Octahedron

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An octahedron is a Platonic solid composed of eight triangular faces, with four meeting at each vertex. The octahedron is a special kind of triangular antiprism and of square bipyramid, and is dual to the cube. Canonical coordinates for the vertices of an octahedron centered at the origin are (±1,0,0), (0,±1,0), (0,0,±1).

The interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. The vertices of the octahedron lie at the midpoints of the faces of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids. One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, and together they define a regular compound.

Octahedra and tetrahedra can be mixed together to form a vertex, edge, and face-uniform tiling of space. This is the only such tiling save the regular tessellation of cubes, and is one of the five Andreini tessellations. Another is a tessellation of octahedra and cuboctahedra.