Great rhombated pentachoron

The great rhombated pentachoron, or grip, also commonly called the cantitruncated 5-cell or cantitruncated pentachoron, is a convex uniform polychoron that consists of 10 triangular prisms, 5 truncated tetrahedra, and 5 truncated octahedra. 1 triangular prism, 1 truncated tetrahedron, and 2 truncated octahedra join at each vertex. As one of its names suggests, it can be obtained by cantitruncating the pentachoron.

Vertex coordinates
The vertices of a great rhombated pentachoron of edge length 1 are given by:


 * $$\left(\frac{\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,\frac{\sqrt3}{2},\,±\frac32\right),$$
 * $$\left(\frac{\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,-\sqrt3,\,0\right),$$
 * $$\left(\frac{\sqrt{10}}{20},\,\frac{5\sqrt6}{12},\,-\frac{\sqrt3}{6},\,±\frac32\right),$$
 * $$\left(\frac{\sqrt{10}}{20},\,\frac{5\sqrt6}{12},\,-\frac{2\sqrt3}{3},\,±1\right),$$
 * $$\left(\frac{\sqrt{10}}{20},\,\frac{5\sqrt6}{12},\,\frac{5\sqrt3}{6},\,±\frac12\right),$$
 * $$\left(\frac{\sqrt{10}}{20},\,-\frac{7\sqrt6}{12},\,\frac{\sqrt3}{3},\,±1\right),$$
 * $$\left(\frac{\sqrt{10}}{20},\,-\frac{7\sqrt6}{12},\,-\frac{2\sqrt3}{3},\,0\right),$$
 * $$\left(-\frac{\sqrt{10}}{5},\,0,\,\frac{\sqrt3}{2},\,±\frac32\right),$$
 * $$\left(-\frac{\sqrt{10}}{5},\,0,\,-\sqrt3,\,0\right),$$
 * $$\left(-\frac{\sqrt{10}}{5},\,\frac{\sqrt6}{3},\,\frac{\sqrt3}{6},\,±\frac32\right),$$
 * $$\left(-\frac{\sqrt{10}}{5},\,\frac{\sqrt6}{3},\,\frac{2\sqrt3}{3},\,±1\right),$$
 * $$\left(-\frac{\sqrt{10}}{5},\,\frac{\sqrt6}{3},\,-\frac{5\sqrt3}{6},\,±\frac12\right),$$
 * $$\left(-\frac{\sqrt{10}}{5},\,-\frac{2\sqrt6}{3},\,\frac{\sqrt3}{6},\,±\frac12\right),$$
 * $$\left(-\frac{\sqrt{10}}{5},\,-\frac{2\sqrt6}{3},\,-\frac{\sqrt3}{3},\,0\right),$$
 * $$\left(\frac{3\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,\frac{\sqrt3}{6},\,±\frac32\right),$$
 * $$\left(\frac{3\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{\sqrt3}{6},\,±\frac32\right),$$
 * $$\left(\frac{3\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,\frac{2\sqrt3}{3},\,±1\right),$$
 * $$\left(\frac{3\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{2\sqrt3}{3},\,±1\right),$$
 * $$\left(\frac{3\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,-\frac{5\sqrt3}{6},\,±\frac12\right),$$
 * $$\left(\frac{3\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,\frac{5\sqrt3}{6},\,±\frac12\right),$$
 * $$\left(\frac{3\sqrt{10}}{10},\,±\frac{\sqrt6}{2},\,0,\,±1\right),$$
 * $$\left(\frac{3\sqrt{10}}{10},\,±\frac{\sqrt6}{2},\,±\frac{\sqrt3}{2},\,±\frac12\right),$$
 * $$\left(-\frac{9\sqrt{10}}{20},\,-\frac{\sqrt6}{12},\,\frac{\sqrt3}{3},\,±1\right),$$
 * $$\left(-\frac{9\sqrt{10}}{20},\,-\frac{\sqrt6}{12},\,-\frac{2\sqrt3}{3},\,0\right),$$
 * $$\left(-\frac{9\sqrt{10}}{20},\,\frac{\sqrt6}{4},\,0,\,±1\right),$$
 * $$\left(-\frac{9\sqrt{10}}{20},\,\frac{\sqrt6}{4},\,±\frac{\sqrt3}{2},\,±\frac12\right),$$
 * $$\left(-\frac{9\sqrt{10}}{20},\,-\frac{5\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),$$
 * $$\left(-\frac{9\sqrt{10}}{20},\,-\frac{5\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right).$$

Much simpler coordinates can be given in five dimensions, as all permutations of:


 * $$\left(\frac{3\sqrt2}{2},\,\sqrt2,\,\frac{\sqrt2}{2},\,0,\,0\right).$$

Representations
The great rhombated pentachoron has the following Coxeter diagrams:


 * x3x3x3o (full symmetry)
 * xuxx3xxux3ooox&#xt (A3 axial, truncated tetrahedron-first)
 * xu(xd)uxo xu(dx)uxx3oo(ox)xux&#xt (A2×A1 symmetry, triangular prism-first)

Semi-uniform variant
The great rhombated pentachoron has a semi-uniform variant of the form x3y3z3o that maintains its full symmetry. This variant uses 5 semi-uniform truncated tetrahedra of form y3z3o, 5 great rhombitetratetrahedra of form x3y3z, and 10 triangular prisms of form x z3o as cells, with 3 edge lengths.

With edges of lengths a, b, and c (such that it is given by a3b3c3o), its circumradius is given by $$\sqrt{\frac{2a^2+3b^2+3c^2+3ab+2ac+4bc}{5}}$$.

Related polychora
Uniform polychoronc ompounds composed of great rhombated pentachoraa include:


 * Great rhombated stellated decachoron (2)
 * Great rhombated medial hexacosichoron (120)