Octahedron

{{Infobox polytope 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.
 * name = Octahedron
 * type=Regular
 * image = Octahedron.png
 * dim = 3
 * off = Octahedron.off
 * 3d = Octahedron.stl
 * obsa = Oct
 * faces = 8 triangles
 * edges = 12
 * vertices = 6
 * verf = Square, edge length 1
 * schlafli = {3,4}
 * coxeter = o4o3x
 * army=Oct
 * reg=Oct
 * symmetry = BC3, order 48
 * circum = $$\frac{\sqrt2}{2} ≈ 0.70711$$
 * inrad = $$\frac{\sqrt6}{6} ≈ 0.40825$$
 * height = $$\frac{\sqrt6}{3} ≈ 0.81650$$
 * volume = $$\frac{\sqrt2}{3} ≈ 0.47140$$
 * dih = $$\arccos\left(-\frac{13\right) ≈ 109.47122°$$
 * pieces = 8
 * loc = 1
 * dual = Cube
 * conjugate = Octahedron
 * conv=Yes
 * orientable=Yes
 * nat=Tame}}

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.

It occurs as cells in one regular polychoron, namely the icositetrachoron.

Vertex coordinates
An octahedron of side length 1 has vertex coordinates given by all permutations of:
 * $$\left(±\frac{\sqrt2}{2},\,0,\,0\right).$$

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 axial, 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)

Variations
Other variants of the octahedron exist, using 8 triangular faces with 6 4-fold vertices. Some of these include:


 * Tetratetrahedron - 2 sets of 4 triangles - just a coloring with no true variants in measures
 * Triangular antiprism - 2 equilateral bases, 6 isosceles sides, vertex transitive
 * Triangular antipodoum - as above with 2 different sized bases and 2 sets of 3 isosceles sides
 * Square bipyramid - 8 isosceles triangles, square prism symmetry
 * Rectangular bipyramid - 2 sets of 4 isosceles triangles
 * Rhombic bipyramid - 8 scalene triangles, digonal prism symmetry

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)

There is also an infinite family of prismatic octahedron compounds, the antiprisms of compounds of triangles:
 * Hexagrammic antiprism (2)
 * Fissal enneagrammic antiprism (3)
 * Tetratriangular antiprism (4)

The octahedron has one stellation, the stella octangula.