Rectified 24-cell

In geometry, the rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by rectification of the 24-cell, reducing its octahedral cells to cubes and cuboctahedra.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC24.

It can also be considered a cantellated 16-cell with the lower symmetries B4 = [3,3,4]. B4 would lead to a bicoloring of the cuboctahedral cells into 8 and 16 each. It is also called a runcicantellated demitesseract in a D4 symmetry, giving 3 colors of cells, 8 for each.

Construction
The rectified 24-cell can be derived from the 24-cell by the process of rectification: the 24-cell is truncated at the midpoints. The vertices become cubes, while the octahedra become cuboctahedra.

Cartesian coordinates
A rectified 24-cell having an edge length of $\sqrt{2}$ has vertices given by all permutations and sign permutations of the following Cartesian coordinates:
 * (0,1,1,2) [4!/2!×23 = 96 vertices]

The dual configuration with edge length 2 has all coordinate and sign permutations of:
 * (0,2,2,2) [4×23 = 32 vertices]
 * (1,1,1,3) [4×24 = 64 vertices]

Symmetry constructions
There are three different symmetry constructions of this polytope. The lowest $${D}_4$$ construction can be doubled into $${C}_4$$ by adding a mirror that maps the bifurcating nodes onto each other. $${D}_4$$ can be mapped up to $${F}_4$$ symmetry by adding two mirror that map all three end nodes together.

The vertex figure is a triangular prism, containing two cubes and three cuboctahedra. The three symmetries can be seen with 3 colored cuboctahedra in the lowest $${D}_4$$ construction, and two colors (1:2 ratio) in $${C}_4$$, and all identical cuboctahedra in $${F}_4$$.

Alternate names

 * Rectified 24-cell, Cantellated 16-cell (Norman Johnson)
 * Rectified icositetrachoron (Acronym rico) (George Olshevsky, Jonathan Bowers)
 * Cantellated hexadecachoron
 * Disicositetrachoron
 * Amboicositetrachoron (Neil Sloane & John Horton Conway)

Related polytopes
The convex hull of the rectified 24-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 192 cells: 48 cubes, 144 square antiprisms, and 192 vertices. Its vertex figure is a triangular bifrustum.

Related uniform polytopes
The rectified 24-cell can also be derived as a cantellated 16-cell: