# Tritruncated 6-cube

Tritruncated 6-cube
Rank6
TypeUniform
Notation
Bowers style acronymXog
Coxeter diagramo4o3x3x3o3o ()
Elements
Peta64 bitruncated 5-simplices, 12 bitruncated 5-orthoplexes
Tera192 truncated pentachora, 384 decachora, 60 truncated hexadecachora
Cells240 tetrahedra, 160 octahedra, 960+960 truncated tetrahedra
Faces960+1280 triangles, 1920 hexagons
Edges1440+1920
Vertices960
Vertex figureSquare tettene, edge lengths 1 (base square and top triangle) and 3 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {26}}{2}}\approx 2.54951}$
Hypervolume${\displaystyle {\frac {17407}{45}}\approx 386.82222}$
Dipetal anglesBittix–tip–bittix: ${\displaystyle \arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }}$
Bittint–deca–bittix: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }}$
Bittit–thex–bittit: 90°
Central density1
Number of external pieces76
Level of complexity20
Related polytopes
ArmyXog
RegimentXog
ConjugateNone
Abstract & topological properties
Flag count921600
Euler characteristic0
OrientableYes
Properties
SymmetryB6, order 46080
ConvexYes
NatureTame

The tritruncated 6-cube, also called the tritruncated 6-orthoplex, tritruncated hexeract, tritruncated hexacontatetrapeton, hexeractihexacontatetrapeton, or xog, is a convex uniform 6-polytope. It consists of 12 bitruncated 5-orthoplexes and 64 bitruncated 5-simplices. 3 bitruncated 5-orthoplexes and 4 bitruncated 5-simplices join at each vertex. As the name suggests, it is the tritruncation of either the 6-cube or its dual 6-orthoplex.

## Vertex coordinates

The vertices of a tritruncated 6-cube of edge length 1 are given by all permutations of:

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

## Representations

A tritruncated 6-cube has the following Coxeter diagrams:

• o4o3x3x3o3o () (full symmetry)
• o3x3x3o3o *b3o () (D6 symmetry)

## Related polytopes

The tritruncated 6-cube is the colonel of a two-member regiment that also includes the tritruncated triacontadihemihexeract.