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.

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:


 * (±1/2, –$\sqrt{21}$/6, –$\sqrt{21}$/12, –$\sqrt{21}$/20, –$\sqrt{35}$/30, –$\sqrt{42}$/42),
 * (0, $\sqrt{7}$/3, –$\sqrt{3}$/12, –$\sqrt{6}$/20, –$\sqrt{10}$/30, –$\sqrt{15}$/42),
 * (0, 0, $\sqrt{21}$/4, –$\sqrt{3}$/20, –$\sqrt{6}$/30, –$\sqrt{10}$/42),
 * (0, 0, 0, $\sqrt{15}$/5, –$\sqrt{21}$/30, –$\sqrt{6}$/42),
 * (0, 0, 0, 0, $\sqrt{10}$/6, –$\sqrt{15}$/42),
 * (0, 0, 0, 0, 0, $\sqrt{21}$/7).

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


 * ($\sqrt{10}$/2, 0, 0, 0, 0, 0, 0).

Representations
A regular heptapeton has the following Coxeter diagrams:


 * x3o3o3o3o3o (full symmetry)
 * ox3oo3oo3oo3oo&#x (A5 axial, hexateric pyarmid)
 * 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 (A2×A1×A1 axial, triangular scalenic scalene)