# Bitruncated 6-simplex

Bitruncated 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymBatal
Coxeter diagramo3x3x3o3o3o ()
Elements
Peta7 truncated hexatera, 7 bitruncated hexatera
Tera21 pentachora, 42 truncated pentachora, 21 decachora
Cells105 tetrahedra, 35+105 truncated tetrahedra
Faces35+210 triangles, 140 hexagons
Edges105+210
Vertices105
Vertex figureTetrahedral scalene , edge lengths 1 (base tetrahedron and top edge) and 3 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {133}}{7}}\approx 1.64751}$
Hypervolume${\displaystyle {\frac {10543{\sqrt {7}}}{5760}}\approx 4.84274}$
Dipetal anglesTix-tip-bittix: ${\displaystyle \arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }}$
Bittix-deca-bittix: ${\displaystyle \arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }}$
Tix-pen-tix: ${\displaystyle \arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }}$
Central density1
Number of external pieces14
Level of complexity15
Related polytopes
ArmyBatal
RegimentBatal
ConjugateNone
Abstract & topological properties
Flag count75600
Euler characteristic0
OrientableYes
Properties
SymmetryA6, order 5040
ConvexYes
NatureTame

The bitruncated 6-simplex (also called the bitruncated heptapeton or batal) is a convex uniform 6-polytope. It consists of 7 truncated hexatera and 7 bitruncated hexatera as facets. 2 truncated hexatera and 4 bitruncated hexatera join at each vertex. As the name suggests, it is the bitruncation of the 6-simplex.

## Vertex coordinates

The vertices of a bitruncated 6-simplex of edge length 1 can be given in seven dimensions as all permutations of:

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

## Representations

A bitruncated 6-simplex has the following Coxeter diagrams:

• o3x3x3o3o3o () (full symmetry)
• xoo3xux3oox3ooo3ooo&#xt (A5 axial, truncated hexateron-first)