Decachoron

The decachoron, or deca, also commonly called the bitruncated 5-cell or bitruncated pentachoron, is a convex noble uniform polychoron that consists of 10 truncated tetrahedra as cells. Four cells join at each vertex. It is the medial stage of the truncation series between a regular pentachoron and its dual.

It is also the 10-3 gyrochoron.

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


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

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


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

Representations
A decachoron has the following Coxeter diagrams:


 * o3x3x3o (full symmetry)
 * oox3xux3xoo&#xt (A3 axial, cell-first)
 * oxuxo ooxux3xuxoo&#xt (A2×A1 axial, triangle-first)

Variations
The following variants of the decachoron exist:


 * Pentapentachoron - semi-uniform with 2 types of cells
 * 10-3 gyrochoron - has step prism symmetry, isochoric

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Truncated tetrahedron (10): Bidecachoron
 * Triangle (20): Biambodecachoron
 * Hexagon (20): Small prismatodecachoron
 * Edge (60): Rectified decachoron