Cantellated 24-cells

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24-cell t0 F4.svg
24-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
24-cell t02 F4.svg
Cantellated 24-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
24-cell t012 F4.svg
Cantitruncated 24-cell
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
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[edit]

Cantellated 24-cell
Type Uniform 4-polytope
Schläfli symbol rr{3,4,3}
s2{3,4,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells 144 24 Small rhombicuboctahedron.png(3.4.4.4)
24 Cuboctahedron.png(3.4.3.4)

96 Triangular prism.png(3.4.4)

Faces 720 288 triangles
432 squares
Edges 864
Vertices 288
Vertex figure Cantellated 24-cell verf.png
Irreg. triangular prism
Symmetry group F4, [3,4,3]
Properties convex
Uniform index 24 25 26

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[edit]

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[edit]

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[edit]

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[edit]

A half-symmetry construction of the cantellated 24-cell, also called a cantic snub 24-cell, as CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png, 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:

Cantellated 24-cell verf.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Cantic snub 24-cell verf.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Coxeter plane F4
Graph 24-cell t02 F4.svg
Dihedral symmetry [12]
Coxeter plane B3 / A2 (a) B3 / A2 (b)
Graph 24-cell t02 B3.svg 24-cell t13 B3.svg
Dihedral symmetry [6] [6]
Coxeter plane B4 B2 / A3
Graph 24-cell t02 B4.svg 24-cell t02 B2.svg
Dihedral symmetry [8] [4]
Schlegel diagrams
Cantel 24cell1.png
Schlegel diagram
Cantel 24cell2.png
Showing 24 cuboctahedra.
Cantel 24cell3.png
Showing 96 triangular prisms.

Cantitruncated 24-cell[edit]

Cantitruncated 24-cell
Cantitruncated 24-cell schlegel halfsolid.png
Schlegel diagram, centered on truncated cuboctahedron
Type Uniform 4-polytope
Schläfli symbol tr{3,4,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells 144 24 4.6.8 Great rhombicuboctahedron.png
96 4.4.3 Triangular prism.png
24 3.8.8 Truncated hexahedron.png
Faces 720 192{3}
288{4}
96{6}
144{8}
Edges 1152
Vertices 576
Vertex figure Cantitruncated 24-cell verf.png
sphenoid
Symmetry group F4, [3,4,3]
Properties convex
Uniform index 27 28 29

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[edit]

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[edit]

orthographic projections
Coxeter plane F4
Graph 24-cell t123 F4.svg
Dihedral symmetry [12]
Coxeter plane B3 / A2 (a) B3 / A2 (b)
Graph 24-cell t123 B3.svg 24-cell t0123 B3.svg
Dihedral symmetry [6] [6]
Coxeter plane B4 B2 / A3
Graph 24-cell t123 B4.svg 24-cell t123 B2.svg
Dihedral symmetry [8] [4]
Stereographic projection
Cantitruncated 24 cell.png

Related polytopes[edit]

References[edit]

  • 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-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)
  • 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 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