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, because its dihedral angle is more than 120° and thus 3 icosahedra cannot fit around an edge in 4D. 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:


 * $$\left(0,\,±\frac{1}{2},\,±\frac{1+\sqrt{5}}{4}\right).$$

Representations
A regular icosahedron can be represented by the following Coxeter diagrams:


 * o5o3x (regular)
 * o4s3s (B3/2 symmetry, alternated truncated octahedron)
 * s3s3s (A3+, snub tetrahedron)
 * oxoo5ooxo&#xt (H2 axial, seen as gyroelongated pentagonal bipyramid)
 * xofo3ofox&#xt (A2 axial, face-first)
 * xofox ofxfo&#xt (A1×A1 axial, edge-first)
 * fxo ofx xof&#zx (A1×A1×A1 subsymmetry)

Variations
An icosahedron has several variations as a snub polyhedron:


 * Pyritohedral icosahedron - formed by alternating a truncated octahedron, has pyritohedral symmetry
 * Snub tetrahedron - formed by alternating a great rhombitetratetrahedron, has chiral tetrahedral symmetry

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.

The icosahedron can be considered to be a snub triangular antiprism, by analogy with the snub disphenoid and snub square antiprism. This can be seen since the icosahedron can be constructed from the octahedron, that is a triangular antiprism, but cutting it into two halves and inserting a set of 12 triangles between the halves.

Two uniform polyhedron compounds are composed of icosahedra, both using it in pyritohedral symmetry:


 * Snub disoctahedron (2)
 * Snub icosicosahedron (5)

The icosahedron has a multitude of stellations, not all of which are true polyhedra. They include the small triambic icosahedron, medial triambic icosahedron, great triambic icosahedron, great icosahedron, chiricosahedron, icosicosahedron, small icosicosahedron, ditrigonal icosahedron and the final stellation of the icosahedron.

The great icosahedron is the icosahedron's conjugate, meaning it is isomorphic. The icosahedron and great icosahedron are only two of the many polyhedra made of equilateral triangles with that abstract structure. For example, one of the pyramids of the icosahedron can be inverted, producing an irregular polyhedron that is concave but with no intersections. More can be found here.