Small prismatohecatonicosidishexacosichoron

The small prismatohecatonicosidishexacosichoron, or sphiddix, is a nonconvex uniform polychoron that consists of 720 pentagrammic prisms, 600 truncated tetrahedra, 600 truncated octahedra, and 120 small icosicosidodecahedra. 1 pentagrammic prism, 1 truncated tetrahedron, 1 small icosicosidodecahedron, and 2 truncated octahedra join at each vertex.

Vertex coordinates
The vertices of a small prismatohecatonicosidishexacosichoron of edge length 1 are given by all permutations of: Plus all even permutations of:
 * $$\left(±\frac12,\,±\frac12,\,±\frac{3+2\sqrt5}{2},\,±\frac{7+2\sqrt5}{2}\right),$$
 * $$\left(±1,\,±1,\,±(2+\sqrt5),\,±(3+\sqrt5)\right),$$
 * $$\left(±\frac{5+2\sqrt5}{2},\,±\rac{5+2\sqrt5}{2},\,±\frac12,\,±\frac32\right),$$
 * $$\left(0,\,±\frac12,\,±\frac{11+5\sqrt5}{4},\,±5\frac{1+\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac{\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,±\frac{11+7\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac{5+\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±\frac{1+3\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac{3+\sqrt5}{2},\,±\frac{5+\sqrt5}{2},\,±(3+\sqrt5)\right),$$
 * $$\left(0,\,±\frac{7+\sqrt5}{4},\,±\frac{11+3\sqrt5}{4},\,±\frac{5+2\sqrt5}{2}\right),$$
 * $$\left(±\frac{\sqrt5-1}{4},\,±\frac12,\,±\frac{13+5\sqrt5}{4},\,±(1+\sqrt5)\right),$$
 * $$\left(±\frac{\sqrt5-1}{4},\,±\frac12,\,±\frac{11+7\sqrt5}{4},\,±\frac{1+\sqrt5}{2}\right),$$
 * $$\left(±\frac{\sqrt5-1}{4},\,±1,\,±3\frac{2+\sqrt5}{2},\,±3\frac{1+\sqrt5}{4}\right),$$
 * $$\left(±\frac{\sqrt5-1}{4},\,±\frac{4+\sqrt5}{2},\,±\frac{11+3\sqrt5}{4},\,±(2+\sqrt5)\right),$$
 * $$\left(±\frac{\sqrt5-1}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{5+\sqrt5}{2},\,±\frac{5+2\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±1,\,±\frac{9+5\sqrt5}{4},\,±\frac{7+5\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±1,\,±\frac{11+7\sqrt5}{4},\,±\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac32,\,±3\frac{2+\sqrt5}{2},\,±\frac{2+\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac{1+3\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±\frac{13+5\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac{4+\sqrt5}{2},\,±\frac{7+2\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac{3+\sqrt5}{2},\,±\frac{11+3\sqrt5}{4},\,±\frac{7+5\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{5+3\sqrt5}{4},\,±\frac{5+\sqrt5}{2},\,±\frac{9+5\sqrt5}{4}\right),$$
 * $$\left(±1,\,±\frac{\sqrt5}{2},\,±\frac{5+3\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),$$
 * $$\left(±1,\,±\frac{3+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{7+2\sqrt5}{2}\right),$$
 * $$\left(±1,\,±\frac{5+\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±\frac{7+5\sqrt5}{4}\right),$$
 * $$\left(±1,\,±\frac{2+\sqrt5}{2},\,±\frac{7+\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),$$
 * $$\left(±1,\,±3\frac{1+\sqrt5}{4},\,±\frac{4+\sqrt5}{2},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{\sqrt5}{2},\,±\frac{4+\sqrt5}{2},\,±\frac{5+2\sqrt5}{2},\,±\frac{3+2\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac32,\,±\frac{9+5\sqrt5}{4},\,±(2+\sqrt5)\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac32,\,±\frac{13+5\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±\frac{7+2\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±\frac{5+2\sqrt5}{2},\,±5\frac{1+\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{4+\sqrt5}{2},\,±\frac{9+5\sqrt5}{4},\,±(1+\sqrt5)\right),$$
 * $$\left(±\frac32,\,±\frac{1+\sqrt5}{2},\,±\frac{7+3\sqrt5}{4},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac{5+\sqrt5}{4},\,±\frac{7+2\sqrt5}{2},\,±\frac{5+3\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac{7+\sqrt5}{4},\,±\frac{9+5\sqrt5}{4},\,±\frac{3+2\sqrt5}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{2},\,±(3+\sqrt5),\,±(1+\sqrt5)\right),$$
 * $$\left(±\frac{1+3\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±(2+\sqrt5),\,±\frac{3+2\sqrt5}{2}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±\frac{5+3\sqrt5}{4},\,±(2+\sqrt5),\,±5\frac{1+\sqrt5}{4}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±(1+\sqrt5),\,±\frac{7+3\sqrt5}{4},\,±\frac{7+5\sqrt5}{4}\right),$$
 * $$\left(±3\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±\frac{7+5\sqrt5}{4},\,±\frac{3+2\sqrt5}{2}\right),$$
 * $$\left(±3\frac{1+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±(1+\sqrt5)\right).$$

Related polychora
The small prismatohecatonicosidishexacosichoron is the colonel of a 3-member regiment that also includes the small hexacosiprismatodishecatonicosachoron and small small hexacosidishecatonicosachoron.