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 tC<sub>24</sub>.
It can also be considered a cantellated 16-cell with the lower symmetries B<sub>4</sub> = [3,3,4]. B<sub>4</sub> would lead to a bicoloring of the cuboctahedral cells into 8 and 16 each. It is also called a runcicantellated demitesseract in a D<sub>4</sub> symmetry, giving 3 colors of cells, 8 for each.
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.
A rectified 24-cell having an edge length of has vertices given by all permutations and sign permutations of the following Cartesian coordinates:
The dual configuration with edge length 2 has all coordinate and sign permutations of:
There are three different symmetry constructions of this polytope. The lowest construction can be doubled into by adding a mirror that maps the bifurcating nodes onto each other. can be mapped up to 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 construction, and two colors (1:2 ratio) in , and all identical cuboctahedra in .
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.
The rectified 24-cell can also be derived as a cantellated 16-cell: