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
[edit]| 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 |
| 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 | |
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 , 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
[edit]| 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 diagram |  Showing 24 cuboctahedra. |  Showing 96 triangular prisms. |
Related polytopes
[edit]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
[edit]| 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 | |
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]| 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] |
 |
Related polytopes
[edit]| 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
[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-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