Truncated icositetrachoron

The truncated icositetrachoron, or tico, also commonly called the truncated 24-cell, is a convex uniform polychoron that consists of 24 cubes and 24 truncated octahedra. 1 cube and 3 truncated octahedra join at each vertex. As the name suggests, it can be obtained as the truncation of an icositetrachoron.

It is also the cantitruncated hexadecachoron, and the omnitruncate of the D4 group. It can be alternated into a snub disicositetrachoron.

Vertex coordinates
The vertices of a truncated icositetrachoron of edge length 1 are given by all permutations of:


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

The vertices of the truncation of the dual icositetrachoron can be given by all permutations of:


 * $$\left(±\frac52,\,±\frac12,\,±\frac12,\,±\frac12\right),$$


 * $$\left(±\frac32,\,±\frac32,\,±\frac32,\,±\frac12\right),$$


 * $$\left(±2,\,±1,\,±1,\,±1\right).$$

Representations
A truncated icositetrachoron has the following Coxeter diagrams:


 * x3x4o3o (full symmetry)
 * o4x3x3x (BC4 symmetry, great rhombated hexadecachoron)
 * x3x3x *b3x (D4 symmetry, omnitruncated demitesseract)
 * s4x3x3x (as snub)
 * oooqooo3xxuxuxx3xuxxxux&#xt (BC3 axial, truncated octahedron-first)
 * xux3qoo3ooo3oqQ&#zx (D4 symmetry, truncated dual ico)
 * xuxu(xd)(xd)uxux4ooqo(oo)(oo)oqoo3oooq(Qo)(Qo)qooo&#xt (BC3 axial, cube-first)

Semi-uniform variant
The truncated icositetrachoron has a semi-uniform variant of the form x3y4o3o that maintains its full symmetry. This variant uses 24 cubes of size y and 24 semi-uniform truncated octahedra of form x3y4o as cells, with 2 edge lengths.

With edges of length a (surrounded by truncated octahedra only) and b (of cubes), its circumradius is given by $$\sqrt{a^2+3b^2+3ab}$$ and its hypervolume is given by $$2a^4+16a^3b+48a^2b^2+64ab^3+29b^4$$.

Variations
The truncated icositetrachoron has two semi-uniform subsymmetrical variations:


 * Great rhombated hexadecachoron - 8 truncated octahedra, 16 great rhombitetratetrahedra, and 24 square prisms with a sphenoid verf
 * Great prismated demitesseract - 3 sets of 8 great rhombitetratetrahedra and 24 cuboids, with an irregular tetrahedron verf