Rectified 24-cell

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Rectified 24-cell
Schlegel half-solid cantellated 16-cell.png
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 CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png or CDel node.pngCDel splitsplit1.pngCDel branch3 11.pngCDel node 1.png
Cells 48 24 3.4.3.4 Cuboctahedron.png
24 4.4.4 Hexahedron.png
Faces 240 96 {3}
144 {4}
Edges 288
Vertices 96
Vertex figure Rectified 24-cell verf.pngCantellated 16-cell verf.pngRuncicantellated demitesseract verf.png
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[edit]

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[edit]

orthographic projections
Coxeter plane F4
Graph 24-cell t1 F4.svg
Dihedral symmetry [12]
Coxeter plane B3 / A2 (a) B3 / A2 (b)
Graph 24-cell t1 B3.svg 24-cell t2 B3.svg
Dihedral symmetry [6] [6]
Coxeter plane B4 B2 / A3
Graph 24-cell t1 B4.svg 24-cell t1 B2.svg
Dihedral symmetry [8] [4]
Stereographic projection
Rectified 24cell.png
Center of stereographic projection
with 96 triangular faces blue

Symmetry constructions[edit]

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 CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
Facets 3: CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
2: CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
2,2: CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
2: CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
1,1,1: CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
2: CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Vertex figure Rectified 24-cell verf.png Cantellated 16-cell verf.png Runcicantellated demitesseract verf.png

Alternate names[edit]

Related uniform polytopes[edit]

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

References[edit]