Square antiprismatic pyramid

The square antiprismatic pyramid, or squappy, is a CRF segmentochoron (designated K-4.17 on Richard Klitzing's list). It has 8 regular tetrahedra, 2 square pyramids and 1 square antiprism as cells. 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 bipyramid.

Vertex coordinates
The vertices of a square antiprismatic pyramid of edge length 1 centered at the origin are given by:
 * $$\left(±\frac{1}{2},\,±\frac{1}{2},\,\frac{\sqrt[4]{8}}{4},\,\sqrt{\frac{4+\sqrt{2}}{56}}\right),$$
 * $$\left(0,\, ±\frac{\sqrt{2}}{2},\, –\frac{\sqrt[4]{8}}{4},\, \sqrt{\frac{4+\sqrt{2}}{56}}\right),$$
 * $$\left(±\frac{\sqrt{2}}{2},\, 0,\, –\frac{\sqrt[4]{8}}{4},\, \sqrt{\frac{4+\sqrt{2}}{56}}\right),$$
 * $$\left(0,\,0,\,0,\,\sqrt{\frac{4+\sqrt{2}}{7}}\right).$$