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|Faces per vertex
|Vertices per face
(plural: octahedra) is a polyhedron with eight faces. A regular
octahedron is a Platonic solid
composed of eight faces each of which is an equilateral triangle four of which meet at each vertex. The regular 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 area A and the volume V of a regular octahedron of edge length a are:
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
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 cubess, and is one of the five Andreini tessellations. Another is a tessellation of octahedra and cuboctahedra.
Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid.