Small rhombated pentachoron

The small rhombated pentachoron, or srip, also commonly called the cantellated 5-cell or cantellated pentachoron, is a convex uniform polychoron that consists of 5 regular octahedra, 10 triangular prisms, and 5 cuboctahedra. 1 octahedron, 2 triangular prisms, and 2 cuboctahedra join at each vertex. As one of its names suggests, it can be obtained by cantellating the pentachoron.

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


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

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


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

Representations
A small rhombated pentachoron has the following Coxeter diagrams:


 * x3o3x3o (full symmetry)
 * oxx3xxo3oox&#xt (A3 axial, octahedron-first)
 * x(uo)xo x(ou)xx3o(xo)xo&#xt (A2×A1 axial, triangular prism-first)

Semi-uniform variant
The small rhombated pentachoron has a semi-uniform variant of the form x3o3y3o that maintains its full symmetry. This variant uses 5 octahedra of size y, 5 rhombitetrahedra of form x3o3y, and 10 triangular prisms of form x y3o as cells, with 2 edge lengths.

With edges of length a (surrounded by one of each cell type) and b (of octahedra), its circumradius is given by $$\sqrt{\frac{2a^2+3b^2+2ab}{5}}$$ and its hypervolume is given by $$(a^4+12a^3b+54a^2b^2+68ab^3+11b^4)\frac{\sqrt5}{96}$$.

Related polychora
When viewed in A3 axial symmetry, the small rhombated pentachoron can be cut into 2 segmentochora, namely cuboctahedron atop truncated tetrahedron and octahedron atop truncated tetrahedron, join at the truncated tetrahedral bases.

The triangular pucofastegium occurs as the triangle-first cap of the small rhombated pentachoron.