Stella octangula

The stella octangula, stellated octahedron, so, or compound of two tetrahedra is a regular polyhedron compound. It's made out of 8 triangles, 3 joining at each vertex. It can be constructed by taking a tetrahedron and overlaying it with its central inversion. Alternatively, it can be created by stellating the octahedron.

It can also be considered an antiprism based on the compound of two digons {4/2}.

Its quotient prismatic equivalent is the hexadecachoron, which is four-dimensional.

Representations

 * (full symmetry)
 * (β2β4o)
 * (β2β2β)
 * xo3oo3ox

Vertex coordinates
The vertices of a stella octangula of edge length 1 can be given by:


 * $$\left(\pm\frac{\sqrt2}{4},\,\pm\frac{\sqrt2}{4},\,\pm\frac{\sqrt2}{4}\right).$$

These arise from the fact that a tetrahedron can be constructed as the alternation of the cube. Taking even changes of sign and odd changes of sign reconstructs the two component tetrahedra.

Alternate coordinates can be derived from those of the triangle, by considering the tetrahedron as a triangular pyramid:


 * $$\pm\left(\pm\frac12,\,-\frac{\sqrt3}{6},\,-\frac{\sqrt6}{12}\right),$$
 * $$\pm\left(0,\,\frac{\sqrt3}{3},\,-\frac{\sqrt6}{12}\right),$$
 * $$\left(0,\,0,\,\pm\frac{\sqrt6}{4}\right).$$

These are more complicated, but generalize to two-simplex compounds of any dimension.