# Truncated 5-simplex

(Redirected from Truncated hexateron)
Truncated 5-simplex
Rank5
TypeUniform
Notation
Bowers style acronymTix
Coxeter diagramx3x3o3o3o ()
Elements
Tera6 pentachora, 6 truncated pentachora
Cells30 tetrahedra, 15 truncated tetrahedra
Faces60 triangles, 20 hexagons
Edges15+60
Vertices30
Vertex figureTetrahedral pyramid, edge lengths 1 (base) and 3 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7}}{2}}\approx 1.32288}$
Hypervolume${\displaystyle {\frac {79{\sqrt {3}}}{160}}\approx 0.85520}$
Diteral anglesTip–tet–pen: ${\displaystyle \arccos \left(-{\frac {1}{5}}\right)\approx 101.53696^{\circ }}$
Tip–tut–tip: ${\displaystyle \arccos \left({\frac {1}{5}}\right)\approx 78.46304^{\circ }}$
Central density1
Number of external pieces12
Level of complexity5
Related polytopes
ArmyTix
RegimentTix
DualPentakis hexateron
ConjugateNone
Abstract & topological properties
Flag count3600
Euler characteristic2
OrientableYes
Properties
SymmetryA5, order 720
Flag orbits5
ConvexYes
NatureTame

The truncated 5-simplex, also called the truncated hexateron or tix, is a convex uniform 5-polytope. It consists of 6 regular pentachora and 6 truncated pentachora. One pentachoron and 4 truncated pentachora join at each vertex. As the name suggests, it is the truncation of the 5-simplex.

## Vertex coordinates

The vertices of a truncated 5-simplex of edge length 1 can be given in 6 dimensions as all permutations of:

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

## Representations

A truncated 5-simplex has the following Coxeter diagrams:

• x3x3o3o3o () (full symmetry)
• xux3oox3ooo3ooo&#xt (A4 axial, pentachoron-first)
• xuxo oxux3ooox3oooo&#xt (A3×A1 axial, edge-first)
• xuxo3xooo ooox3oxux&#xt (A2×A2 axial, hexagon-first)