Rectified 24-cell

Rectified 24-cell

Schlegel diagram
8 of 24 cuboctahedral cells shown
Type 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{31,1,1} = 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
Net

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 {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 = [3,31,1]
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

Alternate names

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 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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