Wikipedia

Cantellated 24-cells


24-cell
Cantellated 24-cell
Cantitruncated 24-cell
Orthogonal projections in F4 Coxeter plane

In four-dimensional geometry, a cantellated 24-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 24-cell.

There are 2 unique degrees of cantellations of the 24-cell including permutations with truncations.

Cantellated 24-cell

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Cantellated 24-cell
Type Uniform 4-polytope
Schläfli symbol rr{3,4,3}
s2{3,4,3}
Coxeter diagram        
       
Cells 144 24  (3.4.4.4)
24  (3.4.3.4)

96  (3.4.4)

Faces 720 288 triangles
432 squares
Edges 864
Vertices 288
Vertex figure  
Wedge
Symmetry group F4, [3,4,3], order 1152
Properties convex
Uniform index 24 25 26
 
Net

The cantellated 24-cell or small rhombated icositetrachoron is a uniform 4-polytope.

The boundary of the cantellated 24-cell is composed of 24 truncated octahedral cells, 24 cuboctahedral cells and 96 triangular prisms. Together they have 288 triangular faces, 432 square faces, 864 edges, and 288 vertices.

Construction

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When the cantellation process is applied to 24-cell, each of the 24 octahedra becomes a small rhombicuboctahedron. In addition however, since each octahedra's edge was previously shared with two other octahedra, the separating edges form the three parallel edges of a triangular prism - 96 triangular prisms, since the 24-cell contains 96 edges. Further, since each vertex was previously shared with 12 faces, the vertex would split into 12 (24*12=288) new vertices. Each group of 12 new vertices forms a cuboctahedron.

Coordinates

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The Cartesian coordinates of the vertices of the cantellated 24-cell having edge length 2 are all permutations of coordinates and sign of:

(0, √2, √2, 2+2√2)
(1, 1+√2, 1+√2, 1+2√2)

The permutations of the second set of coordinates coincide with the vertices of an inscribed runcitruncated tesseract.

The dual configuration has all permutations and signs of:

(0,2,2+√2,2+√2)
(1,1,1+√2,3+√2)

Structure

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The 24 small rhombicuboctahedra are joined to each other via their triangular faces, to the cuboctahedra via their axial square faces, and to the triangular prisms via their off-axial square faces. The cuboctahedra are joined to the triangular prisms via their triangular faces. Each triangular prism is joined to two cuboctahedra at its two ends.

Cantic snub 24-cell

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A half-symmetry construction of the cantellated 24-cell, also called a cantic snub 24-cell, as        , has an identical geometry, but its triangular faces are further subdivided. The cantellated 24-cell has 2 positions of triangular faces in ratio of 96 and 192, while the cantic snub 24-cell has 3 positions of 96 triangles.

The difference can be seen in the vertex figures, with edges representing faces in the 4-polytope:

 
        		 
       

Images

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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]
Schlegel diagrams
 
Schlegel diagram 		 
Showing 24 cuboctahedra. 		 
Showing 96 triangular prisms.
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The convex hull of two cantellated 24-cells in opposite positions is a nonuniform polychoron composed of 864 cells: 48 cuboctahedra, 144 square antiprisms, 384 octahedra (as triangular antipodiums), 288 tetrahedra (as tetragonal disphenoids), and 576 vertices. Its vertex figure is a shape topologically equivalent to a cube with a triangular prism attached to one of its square faces.

Cantitruncated 24-cell

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Cantitruncated 24-cell
 
Schlegel diagram, centered on truncated cuboctahedron
Type Uniform 4-polytope
Schläfli symbol tr{3,4,3}
Coxeter diagram        
Cells 144 24 4.6.8  
96 4.4.3  
24 3.8.8  
Faces 720 192{3}
288{4}
96{6}
144{8}
Edges 1152
Vertices 576
Vertex figure  
sphenoid
Symmetry group F4, [3,4,3], order 1152
Properties convex
Uniform index 27 28 29
 
Net

The cantitruncated 24-cell or great rhombated icositetrachoron is a uniform 4-polytope derived from the 24-cell. It is bounded by 24 truncated cuboctahedra corresponding with the cells of a 24-cell, 24 truncated cubes corresponding with the cells of the dual 24-cell, and 96 triangular prisms corresponding with the edges of the first 24-cell.

Coordinates

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The Cartesian coordinates of a cantitruncated 24-cell having edge length 2 are all permutations of coordinates and sign of:

(1,1+√2,1+2√2,3+3√2)
(0,2+√2,2+2√2,2+3√2)

The dual configuration has coordinates as all permutations and signs of:

(1,1+√2,1+√2,5+2√2)
(1,3+√2,3+√2,3+2√2)
(2,2+√2,2+√2,4+2√2)

Projections

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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
 
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24-cell family polytopes
Name 24-cell truncated 24-cell snub 24-cell rectified 24-cell cantellated 24-cell bitruncated 24-cell cantitruncated 24-cell runcinated 24-cell runcitruncated 24-cell omnitruncated 24-cell
Schläfli
symbol 		{3,4,3} t0,1{3,4,3}
t{3,4,3} 		s{3,4,3} t1{3,4,3}
r{3,4,3} 		t0,2{3,4,3}
rr{3,4,3} 		t1,2{3,4,3}
2t{3,4,3} 		t0,1,2{3,4,3}
tr{3,4,3} 		t0,3{3,4,3} t0,1,3{3,4,3} t0,1,2,3{3,4,3}
Coxeter
diagram 		                                                                               
Schlegel
diagram 		                   
F4                    
B4                    
B3(a)                    
B3(b)            
B2                    

References

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  • 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 [1]
      • (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-Strauss, 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)
  • 3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 24, 25, George Olshevsky.
  • Klitzing, Richard. "4D uniform polytopes (polychora)". x3o4x3o - srico, o3x4x3o - grico
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 polychoron Pentachoron 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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