# Steriruncinated 6-simplex

Steriruncinated 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymCopal
Coxeter diagramx3o3o3x3x3o ()
Elements
Peta35 triangular-tetrahedral duoprisms, 35 triangular-truncated tetrahedral duoprisms, 21 decachoric prisms, 7 bitruncated hexatera, 7 celliprismated hexatera
Tera105 tetrahedral prisms, 140 triangular duoprisms, 140 triangular-hexagonal duoprisms, 42 truncated pentachora, 21 small prismatodecachora, 105+105 truncated tetrahedral prisms, 42 decachora
Cells105+105 tetrahedra, 210+210+210+420 triangular prisms, 420 hexagonal prisms, 105+210 truncated tetrahedra
Faces140+420+420 triangles, 630+630 squares, 420 hexagons
Edges420+630+630
Vertices420
Vertex figureSkewed Triangular antiprismatic scalene, base edge lengths 1 and 2, opposite edge length 1, lacing lengths 2 and 3
Measures (edge length 1)
Circumradius${\displaystyle {\frac {3{\sqrt {21}}}{7}}\approx 1.96396}$
Hypervolume${\displaystyle {\frac {36283{\sqrt {7}}}{1440}}\approx 66.66375}$
Dipetal anglesTratet–triddip–tratut: ${\displaystyle \arccos \left(-{\frac {3}{4}}\right)\approx 138.59038^{\circ }}$
Decap–tuttip–tratut: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {30}}}{15}}\right)\approx 136.91128^{\circ }}$
Bittix–deca–decap: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{6}}\right)\approx 130.20297^{\circ }}$
Cappix–tepe–tratet: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{3}}\right)\approx 118.12551^{\circ }}$
Cappix–thiddip–tratut: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{4}}\right)\approx 110.70481^{\circ }}$
Cappix–tuttip–decap: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{15}}\right)\approx 104.96322^{\circ }}$
Cappix–tip–bittix: ${\displaystyle \arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }}$
Cappix–spid–cappix: ${\displaystyle \arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }}$
Central density1
Number of external pieces105
Level of complexity120
Related polytopes
ArmyCopal
RegimentCopal
ConjugateNone
Abstract & topological properties
Flag count604800
Euler characteristic0
OrientableYes
Properties
SymmetryA6, order 5040
ConvexYes
NatureTame

The steriruncinated 6-simplex, also called the steriruncinated heptapeton, celliprismated heptapeton, or copal, is a convex uniform 6-polytope. It consists of 7 steritruncated 5-simplices, 7 bitruncated 5-simplices, 21 decachoric prisms, 35 triangular-tetrahedral duoprisms, and 35 triangular-truncated tetrahedral duoprisms. 2 steritruncated 5-simplices, 1 bitruncated 5-simplex, 3 decachoric prisms, 1 triangular-tetrahedral duoprism, and 3 triangular-truncated tetrahedral duoprisms join at each vertex. As the name suggests, it is the steriruncination of the 6-simplex.

## Vertex coordinates

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

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

## Related polytopes

The regiment of the celliprismated heptapeton contains 7 uniform members.