Square antiprismatic pyramid

The square antiprismatic pyramid, or squappy, is a CRF segmentochoron (designated K-4.17 on Richard Klitzing's list). It consists of 8 regular tetrahedra, 2 square pyramids, and 1 square antiprism. As the name suggests, it is a pyramid based on the square antiprism.

Two square antiprismatic pyramids can be attached at their bases to form a square antiprismatic tegum.

The segmentochoron octahedron atop cube can be obtained by augmenting the square antiprismatic cells of the square antifastegium with square antiprismatic pyramids.

Vertex coordinates
The vertices of a square antiprismatic pyramid of edge length 1 are given by:
 * $$\left(±\frac12,\,±\frac12,\,\frac{\sqrt[4]8}4,\,0\right),$$
 * $$\left(0,\,±\frac{\sqrt2}2,\,-\frac{\sqrt[4]8}4,\,0\right),$$
 * $$\left(±\frac{\sqrt2}2,\,0,\,-\frac{\sqrt[4]8}4,\,0\right),$$
 * $$\left(0,\,0,\,0,\,\sqrt{\frac{4-\sqrt2}8}\right).$$