Prismatorhombated pentachoron

The prismatorhombated pentachoron, or prip, also commonly called the runcitruncated 5-cell or runcitruncated pentachoron, is a convex uniform polychoron that consists of 10 triangular prisms, 10 hexagonal prisms, 5 truncated tetrahedra, and 5 cuboctahedra. 1 triangular prism, 2 hexagonal prisms, 1 truncated tetrahedron, and 1 cuboctahedron join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the pentachoron.

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


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

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


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

Representations
A prismatorhombated pentachoron has the following coxeter diagrams:


 * x3x3o3x (full symmetry)
 * xxux3xxoo3oxxx&#xt (A3 symmetry, truncated tetrahedron-first)
 * x(ou)(dx)ux x(uo)(ox)xx3x(xu)(xu)xo&#xt (A2×A1 symmetry, hexagonal prism-first)

Semi-uniform variant
The prismatorhombated pentachoron has a semi-uniform variant of the form x3y3o3z that maintains its full symmetry. This variant uses 5 semi-uniform truncated tetrahedra of form x3y3o, 5 rhombitetratetrahedra of form y3o3z, 10 triangular prisms of form x z3o, and 10 ditrigonal prisms of form z x3y as cells, with 3 edge lengths.

With edges of lengths a, b, and c (such that it is given by a3b3o3c), its circumradius is given by $$\sqrt{\frac{2a^2+3b^2+2c^2+3ab+ac+2bc}{5}}$$.

Related polychora
The prismatorhombated pentachoron is the colonel of a 3-member regiment that also includes the prismatotrispentachoron and the rhombiprismic dispentachoron.

The segmentochoron truncated tetrahedron atop truncated octahedron is a cap of the prismatorhombated pentachoron.