# Truncated 6-simplex

Truncated 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymTil
Coxeter diagramx3x3o3o3o3o ()
Elements
Peta7 hexatera, 7 truncated hexatera
Tera42 pentachora, 21 truncated pentachora
Cells105 tetrahedra, 35 truncated tetrahedra
Faces140 triangles, 35 hexagons
Edges21+105
Vertices42
Vertex figurePentachoric pyramid, edge lengths 1 (base) and 3 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {91}}{7}}\approx 1.36277}$
Hypervolume${\displaystyle {\frac {361{\sqrt {7}}}{2880}}\approx 0.33164}$
Dipetal anglesHix-pen-tix: ${\displaystyle \arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }}$
Tix-tip-tix: ${\displaystyle \arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }}$
Central density1
Number of external pieces14
Level of complexity6
Related polytopes
ArmyTil
RegimentTil
DualHexakis heptapeton
ConjugateNone
Abstract & topological properties
Flag count30240
Euler characteristic0
OrientableYes
Properties
SymmetryA6, order 5040
ConvexYes
NatureTame

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:

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

## Representations

A truncated heptapeton has the following Coxeter diagrams:

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