# Birectified 8-simplex

Birectified 8-simplex
Rank8
TypeUniform
Notation
Bowers style acronymBrene
Coxeter diagramo3o3x3o3o3o3o3o ()
Elements
Zetta
Exa
Peta
Tera
Cells
Faces504+1260 triangles
Edges756
Vertices84
Vertex figureTriangular-hexateric duoprism, edge length 1
Measures (edge length 1)
Hypervolume${\displaystyle {\frac {1431}{71680}}\approx 0.019964}$
Dizettal anglesBroc–ril–roc: ${\displaystyle \arccos \left(-{\frac {1}{8}}\right)\approx 97.18076^{\circ }}$
Broc–bril–broc: ${\displaystyle \arccos \left({\frac {1}{8}}\right)\approx 82.81924^{\circ }}$
Roc–hop–roc: ${\displaystyle \arccos \left({\frac {1}{8}}\right)\approx 82.81924^{\circ }}$
Height${\displaystyle {\frac {3}{4}}\approx 0.75}$
Central density1
Number of external pieces18
Level of complexity21
Related polytopes
ArmyBrene
RegimentBrene
ConjugateNone
Abstract & topological properties
Flag count7620480
Euler characteristic0
OrientableYes
Properties
SymmetryA8, order 362880
Flag orbits21
ConvexYes
NatureTame

The birectified 8-simplex, also called the birectified enneazetton, is a convex uniform 8-polytope. It consists of 9 rectified 7-simplices and 9 birectified 7-simplices. 3 rectified 7-simplices and 6 birectified 7-simplices join at each triangular-hexateric duoprismatic vertex. As the name suggests, it is the birectification of the 8-simplex.

It is also a convex segmentozetton, as rectified 7-simplex atop birectified 7-simplex.

A unit birectified 8-simplex can be vertex inscibed into the 421 polytope.

## Vertex coordinates

The vertices of a birectified 8-simplex of edge length 1 can be given in nine dimensions as all permutations of:

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

## Representations

A birectified 8-simplex has the following Coxeter diagrams:

• o3o3x3o3o3o3o3o () (full symmetry)
• oo3xo3ox3oo3oo3oo3oo&#x (A7 axial, rectified octaexon atop birectified octaexon)