Great rhombated icositetrachoron

Great rhombated icositetrachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Grico
Coxeter diagram	x3x4x3o ()
Elements
Cells	96 triangular prisms, 24 truncated cubes, 24 great rhombicuboctahedra
Faces	192 triangles, 288 squares, 96 hexagons, 144 octagons
Edges	288+288+576
Vertices	576
Vertex figure	Sphenoid edge lengths 1 (1), √2 (2), √3 (1), and √2+√2 (2)
Measures (edge length 1)
Circumradius	$\sqrt{10+6\sqrt2} ≈ 4.29945$
Hypervolume	$2(319+224\sqrt2) ≈ 1271.56768$
Dichoral angles	Tic–3–trip: 150°
	Girco–4–trip: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561^\circ$
	Girco–8–tic: 135°
	Girco–6–girco: 120°
Central density	1
Number of external pieces	144
Level of complexity	12
Related polytopes
Army	Grico
Regiment	Grico
Dual	Sphenoidal pentacosiheptacontahexachoron
Conjugate	Great quasirhombated icositetrachoron
Abstract & topological properties
Flag count	13824
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4, order 1152
Convex	Yes
Nature	Tame

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.

Cross-sections

Vertex coordinates

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

• $\left(±3\frac{1+\sqrt2}{2},\,±\frac{1+2\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right),$
• $\left(±\frac{2+3\sqrt2}{2},\,±(1+\sqrt2),\,±\frac{2+\sqrt2}{2},\,0\right).$

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

• $\left(±\frac{5+2\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right),$
• $\left(±\frac{3+2\sqrt2}{2},\,±\frac{3+\sqrt2}{2},\,±\frac{3+\sqrt2}{2},\,±\frac12\right),$
• $\left(±(2+\sqrt2),\,±\frac{2+\sqrt2}{2},\,±\frac{2+\sqrt2}{2},\,±1\right).$

Representations

A great rhombated icositetrachoron has the following Coxeter diagrams:

• x3x4x3o (full symmetry)
• xo4xw3xx3wx&#zx (BC4 symmetry)
• xux4wxx3ooo3xwX (BC4 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 $\sqrt{a^2+3b^2+3c^2+3ab+(2ac+4bc)\sqrt2}$.

Related polychora

Uniform polychoron compounds composed of great rhombated icositetrachora include:

• Great rhombated stellated tetracontoctachoron (2)
• Great rhombated dodecahedronary hexacosichoron (25)

External links

• Bowers, Jonathan. "Category 8: Great Rhombates" (#309).

• Klitzing, Richard. "grico".

• Quickfur. "The Cantitruncated 24-cell".

• Wikipedia Contributors. "Cantitruncated 24-cell".