# Runcinated 6-simplex

Runcinated 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymSpil
Coxeter diagramx3o3o3x3o3o ()
Elements
Peta35 triangular-tetrahedral duoprisms
21 rectified pentachoric prisms
7 dodecatera
7 small prismated hexatera
Tera105 tetrahedral prisms
140 triangular duoprisms
42+42 rectified pentachora
105 octahedral prisms
21 small prismatodecachora
Cells35+105+105 tetrahedra
210+210+420 triangular prisms
210 octahedra
Faces140+420+420 triangles, 630 squares
Edges210+630
Vertices140
Vertex figureTriangle atop alternate triangular duoprism, edge lengths 1 (base duoprism and top triangle) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {2{\sqrt {21}}}{7}}\approx 1.30931}$
Hypervolume${\displaystyle {\frac {3847{\sqrt {7}}}{2880}}\approx 3.53410}$
Dipetal anglesRappip–tepe–tratet: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {30}}}{15}}\right)\approx 136.91128^{\circ }}$
Dot–rap–rappip: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{6}}\right)\approx 130.20297^{\circ }}$
Spix–triddip–tratet: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{4}}\right)\approx 110.70481^{\circ }}$
Spix–ope–rappip: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{15}}\right)\approx 104.96322^{\circ }}$
Spix–rap–dot: ${\displaystyle \arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }}$
Spix–spid–spix: ${\displaystyle \arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }}$
Central density1
Number of external pieces70
Level of complexity56
Related polytopes
ArmySpil
RegimentSpil
ConjugateNone
Abstract & topological properties
Flag count282240
Euler characteristic0
OrientableYes
Properties
SymmetryA6, order 5040
ConvexYes
NatureTame

The runcinated 6-simplex (also called the small prismated heptapeton or spil) is a convex uniform 6-polytope. It consists of 7 dodecatera, 7 small prismated hexatera, 21 rectified pentachoric prisms, and 35 triangular-tetrahedral duoprisms as facets. 1 dodecateron, 3 small prismated hexatera, 3 rectified pentachoric prisms, and 3 triangular-tetrahedral duoprisms join at each vertex. As the name suggests, it is the runcination of the 6-simplex.

## Vertex coordinates

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

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

## Representations

A runcinated 6-simplex has the following Coxeter diagrams:

• x3o3o3x3o3o () (full symmetry)
• ooo3oox3xx3o3ooo3oxx&#xt (A5 axial, dodecateron-first)

## Related polytopes

The runcinated 6-simplex is the colonel of a regiment with 138 uniform members plus 15 fissaries.