# Rectified 24-cell

 Rectified 24-cell Schlegel diagram8 of 24 cuboctahedral cells shown Type Uniform 4-polytope Schläfli symbols r{3,4,3} = rr{3,3,4}=r{31,1,1} = 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 1152B4 [3,3,4], order 384D4 [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 construction can be doubled into by adding a mirror that maps the bifurcating nodes onto each other. can be mapped up to 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 construction, and two colors (1:2 ratio) in , and all identical cuboctahedra in .

Coxeter group = [3,4,3] = [4,3,3] = [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

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• H.S.M. Coxeter:
• Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 23, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) o3x4o3o - rico".
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