Stellated octahedron
| Stellated octahedron | |
|---|---|
| Type | Regular compound Polyhedral compound UC4 W19 |
| Faces | 8 triangles |
| Edges | 12 |
| Vertices | 8 |
| Schläfli symbol | {{3,3}} a{4,3} ß{2,4} ßr{2,2} |
| Coxeter diagram | {4,3}[2{3,3}]{3,4} |
| Symmetry group | octahedral symmetry, pyritohedral symmetry |
| Dual polyhedron | self-dual |
The stellated octahedron is a shape made from two regular tetrahedra crossing each other. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted in Pacioli's 1509 De Divina Proportione.
Two variations of this shape have been considered, one consisting of the two tetrahedra themselves and another consisting of their union, with interior boundaries removed. The two tetrahedra form the simplest of the five regular polyhedral compounds, and the only regular polyhedral compound composed of only two polyhedra. Their union forms the only stellation of the octahedron.
This shape can be seen as a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two crossing equilateral triangles, centrally symmetric to each other, and in the same way, the stellated octahedron can be formed from two centrally symmetric crossing tetrahedra. This can be generalized to any desired number of higher dimensions; the four-dimensional equivalent construction is the compound of two 5-cells.