Birectified 8-simplex

Birectified 8-simplex
(OFF file)
Rank	8
Type	Uniform
Notation
Bowers style acronym	Brene
Coxeter diagram	o3o3x3o3o3o3o3o ()
Elements
Zetta	9 rectified 7-simplices 9 birectified 7-simplices
Exa	36 6-simplices 72 rectified 6-simplices 36 birectified 6-simplices
Peta	252 5-simplices 252 rectified 5-simplices 84 birectified 5-simplices
Tera	756 pentachora 126+504 rectified pentachora
Cells	126+1260 tetrahedra 630 octahedra
Faces	504+1260 triangles
Edges	756
Vertices	84
Vertex figure	Triangular-hexateric duoprism, edge length 1
Measures (edge length 1)
Circumradius	1
Hypervolume	${\frac {1431}{71680}}\approx 0.019964$
Dizettal angles	Broc–ril–roc: $\arccos \left(-{\frac {1}{8}}\right)\approx 97.18076^{\circ }$
	Broc–bril–broc: $\arccos \left({\frac {1}{8}}\right)\approx 82.81924^{\circ }$
	Roc–hop–roc: $\arccos \left({\frac {1}{8}}\right)\approx 82.81924^{\circ }$
Height	${\frac {3}{4}}\approx 0.75$
Central density	1
Number of external pieces	18
Level of complexity	21
Related polytopes
Army	Brene
Regiment	Brene
Conjugate	None
Abstract & topological properties
Flag count	7620480
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A8, order 362880
Flag orbits	21
Convex	Yes
Nature	Tame

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:

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

Representations

A birectified 8-simplex has the following Coxeter diagrams:

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

External links

• Klitzing, Richard. "brene".
• Wikipedia contributors. "Birectified 8-simplex".