Birectified 7-simplex

Birectified 7-simplex
(OFF file)
Rank	7
Type	Uniform
Notation
Bowers style acronym	Broc
Coxeter diagram	o3o3x3o3o3o3o ()
Elements
Exa	8 rectified heptapeta, 8 birectified heptapeta
Peta	28 hexatera, 56 rectified hexatera, 28 dodecatera
Tera	168 pentachora, 56+168 rectified pentachora
Cells	70+420 tetrahedra, 280 octahedra
Faces	280+560 triangles
Edges	420
Vertices	56
Vertex figure	Triangular-pentachoric duoprism, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {15}}{4}}\approx 0.96825$
Hypervolume	${\frac {397}{6720}}\approx 0.059077$
Diexal angles	Bril–rix–ril: $\arccos \left(-{\frac {1}{7}}\right)\approx 98.21321^{\circ }$
	Bril–dot–bril: $\arccos \left({\frac {1}{7}}\right)\approx 81.78679^{\circ }$
	Ril–hix–ril: $\arccos \left({\frac {1}{7}}\right)\approx 81.78679^{\circ }$
Height	${\frac {2{\sqrt {7}}}{7}}\approx 0.75593$
Central density	1
Number of external pieces	16
Level of complexity	15
Related polytopes
Army	Broc
Regiment	Broc
Conjugate	None
Abstract & topological properties
Flag count	604800
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A7, order 40320
Convex	Yes
Nature	Tame

The birectified octaexon, or broc, also called the birectified 7-simplex, is a convex uniform polyexon. It consists of 8 rectified heptapeta and 8 birectified heptapeta. 3 rectified heptapeta and 5 birectified heptapeta join at each triangular-pentachoric duoprismatic vertex. As the name suggests, it is the birectification of the octaexon.

It is also a convex segmentoexon, as rectified heptapeton atop birectified heptapeton.

Vertex coordinates

The vertices of a birectified octaexon of edge length 1 can be given in eight dimensions as all permutations of:

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

Representations

A birectified octaexon has the following Coxeter diagrams:

• o3o3x3o3o3o3o (full symmetry)
• oo3xo3ox3oo3oo3oo&#x (A6 axial, rectified heptapeton atop birectified heptapeton)

External links

• Klitzing, Richard. "broc".
• Wikipedia contributors. "Birectified 7-simplex".