Octahedral pyramid

The octahedral pyramid, or octpy, is a CRF 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. An octahedral pyramid can be further cut in half to produce two square scalenes.

Apart from being a point atop octahedron, it has an alternate segmentochoron representation as a triangle atop gyro tetrahedron seen as a triangular pyramid.

Vertex coordinates
The vertices of an octahedral pyramid of edge length 1 are given by: with all permutations of the first 3 coordinates of:
 * $$\left(0,\,0,\,0,\,\frac{\sqrt2}{2}\right),$$
 * $$\left(0,\,0,\,±\frac{\sqrt2}{2},\,0\right).$$

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)