Heptapeton

{{Infobox polytope 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.
 * type=Regular
 * dim = 6
 * img = 6-simplex_t0.svg
 * off = Heptapeton.off
 * obsa = Hop
 * petons = 7 hexatera
 * terons = 21 pentachora
 * cells = 35 tetrahedra
 * faces = 35 triangles
 * edges = 21
 * vertices = 7
 * verf = Hexateron, edge length 1
 * schlafli = {3,3,3,3,3}
 * coxeter = x3o3o3o3o3o
 * army=Hop
 * reg=Hop
 * symmetry = A6, order 5040
 * circum = $$\frac{\sqrt{21}}{7} ≈ 0.65465$$
 * rad1 = $$\frac{\sqrt{35}}{14] ≈ 0.42258$$
 * rad2 = $$\frac{\sqrt{42}}{21} ≈ 0.30861$$
 * rad3 = $$\frac{\sqrt{42}}{28} ≈ 0.23146$$
 * rad4 = $$\frac{\sqrt{35}}{35} ≈ 0.16903$$
 * inrad = $$\frac{\sqrt{21}}{42} ≈ 0.10911$$
 * height = Point atop hix: $$\frac{\sqrt{21}}{6} ≈ 0.76376$$
 * height2 = Dyad atop perp pen: $$\frac{\sqrt{35}}{10} ≈ 0.59161$$
 * height3 = Trig atop perp tet: $$\frac{\sqrt{42}}{12} ≈ 0.54006$$
 * hypervolume = $$\frac{\sqrt7}{5760} ≈ 0.00045933$$
 * dip= $$\arccos\left(\frac16\right) ≈ 80.40593°$$
 * pieces = 7
 * loc = 1
 * taper = 1{{sup|5}}
 * dual=Heptapeton
 * conjugate=Heptapeton
 * conv = Yes
 * orientable=Yes
 * nat=Tame}}

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 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)