Octahedron
In geometry, an octahedron (pl.: octahedra or octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.
| Regular octahedron | |
|---|---|
(Click here for rotating model) | |
| Type | Platonic solid |
| Elements | F = 8, E = 12 V = 6 (χ = 2) |
| Faces by sides | 8{3} |
| Conway notation | O aT |
| Schläfli symbols | {3,4} |
| r{3,3} or {}+{}+{}=3{} | |
| Face configuration | V4.4.4 |
| Wythoff symbol | 4 | 2 3 |
| Coxeter diagram | |
| Symmetry | Oh, BC3, [4,3], (*432) |
| Rotation group | O, [4,3]+, (432) |
| References | U05, C17, W2 |
| Properties | regular, convexdeltahedron, Hanner polytope |
| Dihedral angle | 109.47122° = arccos(−1⁄3) |
3.3.3.3 (Vertex figure) |
Cube (dual polyhedron) |
Net | |
A regular octahedron is the dual polyhedron of a cube. It is also a rectified tetrahedron, a square bipyramid in any of three orthogonal orientations, and a triangular antiprism in any of four orientations.
An octahedron is the three-dimensional case of the more general concept of a cross polytope.
A regular octahedron is a 3-ball in the Manhattan (ℓ1) metric.
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