Stellated octahedron

Stellated octahedron

Type	Regular compound
Coxeter symbol	{4,3}[2{3,3}]{3,4}[1]
Schläfli symbols	{{3,3}} a{4,3} ß{2,4} ßr{2,2}
Coxeter diagrams	∪
Stellation core	regular octahedron
Convex hull	Cube
Index	UC4, W19
Polyhedra	two tetrahedra
Faces	8 triangles
Edges	12
Vertices	8
Dual polyhedron	self-dual
Symmetry group and Coxeter group	Oh, [4,3], order 48 D4h, [4,2], order 16 D2h, [2,2], order 8 D3d, [2+,6], order 12
Subgroup restricting to one constituent	Td, [3,3], order 24 D2d, [2+,4], order 8 D2, [2,2]+, order 4 C3v, [3], order 6

The stellated octahedron is the only stellation of the octahedron. 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 De Divina Proportione, 1509.^[2]

It is the simplest of five regular polyhedral compounds, and the only regular compound of two tetrahedra. It is also the least dense of the regular polyhedral compounds, having a density of 2.

It can be seen as a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two overlapping equilateral triangles, centrally symmetric to each other, and in the same way the stellated octahedron can be formed from two centrally symmetric overlapping tetrahedra. This can be generalized to any desired amount of higher dimensions; the four-dimensional equivalent construction is the compound of two 5-cells. It can also be seen as one of the stages in the construction of a 3D Koch snowflake, a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure. The first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron.