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| Octahedron | |
|---|---|
Click on picture for large version. Click 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:
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 tilingIn geometry, a tiling (also called tessellation mosaic or dissection of a given shape S consists of a collection of other shapes which precisely cover S. Often the shape S to be tiled is the Euclidean plane, but other shapes and three-dimensional objects of space. This is the only such tiling save the regular tessellation of cubes, and is one of the five Andreini tessellationThe Andreini tessellations are tilings of three-dimensional space using Platonic and Archimedean solids such that all vertices are identical. They are special case of uniform tessellation. There are 28 such tessellations. Grunbaum, Uniform tilings of 3-sps. Another is a tessellation of octahedra and cuboctahedra.
Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid.
Especially in roleplaying, this solid is known as a d8Dice (the plural of the word die probably from the Latin dare to give) are, in general, small polyhedral objects with the faces marked with numbers or other symbols, thrown in order to choose one of the faces randomly. The most common dice are small cubes, one of the more common Polyhedral dice.