6-simplex

6-simplex
(OFF file)
Rank	6
Type	Regular
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
Petrie polygons	360 heptagonal-heptagrammic-great heptagrammic coils
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	${\frac {\sqrt {7}}{5760}}\approx 0.00045933$
Dipetal angle	$\arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }$
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 external pieces	7
Level of complexity	1
Related polytopes
Army	Hop
Regiment	Hop
Dual	Heptapeton
Conjugate	None
Abstract & topological properties
Flag count	5040
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A6, order 5040
Convex	Yes
Nature	Tame

The 6-simplex (also called the heptapeton or hop) is the simplest possible non-degenerate 6-polytope. The full symmetry version has 7 regular hexatera as facets, joining 3 to a tetrahedron peak and 6 to a vertex, and is a regular 6-polytope. It is the 6-dimensional simplex. It is one of two uniform self-dual 6-polytopes, 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 6-segmentotope 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:

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

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

• $\left({\frac {\sqrt {2}}{2}},\,0,\,0,\,0,\,0,\,0,\,0\right)$.

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".