Octahedral pyramid

The octahedral pyramid, or octpy, is a convex segmentochoron (designated K-4.3 on Richard Klitzing's list). It has 8 regular tetrahedra and 1 regular octahedron as cells. As the name suggests, it is a pyramid based on the octahedron.

Two octahedral pyramids can be attached at their bases to form a regular hexadecachoron.

Vertex coordinates
The vertices of an octahedral pyramid of edge length 1 are given by:
 * (±$\sqrt{2}$/2, 0, 0, 0) and all permutations of first 3 coordinates
 * (0, 0, 0, $\sqrt{2}$/2)

Representations
An octahedral pyramid has the following Coxeter diagrams:


 * oo4oo3ox&#x (full symmetry)
 * oo3ox3oo&#x (base is in A3 symmetry, tetratetrahedral pyramid)
 * oxo3oox&#x (base is in A2 symmetry only, triangular antiprismatic pyramid)