Heptapeton

Heptapeton
(OFF file)
Rank	6
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Hop
Coxeter diagram	x3o3o3o3o3o ()
Schläfli symbol	{3,3,3,3,3}
Tapertopic notation	15
Elements
Peta	7 hexatera
Tera	21 pentachora
Cells	35 tetrahedra
Faces	35 triangles
Edges	21
Vertices	7
Vertex figure	Hexateron, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt{21}}{7} \approx 0.65465$
Edge radius	$\frac{\sqrt{35}}{14} \approx 0.42258$
Face radius	$\frac{\sqrt{42}}{21} \approx 0.30861$
Cell radius	$\frac{\sqrt{42}}{28} \approx 0.23146$
Teron radius	$\frac{\sqrt{35}}{35} \approx 0.16903$
Inradius	$\frac{\sqrt{21}}{42} \approx 0.10911$
Hypervolume	$\frac{\sqrt7}{5760} \approx 0.00045933$
Dipetal angle	$\arccos\left(\frac16\right) \approx 80.40593^\circ$
Heights	Point atop hix: $\frac{\sqrt{21}}{6} \approx 0.76376$
	Dyad atop perp pen: $\frac{\sqrt{35}}{10} \approx 0.59161$
	Trig atop perp tet: $\frac{\sqrt{42}}{12} \approx 0.54006$
Central density	1
Number of pieces	7
Level of complexity	1
Related polytopes
Army	Hop
Regiment	Hop
Dual	Heptapeton
Conjugate	None
Abstract properties
Flag count	5040
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	A6, order 5040
Convex	Yes
Nature	Tame

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.

Gallery

• 

  A₅ orthographic projection

• 

  A₄

• 

  A₃

• 

  A₂

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)

External links

• Bowers, Jonathan. "Category 1: Primary Polypeta" (#1).

• Klitzing, Richard. "hop".

• Wikipedia Contributors. "6-simplex".
• Hi.gher.Space Wiki Contributors. "Pyropeton".