Rectified icositetrachoron

The rectified icositetrachoron, or rico, also commonly called the rectified 24-cell, is a convex uniform polychoron that consists of 24 cubes and 24 cuboctahedra. Two cubes and three cuboctahedra join at each triangular prismatic vertex. As the name suggests, it can be obtained by rectifying the icositetrachoron.

It can also be constructed as the cantellated hexadecachoron, under B4 symmetry. This is due to the fact that the regular icositetrachoron coincides with the rectified hexadecachoron.

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


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

The rectification of the dual icositetrachoron has vertices given by all permutations of:


 * $$\left(±\frac32,\,±\frac12,\,±\frac12,\,±\frac12\right),$$
 * $$\left(±1,\,±1,\,±1,\,0\right).$$

Representations
A rectified icositetrachoron has the following Coxeter diagrams:


 * o3x4o3o (full symmetry)
 * o4x3o3x (B4 symetry, small rhombated hexadecachoron)
 * x3o3x *b3x (D4 symetry, as prismatorhombated demitesseract)
 * s4x3o3x (as snub)
 * ooqoo4xxoxx3oxxxo&#xt *B3 axial, cuboctahedron-first)
 * xoxuxox4oqoooqo3ooqoqoo&#xt (B3 axial, cube-first)
 * xo4oq3oo3qo&#zx (B4 symmetry, rectified dual ico)
 * qqo3ooo3qoq *b3oqq&#zx (D4 symmetry)
 * Qqo oqq4xxo3oxx&#zx (B3×A1 symmetry)

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


 * Small rhombated hexadecachoron - 8 cuboctahedra, 16 rhombitetratetrahedra, and 24 square prisms with a wedge verf
 * Prismatorhombated demitesseract - 3 sets of 8 rhombitetratetrahedra and 24 cuboids, with a skewed wedge verf

Related polychora
The rectified icositetrachoron is the colonel of a regiment of 7 members. Of these, 2 more (the facetorectified icositetrachoron and icositetrintercepted icositetrachoron) have full F4 symmetry, while the other 4 (the retrosphenoverted hexadecatesseractioctachoron, small rhombic tesseractihexadecachoron, grand rhombic dishexadecachoron, and hexadecintercepted prismatotesseractihexadecachoron) have only BC4 symmetry. It is also uniform under D4 symmetry, acting as a skewvert, but there are no members with D4 symmetry only.

Uniform polychoron compounds composed of rectified icositetrachora include:


 * Rectified stellated tetracontoctachoron (2)
 * Rectified chirohecatonicosachoron (5)
 * Rectified dodecahedronary hexacosichoron (25)
 * Rectified pentishecatonicosachoron (25)

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Cuboctahedron (24): Icositetrachoron
 * Cube (24): Icositetrachoron
 * Square (144): Semi-uniform small prismatotetracontoctachoron
 * Triangle (96): Rectified icositetrachoron
 * Edge (288): Small rhombated icositetrachoron