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.

A octaexon of edge length 1/2 can be inscribed in the hepteract.

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