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 rhombitetratetrahedra of form x3o3y, and 10 triangular prisms of form x y3o as cells, with 2 edge lengths.

With edges of length a (surrounds 2 rhombitetratetrahedra) 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
The small rhombated pentachoron is the colonel of the largest regiment of uniform polychora with A4 symmetry, which has a total of 7 members. Its facetings include the retrosphenoverted trispentachoron, small rhombic dispentachoron, pseudorhombic prismatopentachoron, grand rhombic prismatopentachoron, prismatopentintercepted dispentachoron, and prismatointercepted prismatodispentachoron.

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.