# Great rhombated tesseract

Great rhombated tesseract Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymGrit
Coxeter diagram       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$\sqrt{\frac{11+5\sqrt2}{2}} ≈ 3.00592$ Hypervolume$\frac{601+424\sqrt2}{6} ≈ 200.10443$ Dichoral anglesTut–3–trip: 150°
Girco–4–trip: $\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439°$ Girco–6–tut: 120°
Girco–8–girco: 90°
Central density1
Number of pieces56
Level of complexity12
Related polytopes
ArmyGrit
RegimentGrit
ConjugateGreat quasirhombated tesseract
Abstract properties
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryB4, order 384
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:

• $\left(±\frac{1+2\sqrt2}{2},\,±\frac{1+2\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$ ## Representations

A great rhombated tesseract has the following Coxeter diagrams:

• x4x3x3o (full symmetry)
• xxwwxx4xuxxux3xoooox&#xt (BC3 symmetry, great rhombicuboctahedron-first)
• wx3xx3xw *b3oo&#zx (D4 symmetry)
• Xwx xxw4xux3xoo&#zx (BC3×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 $\sqrt{\frac{2a^2+3b^2+2c^2+4bc+(3ab+2ac)\sqrt2}{2} }$ .

It has coordinates given by all permutations of:

• $\left(±\frac{a+(b+c)\sqrt2}{2},\,±\frac{a+(b+c)\sqrt2}{2},\,±\frac{a+b\sqrt2}{2},\,±\frac{a}{2}\right).$ 