# Stericantellated 6-simplex

Stericantellated 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymCral
Coxeter diagramx3o3x3o3x3o ()
Elements
Peta35 triangular-octahedral duoprisms, 35 triangular-cuboctahedral duoprisms, 21 small rhombated pentachoric prisms, 7 small birhombidodecatera, 7 cellirhombidodecatera
Tera140+140+140 triangular duoprisms, 210 triangular-square duoprisms, 105 octahedral prisms, 105+105 cuboctahedral prisms, 21+42+42 small rhombated pentachora
Cells210+420+420+420+420+420 triangular prisms, 105+105 octahedra, 630 cubes, 105+210 cuboctahedra
Faces210+420+420+420+420 triangles, 630+1260+1260 squares
Edges630+1260+1260
Vertices630
Vertex figureIrregular edge-diminished tetrahedral antifastegium, edge lengths 1 (base squares and top edge) and 2 (lacing edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {182}}{7}}\approx 1.92725}$
Hypervolume${\displaystyle {\frac {77543{\sqrt {7}}}{2880}}\approx 71.23594}$
Dipetal anglesTroct–triddip–traco: ${\displaystyle \arccos \left(-{\frac {3}{4}}\right)\approx 138.59038^{\circ }}$
Srippip–cope–traco: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {30}}}{15}}\right)\approx 136.91128^{\circ }}$
Srippip–ope–troct: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {30}}}{15}}\right)\approx 136.91128^{\circ }}$
Sibrid–srip–srippip: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{6}}\right)\approx 130.20297^{\circ }}$
Srippip–tisdip–traco: ${\displaystyle \arccos \left(-{\frac {\sqrt {30}}{10}}\right)\approx 123.21091^{\circ }}$
Sibrid–triddip–traco: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{4}}\right)\approx 110.70481^{\circ }}$
Card–triddip–troct: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{4}}\right)\approx 110.70481^{\circ }}$
Card–cope–srippip: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{15}}\right)\approx 104.96322^{\circ }}$
Central density1
Number of external pieces105
Level of complexity185
Related polytopes
ArmyCral
RegimentCral
ConjugateNone
Abstract & topological properties
Flag count932400
Euler characteristic0
OrientableYes
Properties
SymmetryA6, order 5040
ConvexYes
NatureTame

The stericantellated 6-simplex, also called the stericantellated heptapeton, cellirhombated heptapeton, or cral, is a convex uniform 6-polytope. It consists of 7 stericantellated 5-simplices, 7 bicantellated 5-simplices, 21 small rhombated pentachoric prisms, 21 triangular-octahedral duoprisms, and 35 triangular-cuboctahedral duoprisms. 2 stericantellated 5-simplices, 1 bicantellated 5-simplex, 2 small rhombated pentachoric prisms, 1 triangular-octahedral duoprism, and 2 triangular-cuboctahedral duoprisms join at each vertex. As the name suggests, it is the stericantellation of the 6-simplex.

## Vertex coordinates

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

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

## Related polytopes

The regiment of the cellirhombated heptapeton contains 75 uniform members.