Trirectified 8-simplex

Trirectified 8-simplex
(OFF file)
Rank	8
Type	Uniform
Notation
Bowers style acronym	Trene
Coxeter diagram	o3o3o3x3o3o3o3o ()
Elements
Zetta	9 birectified 7-simplices 9 trirectified 7-simplices
Exa	36 rectified 6-simplices 36+72 birectified 6-simplices
Peta	84 5-simplices 84+252 rectified 5-simplices 252 birectified 5-simplices
Tera	126+504 pentachora 504+756 rectified pentachora
Cells	630+1260 tetrahedra 1260 octahedra
Faces	1260+1680 triangles
Edges	1260
Vertices	126
Vertex figure	Tetrahedral-pentachoric duoprism, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {10}}{3}}\approx 1.05409$
Hypervolume	${\frac {15619}{215040}}\approx 0.072633$
Dizettal angles	He–bril–broc: $\arccos \left(-{\frac {1}{8}}\right)\approx 97.18076^{\circ }$
	He–bril–he: $\arccos \left({\frac {1}{8}}\right)\approx 82.81924^{\circ }$
	Broc–ril–broc: $\arccos \left({\frac {1}{8}}\right)\approx 82.81924^{\circ }$
Height	${\frac {3}{4}}\approx 0.75$
Central density	1
Number of external pieces	18
Level of complexity	35
Related polytopes
Army	Trene
Regiment	Trene
Conjugate	None
Abstract & topological properties
Flag count	12700800
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A8, order 362880
Flag orbits	35
Convex	Yes
Nature	Tame

The trirectified 8-simplex, also called the trirectified enneazetton, is a convex uniform 8-polytope. It consists of 9 birectified 7-simplices and 9 trirectified 7-simplices. 4 birectified 7-simplices and 5 trirectified 7-simplices join at each tetrahedral-pentachoric duoprismatic vertex. As the name suggests, it is the trirectification of the 8-simplex.

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

Vertex coordinates

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

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

Representations

A trirectified 8-simplex has the following Coxeter diagrams:

• o3o3o3x3o3o3o3o () (full symmetry)
• oo3oo3xo3ox3oo3oo3oo&#x (A₇ axial, birectified 7-simplex atop trirectified 7-simplex)

External links

• Klitzing, Richard. "trene".
• Wikipedia contributors. "Trirectified 8-simplex".