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.

An octaexon of edge length 1/2 can be inscribed in the 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:


 * (±1/2, –$\sqrt{7}$/6, –$\sqrt{7}$/12, –$\sqrt{7}$/20, –$\sqrt{3}$/30, –$\sqrt{15}$/42, –$\sqrt{3}$/28),
 * (0, $\sqrt{6}$/3, –$\sqrt{10}$/12, –$\sqrt{15}$/20, –$\sqrt{21}$/30, –$\sqrt{7}$/42, –$\sqrt{3}$/28),
 * (0, 0, $\sqrt{6}$/4, –$\sqrt{10}$/20, –$\sqrt{15}$/30, –$\sqrt{21}$/42, –$\sqrt{7}$/28),
 * (0, 0, 0, $\sqrt{6}$/5, –$\sqrt{10}$/30, –$\sqrt{15}$/42, –$\sqrt{21}$/28),
 * (0, 0, 0, 0, $\sqrt{7}$/6, –$\sqrt{10}$/42, –$\sqrt{15}$/28),
 * (0, 0, 0, 0, 0, $\sqrt{21}$/7, –$\sqrt{7}$/28),
 * (0, 0, 0, 0, 0, 0, $\sqrt{15}$/4).

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


 * (1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4),
 * (1/4, 1/4, 1/4, –1/4, –1/4, –1/4, –1/4),
 * (1/4, –1/4, –1/4, –1/4, –1/4, 1/4, 1/4),
 * (1/4, –1/4, –1/4, 1/4, 1/4, –1/4, –1/4),
 * (–1/4, 1/4, –1/4, 1/4, –1/4, 1/4, –1/4),
 * (–1/4, 1/4, –1/4, –1/4, 1/4, –1/4, 1/4),
 * (–1/4, –1/4, 1/4, 1/4, –1/4, –1/4, 1/4),
 * (–1/4, –1/4, 1/4, –1/4, 1/4, 1/4, –1/4).

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


 * ($\sqrt{21}$/2, 0, 0, 0, 0, 0, 0, 0).

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)