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 √2 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:

Images

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

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:

Citations