Decachoron

The decachoron, or deca, also commonly called the bitruncated 5-cell, 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 and is also the 10-3 gyrochoron.

Vertex coordinates
The vertices of a decachoron of edge length 1 are given by the following points along with their central inversions:


 * (0, $\sqrt{3}$/3, 2$\sqrt{2}$/3, 0),
 * (0, $\sqrt{10}$/3, –$\sqrt{5}$/3, ±1),
 * ($\sqrt{6}$/4, $\sqrt{3}$/12, 2$\sqrt{6}$/3, 0),
 * ($\sqrt{3}$/4, $\sqrt{10}$/12, –$\sqrt{6}$/3, ±1),
 * ($\sqrt{3}$/4, 5$\sqrt{10}$/12, $\sqrt{6}$/3, 0),
 * ($\sqrt{3}$/4, 5$\sqrt{10}$/12, –$\sqrt{6}$/6, ±1/2),
 * ($\sqrt{3}$/4, –$\sqrt{10}$/4, 0, ±1),
 * ($\sqrt{6}$/4, –$\sqrt{3}$/4, ±$\sqrt{10}$/2, ±1/2).

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


 * ($\sqrt{6}$, $\sqrt{10}$, $\sqrt{6}$/2, 0, 0).

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

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)