Rectified hexateron

The rectified hexateron, or rix, also called the rectified 5-simplex, is a convex uniform polyteron. It consists of 6 regular pentachora and 6 rectified pentachora. Two pentachora and 4 rectified pentachora join at each tetrahedral prismatic vertex. As the name suggests, it is the rectification of the hexateron.

It is also a convex segmentoteron, as a pentachoron atop rectified pentachoron.

Vertex coordinates
The vertices of a rectified hexateron of edge length 1 are given by:


 * (–2$\sqrt{6}$/15, –$\sqrt{15}$/5, 0, 0, 0),
 * (–2$\sqrt{3}$/15, $\sqrt{15}$/20, –$\sqrt{10}$/4, 0, 0),
 * (–2$\sqrt{15}$/15, $\sqrt{10}$/20, $\sqrt{6}$/12, –$\sqrt{15}$/3, 0),
 * (–2$\sqrt{10}$/15, –3$\sqrt{6}$/20, $\sqrt{3}$/12, $\sqrt{15}$/6, ±1/2),
 * ($\sqrt{10}$/15, –3$\sqrt{6}$/20, –$\sqrt{3}$/4, 0, 0),
 * ($\sqrt{15}$/15, –3$\sqrt{10}$/20, $\sqrt{6}$/12, –$\sqrt{15}$/3, 0),
 * ($\sqrt{10}$/15, –3$\sqrt{6}$/20, $\sqrt{3}$/12, $\sqrt{15}$/6, ±1/2),
 * ($\sqrt{10}$/15, $\sqrt{6}$/10, $\sqrt{3}$/6, $\sqrt{15}$/3, 0),
 * ($\sqrt{10}$/15, $\sqrt{6}$/10, –$\sqrt{3}$/6, –$\sqrt{15}$/3, 0),
 * ($\sqrt{10}$/15, $\sqrt{6}$/10, $\sqrt{3}$/6, –$\sqrt{15}$/6, ±1/2),
 * ($\sqrt{10}$/15, $\sqrt{6}$/10, –$\sqrt{3}$/6, $\sqrt{15}$/6, ±1/2).

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


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

Representations
A rectified hexateron has the following Coxeter diagrams:


 * o3x3o3o3o (full symmetry)
 * xo3ox3oo3oo&#x (A4 axial, pentachoron atop rectified pentachoron)
 * oxo oxo3oox3ooo&#xt (A3×A1 symmetry, vertex-first)
 * x(oo)x3o(ox)o3o(oo)o&#xt (A3 symmetry, tetrahedron-first)
 * oooo3ooxo3oxox&#xr (A3 symmetry)
 * xoo3oxo oxo3oox&#xt (A2×A2 axial, triangle-first)
 * oxoox ooxoo3oxoxo&#xr (A2×A1 symmetry)