# Rectified 6-simplex

Rectified 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymRil
Coxeter diagramo3x3o3o3o3o ()
Elements
Peta7 5-simplices, 7 rectified 5-simplices
Tera42 pentachora, 21 rectified pentachora
Cells105 tetrahedra, 35 octahedra
Faces35+140 triangles
Edges105
Vertices21
Vertex figurePentachoric prism, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {35}}{7}}\approx 0.84515}$
Hypervolume${\displaystyle {\frac {19{\sqrt {7}}}{1920}}\approx 0.026182}$
Dipetal anglesRix–pen–hix: ${\displaystyle \arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }}$
Rix–rap–rix: ${\displaystyle \arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }}$
Height${\displaystyle {\frac {\sqrt {21}}{6}}\approx 0.76376}$
Central density1
Number of external pieces14
Level of complexity5
Related polytopes
ArmyRil
RegimentRil
ConjugateNone
Abstract & topological properties
Flag count25200
Euler characteristic0
OrientableYes
Properties
SymmetryA6, order 5040
Flag orbits5
ConvexYes
NatureTame

The rectified 6-simplex (also called the rectified heptapeton, or ril) is a convex uniform 6-polytope. It consists of 7 regular hexatera and 7 rectified hexatera as facets. Two hexatera and 5 rectified hexatera join at each pentachoric prismatic vertex. As the name suggests, it is the rectification of the heptapeton.

It is isogonal under the 7-2-3 step prism symmetry, and is a 7-2-3 triple step prism.

It is also a convex segmentotope, as a hexateron atop rectified hexateron.

## Vertex coordinates

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

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

## Representations

A rectified 6-simplex has the following Coxeter diagrams:

• o3x3o3o3o3o () (full symmetry)
• xo3ox3oo3oo3oo&#x (A5 axial, hexateron atop rectified hexateron)
• oxo3oox3ooo3ooo oxo&#xt (A4×A1 symmetry, vertex-first)
• oxo3oox3ooo oxo3xoo&#xt (A3×A2 symmetry, triangle-first)
• xoxo3oxoo3oooo3oooo&#xr (A4 axial only)
• oxoox ooooo3ooxoo3oxoxo&#xr (A3×A1 axial only)

## Related polytopes

The rectified 6-simplex is the colonel of a regiment of 7 members.