# Tritruncated 6-simplex

Tritruncated 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymFe
Coxeter diagramo3o3x3x3o3o ()
Elements
Peta14 bitruncated hexatera
Tera42 truncated pentachora, 42 decachora
Cells70 tetrahedra, 210 truncated tetrahedra
Faces280 triangles, 210 hexagons
Edges420
Vertices140
Vertex figureTriangular disphenoid, edge lengths 1 (base triangles) and 3 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {3}}\approx 1.73205}$
Inradius${\displaystyle {\frac {\sqrt {21}}{6}}\approx 0.76376}$
Hypervolume${\displaystyle {\frac {5887{\sqrt {7}}}{1440}}\approx 10.81635}$
Dipetal anglesBittix-deca-bittix: ${\displaystyle \arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }}$
Bittix-tip-bittix: ${\displaystyle \arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }}$
Central density1
Number of external pieces14
Level of complexity10
Related polytopes
ArmyFe
RegimentFe
ConjugateNone
Abstract & topological properties
Flag count100800
Euler characteristic0
OrientableYes
Properties
SymmetryA6×2, order 10080
ConvexYes
NatureTame

The tritruncated 6-simplex (also the tritruncated heptapeton, tetradecapeton, or fe) is a convex noble uniform 6-polypetope. It consists of 14 bitruncated hexatera, with 6 joining at each vertex. As the name suggests, it is the tritruncation of the 6-simplex. It is the medial stage of truncations between the 6-simplex and its dual 6-simplex. It is also the medial vertex-first cross-section of the hepteract. It is also the 14-3-5 gyropeton.

## Vertex coordinates

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

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

## Representations

A tritruncated 6-simplex has the following Coxeter diagrams:

• o3o3x3x3o3o () (full symmetry)
• ooo3xoo3xux3oox3ooo&#xt (A5 axial, facet-first)