# Pentitruncated 6-simplex

Pentitruncated 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymTocal
Coxeter diagramx3x3o3o3o3x ()
Elements
Peta21 pentachoric prisms
35 hexagonal-tetrahedral duoprisms
7 truncated 5-simplices
7 stericated 5-simplices
35 triangular-truncated tetrahedral duoprisms
21 truncated pentachoric prisms
Tera42+42 pentachora
105+105+105 tetrahedral prisms
140 triangular duoprisms
140 triangular-hexagonal duoprisms
42 truncated pentachora
105 truncated tetrahedral prisms
Cells210+210 tetrahedra
210+420+420 triangular prisms
210 hexagonal prisms
105 truncated tetrahedra
Faces420+420 triangles
210+630 squares
140 hexagons
Edges105+420+420
Vertices210
Vertex figureSkew tetrahedral antiprismatic pyramid, edge lengths 1 (base tetrahedra), 2 (sides of antiprism and lacings to one base), and 3 (lacings to other base)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {133}}{7}}\approx 1.64751}$
Hypervolume${\displaystyle {\frac {18701{\sqrt {7}}}{2880}}\approx 17.17993}$
Dipetal anglesTratut–thiddip–hatet: ${\displaystyle \arccos \left(-{\frac {3}{4}}\right)\approx 138.59038^{\circ }}$
Penp–tepe–hatet: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {30}}}{15}}\right)\approx 136.91128^{\circ }}$
Tippip–tuttip–tratut: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {30}}}{15}}\right)\approx 136.91128^{\circ }}$
Scad–pen–penp: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{6}}\right)\approx 130.20297^{\circ }}$
Tix–tip–tippip: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{6}}\right)\approx 130.20297^{\circ }}$
Scad–tepe–hatet: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{3}}\right)\approx 118.12551^{\circ }}$
Scad–triddip–tratut: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{4}}\right)\approx 110.70481^{\circ }}$
Scad–tepe–tippip: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{15}}\right)\approx 104.96322^{\circ }}$
Central density1
Number of external pieces126
Level of complexity96
Related polytopes
ArmyTocal
RegimentTocal
ConjugateNone
Abstract & topological properties
Flag count483840
Euler characteristic0
OrientableYes
Properties
SymmetryA6, order 5040
ConvexYes
NatureTame

The pentitruncated 6-simplex, also called the pentitruncated heptapeton, tericellated heptapeton, or tocal, is a convex uniform 6-polytope. It consists of 7 truncated 5-simplices, 7 stericated 5-simplices, 21 pentachoric prisms, 21 truncated pentachoric prisms, 35 hexagonal-tetrahedral duoprisms, and 35 triangular-truncated tetrahedral duoprisms. 1 truncated 5-simplex, 1 stericated 5-simplex, 1 pentachoric prism, 4 truncated pentachoric prisms, 4 hexagonal-tetrahedral duoprisms, and 6 triangular-truncated tetrahedral duoprisms join at each vertex. As the name suggests, it is the pentitruncation of the 6-simplex.

## Vertex coordinates

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

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

## Related polytopes

The pentitruncated 6-simplex is the colonel of a regiment that includes 15 uniform members.