# Great rhombated tesseract

Great rhombated tesseract
Rank4
TypeUniform
Notation
Bowers style acronymGrit
Coxeter diagramx4x3x3o ()
Elements
Cells32 triangular prisms, 16 truncated tetrahedra, 8 great rhombicuboctahedra
Faces64 triangles, 96 squares, 64 hexagons, 24 octagons
Edges96+96+192
Vertices192
Vertex figureSphenoid edge lengths 1 (1), 2 (2), 3 (2), and 2+2 (1)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {11+5{\sqrt {2}}}{2}}}\approx 3.00592}$
Hypervolume${\displaystyle {\frac {601+424{\sqrt {2}}}{6}}\approx 200.10443}$
Dichoral anglesTut–3–trip: 150°
Girco–4–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Girco–6–tut: 120°
Girco–8–girco: 90°
Central density1
Number of external pieces56
Level of complexity12
Related polytopes
ArmyGrit
RegimentGrit
ConjugateGreat quasirhombated tesseract
Abstract & topological properties
Flag count4608
Euler characteristic0
OrientableYes
Properties
SymmetryB4, order 384
Flag orbits12
ConvexYes
NatureTame

The great rhombated tesseract, or grit, also commonly called the cantitruncated tesseract, is a convex uniform polychoron that consists of 32 triangular prisms, 16 truncated tetrahedra, and 8 great rhombicuboctahedra. 1 triangular prism, 1 truncated tetrahedron, and 2 great rhombicuboctahedra join at each vertex. As one of its names suggests, it can be obtained by cantitruncating the tesseract.

## Vertex coordinates

The vertices of a great rhombated tesseract of edge length 1 are given by all permutations of:

• ${\displaystyle \left(\pm {\frac {1+2{\sqrt {2}}}{2}},\,\pm {\frac {1+2{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right)}$.

## Representations

A great rhombated tesseract has the following Coxeter diagrams:

• x4x3x3o () (full symmetry)
• xxwwxx4xuxxux3xoooox&#xt (B3 symmetry, great rhombicuboctahedron-first)
• wx3xx3xw *b3oo&#zx (D4 symmetry)
• Xwx xxw4xux3xoo&#zx (B3×A1 symmetry)

## Semi-uniform variant

The great rhombated tesseract has a semi-uniform variant of the form x4y3z3o that maintains its full symmetry. This variant uses 16 truncated tetrahedra of form y3z3o, 8 great rhombicuboctahedra of form x4y3z, and 32 triangular prisms of form x z3o as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a4b3c3o), its circumradius is given by ${\displaystyle {\sqrt {\frac {2a^{2}+3b^{2}+2c^{2}+4bc+(3ab+2ac){\sqrt {2}}}{2}}}}$.

It has coordinates given by all permutations of:

• ${\displaystyle \left(\pm {\frac {a+(b+c){\sqrt {2}}}{2}},\,\pm {\frac {a+(b+c){\sqrt {2}}}{2}},\,\pm {\frac {a+b{\sqrt {2}}}{2}},\,\pm {\frac {a}{2}}\right)}$.

## Related polychora

Uniform polychoron compounds composed of great rhombated tesseracts include: