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.

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:
 * $$(±\frac{\sqrt{2}}{2},\,0,\,0).$$

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)