# Bicantitruncated 6-simplex

Bicantitruncated 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymGabril
Coxeter diagramo3x3x3x3o3o ()
Elements
Peta35 triangular-tetrahedral duoprisms
7 cantitruncated 5-simplices
7 bicantitruncated 5-simplices
Tera105 tetrahedral prisms
140 triangular duoprisms
21 truncated pentachora
21+42 great rhombated pentachora
Cells105 tetrahedra
210+420 triangular prisms
35+105 truncated tetrahedra
105 truncated octahedra
Faces140+420 triangles
630 squares
140+210 hexagons
Edges210+420+630
Vertices420
Vertex figureTriangular pyramidal scalene, edge lengths 1 (triangle and top dyad), 2 (lacings between triangle and dyad), and 3 (remaining lacings)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {2{\sqrt {70}}}{7}}\approx 2.39046}$
Hypervolume${\displaystyle {\frac {6869{\sqrt {7}}}{180}}\approx 100.96481}$
Dipetal anglesGarx–tepe–tratet: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{3}}\right)\approx 118.12551^{\circ }}$
Gibrid–triddip–tratet: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{4}}\right)\approx 110.70481^{\circ }}$
Gibrid–grip–garx: ${\displaystyle \arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }}$
Gibrid–grip–gibrid: ${\displaystyle \arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }}$
Garx–tip–garx: ${\displaystyle \arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }}$
Central density1
Number of external pieces49
Level of complexity60
Related polytopes
ArmyGabril
RegimentGabril
ConjugateNone
Abstract & topological properties
Flag count302400
Euler characteristic0
OrientableYes
Properties
SymmetryA6, order 5040
ConvexYes
NatureTame

The bicantitruncated 6-simplex, also called the bicantitruncated heptapeton, great birhombated heptapeton, or gabril, is a convex uniform 6-polytope. It consists of 7 bicantitruncated 5-simplices, 7 cantitruncated 5-simplices, and 35 triangular-tetrahedral duoprisms. 3 bicantitruncated 5-simplices, 2 cantitruncated 5-simplices, and 1 triangular-tetrahedral duoprism join at each vertex. As the name suggests, it is the bicantitruncation of the 6-simplex.

## Vertex coordinates

The vertices of a bicantitruncated 6-simplex of edge length 1 can be given in 7 dimensions as all permutations of:

• ${\displaystyle \left({\frac {3{\sqrt {2}}}{2}},\,{\frac {3{\sqrt {2}}}{2}},\,{\sqrt {2}},\,{\frac {\sqrt {2}}{2}},\,0,\,0,\,0\right)}$.