# Trirectified 8-simplex

Trirectified 8-simplex
Rank8
TypeUniform
Notation
Bowers style acronymTrene
Coxeter diagramo3o3o3x3o3o3o3o ()
Elements
Zetta
Exa
Peta
Tera
Cells
Faces1260+1680 triangles
Edges1260
Vertices126
Vertex figureTetrahedral-pentachoric duoprism, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {10}}{3}}\approx 1.05409}$
Hypervolume${\displaystyle {\frac {15619}{215040}}\approx 0.072633}$
Dizettal anglesHe–bril–broc: ${\displaystyle \arccos \left(-{\frac {1}{8}}\right)\approx 97.18076^{\circ }}$
He–bril–he: ${\displaystyle \arccos \left({\frac {1}{8}}\right)\approx 82.81924^{\circ }}$
Broc–ril–broc: ${\displaystyle \arccos \left({\frac {1}{8}}\right)\approx 82.81924^{\circ }}$
Height${\displaystyle {\frac {3}{4}}\approx 0.75}$
Central density1
Number of external pieces18
Level of complexity35
Related polytopes
ArmyTrene
RegimentTrene
ConjugateNone
Abstract & topological properties
Flag count12700800
Euler characteristic0
OrientableYes
Properties
SymmetryA8, order 362880
Flag orbits35
ConvexYes
NatureTame

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:

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

## Representations

A trirectified 8-simplex has the following Coxeter diagrams:

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