Octahedron

The octahedron, or oct, is one of the five Platonic solids. It consists of 8 equilateral triangles, joined 4 to a square vertex. It is the 3 dimensional orthoplex.

It can be built by joining two square pyramids by their square face, which makes it the square bipyramid.

It is also the uniform triangular antiprism, and is a segmentohedron in this form.

Vertex coordinates
An octahedron of side length 1 has vertex coordinates given by all permutations of:
 * $$\left(±\frac{\sqrt{2}}{2},\,0,\,0\right).$$

Tetratetrahedron
The tetratetrahedron, or tatet, is a variant of the octahedron with A3 symmetry. It consists of two types of equilateral triangles. It can be constructed as a rectification of the tetrahedron. It can be represented as o3x3o.

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


 * o4o3x (regular)
 * o3x3o (A3 symmetry, tetratetrahedron)
 * s2s3s (generally a Triangular antiprism)
 * s2s6o (similar to above, as alternated hexagonal prism)
 * xo3ox&#x (A2 axisal, generally a triangular antipodium)
 * oxo4ooo&#xt (BC2 axial, generally a square bipyramid)
 * oxo oxo&#xt (generally a rectangular bipyramid)
 * xox oqo&#xt (A1×A1 axial, edge-first)
 * oxox&#xr (single symmetry axis only)
 * qo ox4oo&#xt (BC2 prism symmetry square bipyramid)
 * qo ox ox&#xt (brick symmetry rectangle bipyramid)
 * qoo oqo ooq&#zx (brick symmetry, rhombic bipyramid)

Related polyhedra
The octahedron is the colonel of a two-member regiment that also includes the tetrahemihexahedron.

The octahedron is the regular-faced square bipyramid. If a cube, seen as a square prism, is inserted between the two haves, the result is an elongated square bipyramid.

a number of uniform polyhedron compounds are composed of octahedra, all but one of them featured octahedra in triangular antiprism symmetry:


 * Small icosicosahedron (5)
 * Snub octahedron (4)
 * Inner disnub octahedron (8, with rotational freedom)
 * Hexagrammic disnub octahedron (8)
 * Outer disnub octahedron (8, with rotational freedom)
 * Inner disnub tetrahedron (4, with rotational freedom)
 * Hexagrammic disnub tetrahedron (4)
 * Outer disnub tetrahedron (4, with rotational freedom)
 * Snub icosahedron (10)
 * Great snub icosahedron (10)
 * Outer disnub icosahedron (20, with rotational freedom)
 * Inner disnub icosahedron (20, with rotational freedom)
 * Great disnub icosahedron (20, with rotational freedom)
 * Disnub icosahedron (20)

The octahedron has one stellation, the stella octangula.