahedron is related to many Johnson solids. Most obviously, it can be constructed by joining two pentagonal pyramids to a pentagonal antiprism. This means the icosahedron could also be called a gyroelongated pentagonal bipyramid. Joining a single pentagonal pyramid, or diminishing one vertex from the icosahedron, yields the gyroelongated pentagonal pyramid, and replacing the antiprism by a pentagonal prism yields the elongated pentagonal pyramid and the elongated pentagonal bipyramid. Cutting off two pyramids from two non-parallel, non-adjacent vertices yields the metabidiminished icosahedron, and cutting off a further non-adjacent pyramid yields the tridiminished icosahedron.

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.