Rectified 24-cell
In geometry, the rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional 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 = . 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 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:
Symmetry constructions
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 in, and all identical cuboctahedra in.
| Coxeter group | = | = | = |
| Order | 1152 | 384 | 192 |
| Full symmetry group | <> = | ||
| Coxeter diagram | |||
| Facets | 3: 2: | 2,2: 2: | 1,1,1: 2: |
| Vertex figure |
Alternate names
- Rectified 24-cell, Cantellated 16-cell
- Rectified icositetrachoron
- * Cantellated hexadecachoron
- Disicositetrachoron
- Amboicositetrachoron