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:


 * $$\left(-\frac{2\sqrt{15}}{15},\,-\frac{\sqrt{10}}{5},\,0,\,0,\,0\right),$$
 * $$\left(-\frac{2\sqrt{15}}{15},\,\frac{\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,0,\,0\right),$$
 * $$\left(-\frac{2\sqrt{15}}{15},\,\frac{\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right),$$
 * $$\left(-\frac{2\sqrt{15}}{15},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),$$
 * $$\left(\frac{\sqrt{15}}{15},\,-\frac{3\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,0,\,0\right),$$
 * $$\left(\frac{\sqrt{15}}{15},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right),$$
 * $$\left(\frac{\sqrt{15}}{15},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),$$
 * $$\left(\frac{\sqrt{15}}{15},\,\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,\frac{\sqrt3}{3},\,0\right),$$
 * $$\left(\frac{\sqrt{15}}{15},\,\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,-\frac{\sqrt3}{3},\,0\right),$$
 * $$\left(\frac{\sqrt{15}}{15},\,\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{\sqrt3}{6},\,±\frac12\right),$$
 * $$\left(\frac{\sqrt{15}}{15},\,\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,\frac{\sqrt3}{6},\,±\frac12\right).$$

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


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

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)

Related polytopes
A rectified hexateron can be diminished by removing tetrahedral prismatic pyramids. 3 such pyramids from non-adjacent vertices can be removed to form a scaliform tridiminished rectified hexateron.

The rectified hexateron is the colonel of a 7-member regiment. Its other members include the cellibiprismatointercepted hexateron, facetorectified hexateron, cellihexateron, biprismatodishexateron, cellintercepted dishexateron, and spinobiprismatohexateron.