# Rectified 24-cell

Rectified 24-cell :
| Schlegel diagram 8 of 24 cuboctahedral cells shown |||
| Type | Uniform 4-polytope | Uniform 4-polytope |
| Schläfli symbols | r{3,4,3} = $\left\{{\begin{array}{l}3\\4,3\end{array}}\right\}$ rr{3,3,4}=$r\left\{{\begin{array}{l}3\\3,4\end{array}}\right\}$ r{3$^{1,1,1}$} = $r\left\{{\begin{array}{l}3\\3\\3\end{array}}\right\}$ | r{3,4,3} = $\left\{{\begin{array}{l}3\\4,3\end{array}}\right\}$ rr{3,3,4}=$r\left\{{\begin{array}{l}3\\3,4\end{array}}\right\}$ r{3$^{1,1,1}$} = $r\left\{{\begin{array}{l}3\\3\\3\end{array}}\right\}$ |
| Coxeter diagrams | or | or |
| Cells | 48 | 24 3.4.3.4 24 4.4.4 |
| Faces | 240 | 96 {3} 144 {4} |
| Edges | 288 | 288 |
| Vertices | 96 | 96 |
| Vertex figure | Triangular prism | Triangular prism |
| Symmetry groups | F$_4$ [3,4,3], order 1152 B$_4$ [3,3,4], order 384 D$_4$ [3$^{1,1,1}$], order 192 | F$_4$ [3,4,3], order 1152 B$_4$ [3,3,4], order 384 D$_4$ [3$^{1,1,1}$], order 192 |
| Properties | convex, edge-transitive | convex, edge-transitive |
| Uniform index | 22 23 24 | 22 23 24 |
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$_{24}$.
It can also be considered a cantellated 16-cell with the lower symmetries B$_4$ = [3,3,4]. B$_4$ would lead to a bicoloring of the cuboctahedral cells into 8 and 16 each. It is also called a runcicantellated demitesseract in a D$_4$ 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:
    (0,1,1,2) [4!/2!×2$^3$ = 96 vertices]
The dual configuration with edge length 2 has all coordinate and sign permutations of:
    (0,2,2,2) [4×2$^3$ = 32 vertices]
    (1,1,1,3) [4×2$^4$ = 64 vertices]

## Images

orthographic projections :
| Coxeter plane | F$_4$ | F$_4$ |
| Graph | | |
| Dihedral symmetry | [12] | [12] |
| Coxeter plane | B$_3$ / A$_2$ (a) | B$_3$ / A$_2$ (b) |
| Graph | | |
| Dihedral symmetry | [6] | [6] |
| Coxeter plane | B$_4$ | B$_2$ / A$_3$ |
| Graph | | |
| Dihedral symmetry | [8] | [4] |
* Stereographic projection
* Center of stereographic projection with 96 triangular faces blue

## 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 ${F}_{4}$.
| Coxeter group | ${F}_{4}$ = [3,4,3] | ${C}_{4}$ = [4,3,3] | ${D}_{4}$ = [3,3$^{1,1}$] |
| Order | 1152 | 384 | 192 |
| Full symmetry group | [3,4,3] | [4,3,3] | <[3,3$^{1,1}$]> = [4,3,3] [3[3$^{1,1,1}$]] = [3,4,3] |
| Coxeter diagram | | | |
| Facets | 3: 2: | 2,2: 2: | 1,1,1: 2: |
| Vertex figure | | | |

## 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

D$_4$ uniform polychora :
| {3,3$^{1,1}$} h{4,3,3} | 2r{3,3$^{1,1}$} h$_3${4,3,3} | t{3,3$^{1,1}$} h$_2${4,3,3} | 2t{3,3$^{1,1}$} h$_{2,3}${4,3,3} | r{3,3$^{1,1}$} {3$^{1,1,1}$}={3,4,3} | rr{3,3$^{1,1}$} r{3$^{1,1,1}$}=r{3,4,3} | tr{3,3$^{1,1}$} t{3$^{1,1,1}$}=t{3,4,3} | sr{3,3$^{1,1}$} s{3$^{1,1,1}$}=s{3,4,3} |
24-cell family polytopes :
| Name | 24-cell | truncated 24-cell | snub 24-cell | rectified 24-cell | cantellated 24-cell | bitruncated 24-cell | cantitruncated 24-cell | runcinated 24-cell | runcitruncated 24-cell | omnitruncated 24-cell |
| Schläfli symbol | {3,4,3} | t$_{0,1}${3,4,3} t{3,4,3} | s{3,4,3} | t$_1${3,4,3} r{3,4,3} | t$_{0,2}${3,4,3} rr{3,4,3} | t$_{1,2}${3,4,3} 2t{3,4,3} | t$_{0,1,2}${3,4,3} tr{3,4,3} | t$_{0,3}${3,4,3} | t$_{0,1,3}${3,4,3} | t$_{0,1,2,3}${3,4,3} |
| Coxeter diagram | | | | | | | | | | |
| Schlegel diagram | | | | | | | | | | |
| F$_4$ | | | | | | | | | | |
| B$_4$ | | | | | | | | | | |
| B$_3$(a) | | | | | | | | | | |
| B$_3$(b) | | | | | | | | | | |
| B$_2$ | | | | | | | | | | |
The rectified 24-cell can also be derived as a cantellated 16-cell:
B4 symmetry polytopes :
| Name | tesseract | rectified tesseract | truncated tesseract | cantellated tesseract | runcinated tesseract | bitruncated tesseract | cantitruncated tesseract | runcitruncated tesseract | omnitruncated tesseract |
| Coxeter diagram | | = | | | | = | | | |
| Schläfli symbol | {4,3,3} | t$_1${4,3,3} r{4,3,3} | t$_{0,1}${4,3,3} t{4,3,3} | t$_{0,2}${4,3,3} rr{4,3,3} | t$_{0,3}${4,3,3} | t$_{1,2}${4,3,3} 2t{4,3,3} | t$_{0,1,2}${4,3,3} tr{4,3,3} | t$_{0,1,3}${4,3,3} | t$_{0,1,2,3}${4,3,3} |
| Schlegel diagram | | | | | | | | | |
| B$_4$ | | | | | | | | | |
| Name | 16-cell | rectified 16-cell | truncated 16-cell | cantellated 16-cell | runcinated 16-cell | bitruncated 16-cell | cantitruncated 16-cell | runcitruncated 16-cell | omnitruncated 16-cell |
| Coxeter diagram | = | = | = | = | | = | = | | |
| Schläfli symbol | {3,3,4} | t$_1${3,3,4} r{3,3,4} | t$_{0,1}${3,3,4} t{3,3,4} | t$_{0,2}${3,3,4} rr{3,3,4} | t$_{0,3}${3,3,4} | t$_{1,2}${3,3,4} 2t{3,3,4} | t$_{0,1,2}${3,3,4} tr{3,3,4} | t$_{0,1,3}${3,3,4} | t$_{0,1,2,3}${3,3,4} |
| Schlegel diagram | | | | | | | | | |
| B$_4$ | | | | | | | | | |

## Citations

* ↑ Coxeter 1973, p. 154, §8.4.