Pentagrammic double antiprismoid

The pentagrammatic double antiprismoid, or padiap, is a nonconvex uniform polychoron that consists of 300 tetrahedra and 20 pentagrammic retroprisms. 12 tetrahedra and 2 pentagrammic retroprisms join at each vertex.

This polychoron can be formed as a subsymmetrical faceting of the grand hexacosichoron in a similar way as its conjugate, the convex grand antiprism, can be form from the hexacosichoron. The pentagrammic retroprisms are facetings of the great icosahedra which form the grand hexacosichoron's vertex figures.

Vertex coordinates
The vertices of a pentagrammatic double antiprismoid of edge length 1 are given by:


 * $$±\left(±\frac{3-\sqrt5}{4},\,0,\,\frac{\sqrt5-1}{4},\,-\frac12\right),$$
 * $$±\left(±\frac12,\,0,\,\frac{3-\sqrt5}{4},\,-\frac{\sqrt5-1}{4}\right),$$
 * $$±\left(0,\,±\frac{3-\sqrt5}{4},\,\frac12,\,\frac{\sqrt5-1}{4}\right),$$
 * $$±\left(0,\,±\frac12,\,\frac{\sqrt5-1}{4},\,\frac{3-\sqrt5}{4}\right),$$
 * $$\left(0,\,0,\,±\frac{\sqrt5-1}{2},\,0\right),$$
 * $$\left(0,\,0,\,0,\,±\frac{\sqrt5-1}{2}\right),$$
 * $$\left(±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4}\right),$$
 * $$\left(±\frac{\sqrt5-1}{4},\,0,\,±\frac12,\,±\frac{3-\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac{\sqrt5-1}{4},\,±\frac{3-\sqrt5}{4},\,±\frac12\right),$$
 * $$\left(±\frac{\sqrt5-1}{4},\,±\frac12,\,±\frac{3-\sqrt5}{4},\,0\right),$$
 * $$\left(±\frac{3-\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,±\frac12,\,0\right),$$
 * $$\left(±\frac12,\,±\frac{3-\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,0\right),$$
 * $$\left(±\frac{3-\sqrt5}{4},\,±\frac12,\,0,\,±\frac{\sqrt5-1}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt5-1}{4},\,0,\,±\frac{3-\sqrt5}{4}\right),$$
 * $$\left(±\frac{\sqrt5-1}{4},\,±\frac{3-\sqrt5}{4},\,0,\,±\frac12\right).$$