Rectified hexateron

{{Infobox polytope 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.
 * img=5-simplex_t1.svg
 * off=Rectified_hexateron.off
 * type=Uniform
 * dim = 5
 * obsa = Rix
 * terons = 6 pentachora, 6 rectified pentachora
 * cells = 30 tetrahedra, 15 octahedra
 * faces = 20+60 triangles
 * edges = 60
 * vertices = 15
 * verf = Tetrahedral prism, edge length 1
 * coxeter = o3x3o3o3o ({{CDD|node|3|node_1|3|node|3|node|3|node}})
 * army=Rix
 * reg=Rix
 * symmetry = A5, order 720
 * circum = $$\frac{\sqrt6}{3] ≈ 0.81650$$
 * height = $$\frac{\sqrt{15}}{5} ≈ 0.77460$$
 * hypervolume = $$\frac{13\sqrt3}{240} ≈ 0.093819$$
 * dit = Rap–tet–pen: $$\arccos\left(-\frac15\right) ≈ 101.53696°$$
 * dit2 = Rap–oct–rap: $$\arccos\left(\frac15\right) ≈ 78.46304°$$
 * pieces = 12
 * loc = 4
 * dual = Joined hexateron
 * conjugate=Rectified hexateron
 * conv = Yes
 * orientable=Yes
 * nat=Tame}}

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