Great rhombated icositetrachoron

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:


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

The cantitruncation of the dual icositetrachron 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 $$\sqrt{a^2+3b^2+3c^2+3ab+(2ac+4bc)\sqrt2}$$.