Hexacosihecatonicosachoron

The hexacosihecatonicosachoron, or xhi, also commonly called the bitruncated 120-cell, is a convex uniform polychoron that consists of 600 truncated tetrahedra and 120 truncated icosahedra. 2 truncated tetrahedra and 2 truncated icosahedra join at each vertex. It is the medial stage of the truncation series between a hecatonicosachoron and its dual hexacosichoron. As such, it could also be called a bitruncated 600-cell.

Vertex coordinates
Coordinates for the vertices of a hexacosihecatonicosachoron of edge length 1 are given by all permutations of: together with all even permutations of:
 * $$\left(0,\,0,\,±(1+\sqrt5),\,±\frac{7+3\sqrt5}{2}\right),$$
 * $$\left(±\frac{5+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,±\frac{9+5\sqrt5}{4},\,±\frac{9+5\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac12,\,±\frac{13+5\sqrt5}{4},\,±\frac{7+5\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac12,\,±3\frac{1+\sqrt5}{4},\,±\frac{13+7\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac{1+\sqrt5}{4},\,±5\frac{3+\sqrt5}{4},\,±\frac{3+2\sqrt5}{2}\right),$$
 * $$\left(0,\,±3\frac{3+\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±\frac{11+3\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{1+\sqrt5}{2},\,±\frac{13+7\sqrt5}{4},\,±\frac{5+\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{1+\sqrt5}{2},\,±5\frac{3+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±1,\,±\frac{11+5\sqrt5}{4},\,±\frac{9+5\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{3+\sqrt5}{2},\,±\frac{11+5\sqrt5}{4},\,±\frac{11+3\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±1,\,±\frac{2+\sqrt5}{2},\,±\frac{13+7\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{9+5\sqrt5}{4},\,±\frac{11+3\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±\frac{9+5\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac{7+3\sqrt5}{4},\,±\frac{7+5\sqrt5}{4},\,±\frac{5+2\sqrt5}{2}\right),$$
 * $$\left(±1,\,±\frac{1+\sqrt5}{2},\,±\frac{7+3\sqrt5}{2},\,±\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±1,\,±\frac{2+\sqrt5}{2},\,±\frac{13+5\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±1,\,±\frac{5+\sqrt5}{4},\,±\frac{11+5\sqrt5}{4},\,±\frac{5+2\sqrt5}{2}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±(1+\sqrt5),\,±\frac{11+5\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±\frac{3+\sqrt5}{2},\,±\frac{7+5\sqrt5}{4},\,±\frac{9+5\sqrt5}{4}\right),$$
 * $$\left(±3\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{5+\sqrt5}{4},\,±\frac{2+\sqrt5}{2},\,±5\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac{5+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{13+5\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right).$$

Semi-uniform variant
The hexacosihecatonicosachoron has a semi-uniform variant of the form o5x3y3o that maintains its full symmetry. This variant uses 600 semi-uniform truncated tetrahedra of form x3y3o and 120 semi-uniform truncated icosahedra of form o5x3y as cells, with 2 edge lengths.

With edges of length a (of pentagonal faces) and b (of triangular faces), its circumradius is given by $$\sqrt{\frac{21a^2+10b^2+28ab+(9a^2+4b^2+12ab)\sqrt5}{2}}$$.