Runcicantellated 6-simplex

Runcicantellated 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymPril
Coxeter diagramx3o3x3x3o3o ()
Elements
Peta35 triangular-tetrahedral duoprisms
21 truncated pentachoric prisms
7 bitruncated 5-simplices
7 runcicantellated 5-simplices
Tera105 tetrahedral prisms
140 triangular duoprisms
42 truncated pentachora
105 truncated tetrahedral prisms
42 decachora
21 prismatorhombated pentachora
Cells105 tetrahedra
210+420 triangular prisms
210 hexagonal prisms
105+210 truncated tetrahedra
35 cuboctahedra
Faces140+140+420 triangles, 210+630 squares, 420 hexagons
Edges420+420+630
Vertices420
Vertex figureSkewed rectangular tettene, base edge lengths 1 and 2, top triangle edge length 1, lacing lengths 2 and 3
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {217}}{7}}\approx 2.10442}$
Hypervolume${\displaystyle {\frac {3433{\sqrt {7}}}{144}}\approx 63.07545}$
Dipetal anglesTippip–tepe–tratet: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {30}}}{15}}\right)\approx 136.91128^{\circ }}$
Bittix–tip–tippip: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{6}}\right)\approx 130.20297^{\circ }}$
Pirx–triddip–tratet: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{4}}\right)\approx 110.70481^{\circ }}$
Pirx–tuttip–tippip: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{15}}\right)\approx 104.96322^{\circ }}$
Pirx–deca–bittix: ${\displaystyle \arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }}$
Pirx–prip–pirx: ${\displaystyle \arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }}$
Central density1
Number of external pieces70
Level of complexity80
Related polytopes
ArmyPril
RegimentPril
ConjugateNone
Abstract & topological properties
Flag count403200
Euler characteristic0
OrientableYes
Properties
SymmetryA6, order 5040
ConvexYes
NatureTame

The runcicantellated 6-simplex, also called the runcicantellated heptapeton, prismatorhombated heptapeton, or pril, is a convex uniform 6-polytope. It consists of 7 runcicantellated 5-simplices, 7 bitruncated 5-simplices, 21 truncated pentachoric prisms, and 35 triangular-tetrahedral duoprisms. 3 runcicantellated 5-simplices, 1 bitruncated 5-simplex, 2 truncated pentachoric prisms, and 1 triangular-tetrahedral duoprism join at each vertex. As the name suggests, it is the runcicantellation of the 6-simplex.

Vertex coordinates

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

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

Related polytopes

The regiment of the runcicantellated 6-simplex contains two other uniform members.