Truncated 6-simplex

Truncated 6-simplex
(OFF file)
Rank	6
Type	Uniform
Notation
Bowers style acronym	Til
Coxeter diagram	x3x3o3o3o3o ()
Elements
Peta	7 hexatera, 7 truncated hexatera
Tera	42 pentachora, 21 truncated pentachora
Cells	105 tetrahedra, 35 truncated tetrahedra
Faces	140 triangles, 35 hexagons
Edges	21+105
Vertices	42
Vertex figure	Pentachoric pyramid, edge lengths 1 (base) and √3 (sides)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {91}}{7}}\approx 1.36277$
Hypervolume	${\frac {361{\sqrt {7}}}{2880}}\approx 0.33164$
Dipetal angles	Hix-pen-tix: $\arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }$
	Tix-tip-tix: $\arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }$
Central density	1
Number of external pieces	14
Level of complexity	6
Related polytopes
Army	Til
Regiment	Til
Dual	Hexakis heptapeton
Conjugate	None
Abstract & topological properties
Flag count	30240
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A6, order 5040
Convex	Yes
Nature	Tame

The truncated 6-simplex (also called the truncated heptapeton, or til) is a convex uniform 6-polytope. It consists of 7 hexatera and 7 truncated hexatera as facets. 1 hexateron and 5 truncated hexatera join at each vertex. As the name suggests, it is the truncation of the 6-simplex.

Vertex coordinates

The vertices of a truncated heptapeton of edge length 1 can be given in seven dimensions as all permutations of:

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

Representations

A truncated heptapeton has the following Coxeter diagrams:

• x3x3o3o3o3o () (full symmetry)
• xux3oox3ooo3ooo3ooo&#xt (A₅ axial, hexateron-first)

Gallery

• 

  A₅ orthographic projection

• 

  A₄

• 

  A₃

• 

  A₂

External links

• Klitzing, Richard. "til".
• Wikipedia contributors. "Truncated 6-simplex".