Dodecateron

The dodecateron, or dot, also called the birectified 5-simplex, is a convex noble uniform polyteron. It consists of 12 rectified pentachora as facets. 6 rectified pentachora join at each triangular duoprismatic vertex. As the name suggests, it is the birectification of the hexateron. It is the medial stage in the series of truncations between the regular hexateron and its dual.

It can be seen as a segmentoteron as rectified pentachoron atop inverted rectified pentachoron, that is a kind of rectified pentachoric alterprism.

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


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

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


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

Representations
A dodecateron has the following Coxeter diagrams:


 * o3o3x3o3o (full symmetry)
 * oo3xo3ox3oo&#x (A4 axial, rectified pentachoric alterprism)
 * oxoo3ooxo oxoo3ooxo&#xt (A2×A2 axial, vertex-first)
 * oxo xoo3oxooox&#xt (A3×A1 axial, tetrahedron-first)
 * oxoo3xoxo3ooox&#xt (A3 symmetry)

Related polytopes
The dodecateron is the colonel of a five-member regiment that also includes the biprismatododecateron, biprismatointercepted dodecateron, cellibiprismatohexateron, and cellidishexateron.