Heptapeton

The heptapeton, or hop, also commonly called the 6-simplex, is the simplest possible non-degenerate polypeton. The full symmetry version has 7 regular hexatera as facets, joining 3 to a tetrahedron peak and 6 to a vertex, and is one of the 3 regular polypeta. It is the 6-dimensional simplex. It is one of two uniform self-dual polypeta, the other being the great icosiheptapeton. It is also the 7-2-3 step prism and gyropeton, making it the simplest 6D step prism.

It can be obtained as a segmentopeton in three ways: as a hexateric pyramid, dyad atop perpendicular pentachoron, or triangle atop perpendicular tetrahedron.

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


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

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


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

Representations
A regular heptapeton has the following Coxeter diagrams:


 * x3o3o3o3o3o (full symmetry)
 * ox3oo3oo3oo3oo&#x (A5 axial, hexateric pyramid)
 * xo ox3oo3oo3oo&#x (A4×A1 axial, pentachric scalene)
 * xo3oo ox3oo3oo&#x (A3×A2 axial, tetrahedral tettene)
 * oxo3ooo3ooo3ooo&#x (A4 only, pentachoric pyramidal pyramid)
 * oxo oox3ooo3ooo&#xt (A3×A1 axial, tetrahedral scalenic pyramid)
 * oxo3ooo oox3ooo&#x (A2×A2 axial, triangular disphenoidal pyramid)
 * xoo oxo oox3ooo&#x (A1×A2×A1 axial, triangular scalenic scalene)