Great prismatodecachoron

The great prismatodecachoron, or gippid, also commonly called the omnitruncated 5-cell or omnitruncated pentachoron, is a convex uniform polychoron that consists of 20 hexagonal prisms and 10 truncated octahedra. 2 hexagonal prisms and 2 truncated octahedra join at each vertex. It is the omnitruncate of the A4 family of uniform polychora.

This polychoron can be alternated into a snub decachoron, although it cannot be made uniform.

Like the omnitruncated simplex of any dimension, this polychoron can tile 4D space. The resulting tetracomb is called the omnitruncated cyclopentachoric tetracomb.

Vertex coordinates
The vertices of a great prismatodecachoron of edge length 1 are given by the following points:


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

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


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

Representations
A great prismatodecachoron has the following Coxeter diagrams:


 * x3x3x3x (full symmetry)
 * xxxux3xxuxx3xuxxx&#xt (A3 axial, truncated octahedron-first)

Semi-uniform variant
The truncated pentachoron has a semi-uniform variant of the form x3y3o3o that maintains its full symmetry. This variant uses 5 tetrahedra of size y and 5 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 tetrahedra), its circumradius is given by $$\sqrt{\frac{2a^2+3b^2+3ab}{5}}$$ and its hypervolume is given by $$(a^4+8a^3b+24a^2b^2+32ab^3+11b^4)\frac{\sqrt5}{96}$$.

Semi-uniform variant
The great prismatodecachoron has a semi-uniform variant of the form x3y3y3x that maintains its full symmetry. This variant uses 10 great rhombitetratetrahedra of form x3y3y and 20 ditrigonal prisms of form x x3y as cells, with 2 edge lengths.

With edges of length a and b (so that it is represented by a3b3b3a), its circumradius is given by $$\sqrt{a^2+2b^2+2ab}$$.

If it has only single pentachoric symmetry, the variant is called a great disprismatopentapentachoron.