| bgcolor=#e7dcc3 align=center colspan=3 | Rectified 24-cell | ||
| bgcolor=#ffffff align=center colspan=3 | Schlegel diagram 8 of 24 cuboctahedral cells shown | ||
| Type | Uniform 4-polytope | ||
| Schläfli symbols | r = \left\{\begin{array}{l}3\\4,3\end{array}\right\} rr= r\left\{\begin{array}{l}3\\3,4\end{array}\right\} r = r\left\{\begin{array}{l}3\\3\\3\end{array}\right\} | ||
| Coxeter diagrams | or | ||
| Cells | 48 | ||
| Faces | 240 | 96 144 | |
| Edges | 288 | ||
| Vertices | 96 | ||
| Vertex figure | Triangular prism | ||
| Symmetry groups | F4 [3,4,3], order 1152 B4 [3,3,4], order 384 D4 [3<sup>1,1,1</sup>], order 192 | ||
| Properties | convex, edge-transitive | ||
| Uniform index | 22 23 24 | ||
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.
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:
(0,1,1,2) [4!/2!×2<sup>3</sup> = 96 vertices]
The dual configuration with edge length 2 has all coordinate and sign permutations of:
(0,2,2,2) [4×2<sup>3</sup> = 32 vertices]
(1,1,1,3) [4×2<sup>4</sup> = 64 vertices]
| Stereographic projection | |
|---|---|
| Center of stereographic projection with 96 triangular faces blue |
There are three different symmetry constructions of this polytope. The lowest
{D}4
{C}4
{D}4
{F}4
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
{C}4
{F}4
| Coxeter group | {F}4 | {C}4 | {D}4 | |
|---|---|---|---|---|
| Order | 1152 | 384 | 192 | |
| Full symmetry group | [3,4,3] | [4,3,3] | <[3,3<sup>1,1</sup>]> = [4,3,3] [3[3<sup>1,1,1</sup>]] = [3,4,3] | |
| Coxeter diagram | ||||
| Facets | 3: 2: | 2,2: 2: | 1,1,1: 2: | |
| Vertex figure |
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: