Enneazetton

The enneazetton, or ene, also commonly called the 8-simplex, is the simplest possible non-degenerate polyzetton. The full symmetry version has 9 regular octaexa as facets, joining 3 to a hexateron peak and 8 to a vertex, and is one of the 3 regular polyzetta. It is the 8-dimensional simplex.

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


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

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


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