# Birectified 6-simplex

Birectified 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymBril
Coxeter diagramo3o3x3o3o3o ()
Elements
Peta7 rectified 5-simplices, 7 birectified 5-simplices
Tera21 pentachora, 21+42 rectified pentachora
Cells35+105 tetrahedra, 105 octahedra
Faces140+210 triangles
Edges210
Vertices35
Vertex figureTriangular-tetrahedral duoprism, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {42}}{7}}\approx 0.92582}$
Hypervolume${\displaystyle {\frac {151{\sqrt {7}}}{2880}}\approx 0.13872}$
Dipetal anglesDot–rap–rix: ${\displaystyle \arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }}$
Dot–rap–dot: ${\displaystyle \arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }}$
Rix–pen–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 complexity10
Related polytopes
ArmyBril
RegimentBril
ConjugateNone
Abstract & topological properties
Flag count50400
Euler characteristic0
OrientableYes
Properties
SymmetryA6, order 5040
Flag orbits10
ConvexYes
NatureTame

The birectified 6-simplex (also called the birectified heptapteon or bril) is a convex uniform 6-polytope. It has 7 rectified hexatera and 7 dodecatera as facets. 3 rectified hexatera and 4 dodecatera join at each triangular-tetrahedral duoprismatic vertex. As the name suggests, it is the birectification of the 6-simplex.

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

## Vertex coordinates

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

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

## Representations

A birectified 6-simplex has the following Coxeter diagrams:

• o3o3x3o3o3o () (full symmetry)
• oo3xo3ox3oo3oo&#x (A5 axial, rectified hexateron atop dodecateron)
• oxo xoo3oxo3oox3ooo&#xt (A4×A1 axial, pentachoron-first)
• oxoo3ooxo oxoo3ooxo3ooox&#xt (A3×A2 axial, vertex-first)

## Related polytopes

The birectified 6-simplex is the colonel of a regiment of 18 members.