Truncated hexacosichoron

The truncated hexacosichoron, or tex, also commonly called the truncated 600-cell, is a convex uniform polychoron that consists of 120 regular icosahedra and 600 truncated tetrahedra. 1 icosahedron and five truncated tetrahedra join at each vertex. As the name suggests, it can be obtained as the truncation of a hexacosichoron.

It is also isogonal under H4/5 symmetry, with the icosahedra having the symmetry of snub tetrahedra, and 480 of the truncated tetrahedra having trigonal symmetry only.

Vertex coordinates
The vertices of a truncated hexacosichoron of edge length 1 are given by all even permutations of:
 * $$\left(0,\,±\frac12,\,±\frac{1+\sqrt5}{4},\,±\frac{7+5\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac12,\,±3\frac{1+\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),$$
 * $$\left(0,\,±1,\,±\frac{1+\sqrt5}{2},\,±(2+\sqrt5)\right),$$
 * $$\left(0,\,±\frac{5+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±\frac{4+\sqrt5}{2}\right),$$
 * $$\left(0,\,±\frac{2+\sqrt5}{2},\,±\frac{7+3\sqrt5}{4},\,±\frac{7+\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±1,\,±\frac{5+3\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{1+\sqrt5}{2},\,±3\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±1,\,±\frac{2+\sqrt5}{2},\,±3\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt5}{2},\,±3\frac{1+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4}\right).$$

Semi-uniform variant
The truncated hexacosichoron has a semi-uniform variant of the form o5o3y3x that maintains its full symmetry. This variant uses 120 icosahedra of size y and 600 semi-uniform truncated tetrahedra of form x3y3o as cells, with 2 edge lengths.

With edges of length a (surrounded by truncated tetrahedra only) and b (of icosahedra), its circumradius is given by $$\sqrt{\frac{3a^2+10b^2+10ab+(a^2+4b^2+4ab)\sqrt5}{2}}$$.

Related polychora
The truncated hexacosichoron is the colonel of a two-member regiment that also includes the truncated faceted hexacosichoron.

Uniform polychoron compounds composed of truncated hexacosichora include:


 * Truncated chirotetrahedral dishexacosichoron (2)
 * Truncated snub pentishecatonicosachoron (5)
 * Truncated snub decahecatonicosachoron (10)