# Pentellated 6-simplex

Pentellated 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymStaf
Coxeter diagramx3o3o3o3o3x ()
Elements
Peta14 hexatera, 42 pentachoric prisms, 70 triangular-tetrahedral duoprisms
Tera84 pentachora, 210 tetrahedral prisms, 140 triangular duoprisms
Cells210 tetrahedra, 420 triangular prisms
Faces280 triangles, 210 squares
Edges210
Vertices42
Vertex figurePentachoric antiprism, edge lengths 1 (base) and 2 (sides)
Measures (edge length 1)
Hypervolume${\displaystyle {\frac {77{\sqrt {7}}}{480}}\approx 0.42442}$
Dipetal anglesTratet–triddip–tratet: ${\displaystyle \arccos \left(-{\frac {3}{4}}\right)\approx 138.59038^{\circ }}$
Penp–tepe–tratet: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {30}}}{15}}\right)\approx 136.91128^{\circ }}$
Hix–pen–penp: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{6}}\right)\approx 130.20297^{\circ }}$
Central density1
Number of external pieces126
Level of complexity16
Related polytopes
ArmyStaf
RegimentStaf
ConjugateNone
Abstract & topological properties
Flag count161280
Euler characteristic0
OrientableYes
Properties
SymmetryA6×2, order 10080
ConvexYes
NatureTame

The pentellated 6-simplex (also called the pentellated heptapeton, small teritetradecapeton or staf) is a convex uniform 6-polytope. It consists of 14 hexatera, 42 pentachoric prisms, and 70 triangular-tetrahedral duoprisms as facets. 2 hexatera, 10 pentachoric prisms, and 20 triangular-tetrahedral duoprisms join at each vertex. As the name suggests, it is the pentellation of the 6-simplex. It can be constructed by expanding the facets of the 6-simplex outward and filling in the gaps with new facets.

## Vertex coordinates

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

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

## Representations

A pentellated 6-simplex has the following Coxeter diagrams:

• x3o3o3o3o3x () (full symmetry)
• xxo3ooo3ooo3ooo3oxx&#xt (A5 axial, hexateron-first)
• x(xo)o3o(xo)x x(ox)o3o(oo)o3o(ox)x&#xt (A3×A2 axial, triangular-tetrahedral duoprism first)

## Related polytopes

The pentellated 6-simplex is the colonel of a regiment of 19 members.