# Great rhombated icositetrachoron

Great rhombated icositetrachoron
Rank4
TypeUniform
Notation
Bowers style acronymGrico
Coxeter diagramx3x4x3o ()
Elements
Cells96 triangular prisms, 24 truncated cubes, 24 great rhombicuboctahedra
Faces192 triangles, 288 squares, 96 hexagons, 144 octagons
Edges288+288+576
Vertices576
Vertex figureSphenoid edge lengths 1 (1), 2 (2), 3 (1), and 2+2 (2)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {10+6{\sqrt {2}}}}\approx 4.29945}$
Hypervolume${\displaystyle 2(319+224{\sqrt {2}})\approx 1271.56768}$
Dichoral anglesTic–3–trip: 150°
Girco–4–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Girco–8–tic: 135°
Girco–6–girco: 120°
Central density1
Number of external pieces144
Level of complexity12
Related polytopes
ArmyGrico
RegimentGrico
DualSphenoidal pentacosiheptacontahexachoron
ConjugateGreat quasirhombated icositetrachoron
Abstract & topological properties
Flag count13824
Euler characteristic0
OrientableYes
Properties
SymmetryF4, order 1152
Flag orbits12
ConvexYes
NatureTame

The great rhombated icositetrachoron, or grico, also commonly called the cantitruncated 24-cell, is a convex uniform polychoron that consists of 96 triangular prisms, 24 truncated cubes, and 24 great rhombicuboctahedra. 1 triangular prism, 1 truncated cube, and 2 great rhombicuboctahedra join at each vertex. As one of its names suggests, it can be obtained by cantitruncating the icositetrachoron.

## Vertex coordinates

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

• ${\displaystyle \left(\pm 3{\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+2{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {2+3{\sqrt {2}}}{2}},\,\pm (1+{\sqrt {2}}),\,\pm {\frac {2+{\sqrt {2}}}{2}},\,0\right)}$.

The cantitruncation of the dual icositetrachoron has coordinates given by all permutations of:

• ${\displaystyle \left(\pm {\frac {5+2{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+2{\sqrt {2}}}{2}},\,\pm {\frac {3+{\sqrt {2}}}{2}},\,\pm {\frac {3+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(\pm (2+{\sqrt {2}}),\,\pm {\frac {2+{\sqrt {2}}}{2}},\,\pm {\frac {2+{\sqrt {2}}}{2}},\,\pm 1\right)}$.

## Representations

A great rhombated icositetrachoron has the following Coxeter diagrams:

• x3x4x3o () (full symmetry)
• xo4xw3xx3wx&#zx (B4 symmetry)
• xux4wxx3ooo3xwX (B4 symmetry, dual ico)

## Semi-uniform variant

The great rhombated icositetrachoron has a semi-uniform variant of the form x3y4z3o that maintains its full symmetry. This variant uses 24 great rhombicuboctahedra of form z4y3x, 24 truncated cubes of form y4z3o, and 96 triangular prisms of form x z3o as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a3b4c3o), its circumradius is given by ${\displaystyle {\sqrt {a^{2}+3b^{2}+3c^{2}+3ab+(2ac+4bc){\sqrt {2}}}}}$.

## Related polychora

Uniform polychoron compounds composed of great rhombated icositetrachora include: