Octaexon

The octaexon, or oca, also commonly called the 7-simplex, is the simplest possible non-degenerate polyexon. The full symmetry version has 8 regular heptapeta as facets, joining 3 to a pentachoron peak and 7 to a vertex, and is one of the 3 regular polyexa. It is the 7-dimensional simplex. It is also a pyramid based on the heptapeton.

A regular octaexon of edge length 2 can be inscribed in the unit hepteract. The next largest simplex that can be inscribed in a hypercube is the dodecadakon.

Vertex coordinates
The vertices of a regular octaexon of edge length 1, centered at the origin, are given by:


 * $$\left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),$$
 * $$\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),$$
 * $$\left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),$$
 * $$\left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,\frac{\sqrt{15}}{6},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{21}}{7},\,-\frac{\sqrt7}{28}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt7}{4}\right).$$

Much simpler sets of coordinates can be found by inscribing the octaexon into the hepteract. One such set is given by:


 * $$\left(\frac14,\,\frac12,\,\frac14,\,\frac14,\,\frac14,\,\frac14,\,\frac14\right),$$
 * $$\left(\frac14,\,\frac14,\,\frac14,\,-\frac14,\,-\frac14,\,-\frac14,\,-1/4\right),$$
 * $$\left(\frac14,\,-\frac14,\,-\frac14,\,-\frac14,\,-\frac14,\,\frac14,\,\frac14\right),$$
 * $$\left(\frac14,\,-\frac14,\,-\frac14,\,\frac14,\,\frac14,\,-\frac14,\,-\frac14\right),$$
 * $$\left(-\frac14,\,\frac14,\,-\frac14,\,\frac14,\,-\frac14,\,\frac14,\,-\frac14\right),$$
 * $$\left(-\frac14,\,\\frac14,\,-\frac14,\,-\frac14,\,\frac14,\,-\frac14,\,\frac14\right),$$
 * $$\left(-\frac14,\,-\frac14,\,\frac14,\,\frac14,\,-\frac14,\,-\frac14,\,\frac14\right),$$
 * $$\left(-\frac14,\,-\frac14,\,\frac14,\,\frac14,\,\frac14,\,\frac14,\,-\frac14\right).$$

Even simpler coordinates can be given in eight dimensions, as all permutations of:


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

Representations
An octaexon has the following Coxeter diagrams:


 * x3o3o3o3o3o3o (full symmetry)
 * ox3oo3oo3oo3oo3oo&#x (A6 axial, heptapetal pyramid)
 * xo ox3oo3oo3oo3oo&#x (A5×A1 axial, hexateral scalene)
 * xo3oo ox3oo3oo&#x (A4×A2 axial, pentachoric tettene)
 * xo3oo3oo ox3oo3oo&#x (A3×A3 axial, tetrahedral disphenoid)
 * oxo3ooo oox3ooo3ooo&#x (A3×A2 symmetry, tetrahedral tettene pyramid)
 * oxo xoo3ooo ooxooo&#x (A2×A2×A1 symmetry, trigonal disphenoid scalene)
 * xoo oox oxo3ooo3ooo&#x (A3×A1×A1 symmetry, tetrahedral scalenic scalene)
 * oxoo3oooo ooxo3oooo&#x (A2×A2 symmetry, trigonal pyramidal disphenoid)