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:


 * $$\left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12}\right),$$
 * $$\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12}\right),$$
 * $$\left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12}\right),$$
 * $$\left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,\frac{\sqrt{15}}{6},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{21}}{7},\,-\frac{\sqrt7}{28},\,-\frac{1}{12}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt7}{4},\,-\frac{1}{12}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac23\right),$$

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


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

Representations
A regular enneazetton has the following Coxeter diagrams:


 * x3o3o3o3o3o3o3o (full symmetry)
 * ox3oo3oo3oo3oo3oo3oo&#x (A7 axial, octaexal pyramid)