# Runcitruncated 6-simplex

Runcitruncated 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymPatal
Coxeter diagramx3x3o3x3o3o ()
Elements
Peta21 rectified pentachoric prisms
35 hexagonal-tetrahedral duoprisms
7 small rhombated hexatera
7 prismatotruncated hexatera
Tera105+105 tetrahedral prisms
42 rectified pentachora
105 octahedral prisms
140 triangular-hexagonal duoprisms
42 small rhombated pentachora
21 prismatorhombated pentachora
Cells210 tetrahedra
210+420+420 triangular prisms
210 octahedra
210 hexagonal prisms
35 truncated tetrahedra
105 cuboctahedra
Faces840+420+140 triangles
630+630 squares
140 hexagons
Edges1260+420+210
Vertices420
Vertex figureTriangular pyramidal prismatic pyramid, edge lengths 1 (base triangular prism and top edge of base), 2 (sides of base and lacings to prism), and 3 (lacings to top edge)
Measures (edge length 1)
Hypervolume${\displaystyle {\frac {41867{\sqrt {7}}}{2880}}\approx 38.46169}$
Dipetal anglesRappip–tepe–hatet: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {30}}}{15}}\right)\approx 136.91128^{\circ }}$
Sarx–rap–rappip: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{6}}\right)\approx 130.20297^{\circ }}$
Sarx–tepe–hatet: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{3}}\right)\approx 118.12551^{\circ }}$
Pattix–thiddip–hatet: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{4}}\right)\approx 110.70481^{\circ }}$
Pattix–ope–rappip: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{15}}\right)\approx 104.96322^{\circ }}$
Pattix–srip–sarx: ${\displaystyle \arccos \left(-{\frac {1}{6}}\right)\approx 101.53696^{\circ }}$
Pattix–prip–pattix: ${\displaystyle \arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }}$
Central density1
Number of external pieces70
Level of complexity96
Related polytopes
ArmyPatal
RegimentPatal
ConjugateNone
Abstract & topological properties
Flag count483840
Euler characteristic0
OrientableYes
Properties
SymmetryA6, order 5040
ConvexYes
NatureTame

The runcitruncated 6-simplex (also called the runcitruncated heptapeton, prismatotruncated heptapeton or patal) is a convex uniform 6-polytope. It consists of 7 prismatotruncated hexatera, 7 small rhombated hexatera, 21 rectified pentachoric prisms, and 35 hexagonal-tetrahedral duoprisms as facets. 3 prismatotruncated hexatera, 1 small rhombated hexateron, 1 rectified pentachoric prism, and 2 hexagonal-tetrahedral duoprisms join at each vertex. As the name suggests, it is the runcitruncation of the 6-simplex.

## Vertex coordinates

The vertices of a prismatotruncated heptapeton of edge length 1 can be given in 7 dimensions as all permutations of:

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

## Related polytopes

The regiment of the prismatotruncated heptapeton contains 15 uniform members.