24-cell

In 4-dimensional geometry, there are 9 uniform 4-polytopes with F4 symmetry, and one chiral half symmetry, the snub 24-cell. There is one self-dual regular form, the 24-cell with 24 vertices.

Visualization

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Each can be visualized as symmetric orthographic projections in Coxeter planes of the F4 Coxeter group, and other subgroups.

The 3D picture are drawn as Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at position 0 are shown solid.

F4, [3,4,3] symmetry polytopes
# Name
Coxeter diagram
Schläfli symbol
Graph
Schlegel diagram Net
F4
[12]
B4
[8]
B3
[6]
B2
[4]
Octahedron
centered
Dual octahedron
centered
1 24-cell
(rectified 16-cell)
        =        
{3,4,3} = r{3,3,4}
             
2 rectified 24-cell
(cantellated 16-cell)
        =        
r{3,4,3} = rr{3,3,4}
             
3 truncated 24-cell
(cantitruncated 16-cell)
        =        
t{3,4,3} = tr{3,3,4}
             
4 cantellated 24-cell
       
rr{3,4,3}
             
5 cantitruncated 24-cell
       
tr{3,4,3}
             
6 runcitruncated 24-cell
       
t0,1,3{3,4,3}
             
[[3,3,3]] extended symmetries of F4
# Name
Coxeter diagram
Schläfli symbol
Graph
Schlegel diagram Net
F4
[[12]] = [24]
B4
[8]
B3
[6]
B2
[[4]] = [8]
Octahedron
centered
7 *runcinated 24-cell
       
t0,3{3,4,3}
           
8 *bitruncated 24-cell
       
2t{3,4,3}
           
9 *omnitruncated 24-cell
       
t0,1,2,3{3,4,3}
           
[3+,4,3] half symmetries of F4
# Name
Coxeter diagram
Schläfli symbol
Graph
Schlegel diagram Orthogonal
Projection
Net
F4
[12]+
B4
[8]
B3
[6]+
B2
[4]
Octahedron
centered
Dual octahedron
centered
Octahedron
centered
10 snub 24-cell
       
s{3,4,3}
             
11
Nonuniform
runcic snub 24-cell
       
s3{3,4,3}
       

Coordinates

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Vertex coordinates for all 15 forms are given below, including dual configurations from the two regular 24-cells. (The dual configurations are named in bold.) Active rings in the first and second nodes generate points in the first column. Active rings in the third and fourth nodes generate the points in the second column. The sum of each of these points are then permutated by coordinate positions, and sign combinations. This generates all vertex coordinates. Edge lengths are 2.

The only exception is the snub 24-cell, which is generated by half of the coordinate permutations, only an even number of coordinate swaps. φ=(5+1)/2.

24-cell family coordinates
# Base point(s)
t(0,1)
Base point(s)
t(2,3)
Schläfli symbol Name
Coxeter diagram
 
1 (0,0,1,1)2 {3,4,3} 24-cell        
2 (0,1,1,2)2 r{3,4,3} rectified 24-cell        
3 (0,1,2,3)2 t{3,4,3} truncated 24-cell        
10 (0,1,φ,φ+1)2 s{3,4,3} snub 24-cell        
 
2 (0,2,2,2)
(1,1,1,3)
r{3,4,3} rectified 24-cell        
4 (0,2,2,2) +
(1,1,1,3) +
(0,0,1,1)2
"
rr{3,4,3} cantellated 24-cell        
8 (0,2,2,2) +
(1,1,1,3) +
(0,1,1,2)2
"
2t{3,4,3} bitruncated 24-cell        
5 (0,2,2,2) +
(1,1,1,3) +
(0,1,2,3)2
"
tr{3,4,3} cantitruncated 24-cell        
 
1 (0,0,0,2)
(1,1,1,1)
{3,4,3} 24-cell        
7 (0,0,0,2) +
(1,1,1,1) +
(0,0,1,1)2
"
t0,3{3,4,3} runcinated 24-cell        
4 (0,0,0,2) +
(1,1,1,1) +
(0,1,1,2)2
"
t1,3{3,4,3} cantellated 24-cell        
6 (0,0,0,2) +
(1,1,1,1) +
(0,1,2,3)2
"
t0,1,3{3,4,3} runcitruncated 24-cell        
 
3 (1,1,1,5)
(1,3,3,3)
(2,2,2,4)
t{3,4,3} truncated 24-cell        
6 (1,1,1,5) +
(1,3,3,3) +
(2,2,2,4) +
(0,0,1,1)2
"
"
t0,2,3{3,4,3} runcitruncated 24-cell        
5 (1,1,1,5) +
(1,3,3,3) +
(2,2,2,4) +
(0,1,1,2)2
"
"
tr{3,4,3} cantitruncated 24-cell        
9 (1,1,1,5) +
(1,3,3,3) +
(2,2,2,4) +
(0,1,2,3)2
"
"
t0,1,2,3{3,4,3} Omnitruncated 24-cell        

References

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  • J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
  • H.S.M. Coxeter:
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • 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 Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
    • (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]
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
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Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 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