# Stericated 6-simplex

Stericated 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymScal
Coxeter diagramx3o3o3o3x3o ()
Elements
Peta35 triangular-tetrahedral duoprisms
35 triangular-octahedral duoprisms
7 rectified hexatera
21 rectified pentachoric prisms
7 small cellidodecatera
Tera21+42 pentachora
105+105 tetrahedral prisms
140+140 triangular duoprisms
42 rectified pentachora
105 octahedral prisms
Cells105+210 tetrahedra
210+420+420 triangular prisms
105 octahedra
Faces140+210+420 triangles, 630 squares
Edges210+420
Vertices105
Vertex figureTetrahedral antifastegium, edge lengths 1 (base tetrahedral prism and top tetrahedron) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {70}}{7}}\approx 1.19523}$
Hypervolume${\displaystyle {\frac {1151{\sqrt {7}}}{1152}}\approx 2.64345}$
Dipetal anglesTratet–triddip–troct: ${\displaystyle \arccos \left(-{\frac {3}{4}}\right)\approx 138.59038^{\circ }}$
Rappip–ope–troct: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {30}}}{15}}\right)\approx 136.91128^{\circ }}$
Rix–rap–rappip: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{6}}\right)\approx 130.20297^{\circ }}$
Scad–tepe–tratet: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{3}}\right)\approx 118.12551^{\circ }}$
Scad–triddip–troct: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{4}}\right)\approx 110.70481^{\circ }}$
Scad–tepe–rappip: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{15}}\right)\approx 104.96322^{\circ }}$
Scad–pen–rix: ${\displaystyle \arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }}$
Scad–pen–scad: ${\displaystyle \arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }}$
Central density1
Number of external pieces105
Level of complexity65
Related polytopes
ArmyScal
RegimentScal
ConjugateNone
Abstract & topological properties
Flag count327600
Euler characteristic0
OrientableYes
Properties
SymmetryA6, order 5040
ConvexYes
NatureTame

The stericated 6-simplex (also called the stericated heptapeton, small cellated heptapeton, or scal) is a convex uniform 6-polytope. It consists of 7 rectified hexatera, 7 small cellidodecatera, 21 rectified pentachoric prisms, 35 triangular-tetrahedral duoprisms, and 35 triangular-octahedral duoprisms as facets. 1 rectified hexateron, 2 small cellidodecatera, 4 rectified pentachoric prisms, 4 triangular-tetrahedral duoprisms, and 6 triangular-octahedral duoprisms join at each vertex. As the name suggests, it is the sterication of the 6-simplex.

## Vertex coordinates

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

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

## Representations

A stericated 6-simplex has the following Coxeter diagrams:

• x3o3o3o3x3o () (full symmetry)
• oox3xxo3ooo3ooo3oxx&#xt (A5 axial, rectified hexateron-first)

## Related polytopes

The stericated 6-simplex is the colonel of a regiment with 125 uniform members plus 11 fissaries.