| Octahedron |
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Click here for spinning version. |
| Type |
Platonic |
| Face polygon |
triangle |
| Faces |
8 |
| Edges |
12 |
| Vertices |
6 |
| Faces per vertex |
4 |
| Vertices per face |
3 |
| Symmetry group |
octahedral (Oh) |
| Dual polyhedron |
cube |
| Properties |
regular, convex |
An octahedron (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:
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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.
Using the standard nomenclature for Johnson solids, an octahedron
would be called a square bipyramid.
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