# Rectified 24-cell

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 Rectified 24-cell Schlegel diagram 8 of 24 cuboctahedral cells shown Type Uniform 4-polytope Schläfli symbols r{3,4,3} = ${\displaystyle \left\{{\begin{array}{l}3\\4,3\end{array}}\right\}}$ rr{3,3,4}=${\displaystyle r\left\{{\begin{array}{l}3\\3,4\end{array}}\right\}}$ r{31,1,1} = ${\displaystyle r\left\{{\begin{array}{l}3\\3\\3\end{array}}\right\}}$ Coxeter diagrams or Cells 48 24 3.4.3.4 24 4.4.4 Faces 240 96 {3} 144 {4} Edges 288 Vertices 96 Vertex figure Triangular prism Symmetry groups F4 [3,4,3], order 1152 B4 [3,3,4], order 384 D4 [31,1,1], order 192 Properties convex, edge-transitive Uniform index 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 reducing the 24-cell's cells to cubes or 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.

## 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!×23 = 96 vertices]

The dual configuration with edge length 2 has all coordinate and sign permutations of:

(0,2,2,2) [4×23 = 32 vertices]
(1,1,1,3) [4×24 = 64 vertices]

## Images

orthographic projections
Coxeter plane F4
Graph
Dihedral symmetry [12]
Coxeter plane B3 / A2 (a) B3 / A2 (b)
Graph
Dihedral symmetry [6] [6]
Coxeter plane B4 B2 / A3
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 ${\displaystyle {D}_{4}}$ construction can be doubled into ${\displaystyle {C}_{4}}$ by adding a mirror that maps the bifurcating nodes onto each other. ${\displaystyle {D}_{4}}$ can be mapped up to ${\displaystyle {F}_{4}}$ 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 ${\displaystyle {D}_{4}}$ construction, and two colors (1:2 ratio) in ${\displaystyle {C}_{4}}$, and all identical cuboctahedra in ${\displaystyle {F}_{4}}$.

Coxeter group ${\displaystyle {F}_{4}}$ = [3,4,3] ${\displaystyle {C}_{4}}$ = [4,3,3] ${\displaystyle {D}_{4}}$ = [3,31,1]
Order 1152 384 192
Full
symmetry
group
[3,4,3] [4,3,3] <[3,31,1]> = [4,3,3]
[3[31,1,1]] = [3,4,3]
Coxeter diagram
Facets 3:
2:
2,2:
2:
1,1,1:
2:
Vertex figure

## Related uniform polytopes

The rectified 24-cell can also be derived as a cantellated 16-cell:

## References

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds