- Cartesian coordinates
- Images
- Symmetry constructions
- Alternate names
- Related uniform polytopes
- References
| Rectified 24-cell | ||
Schlegel diagram 8 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 | {{CDD|node|3|node_1|4|node|3|node}} {{CDD|node_1|3|node|3|node_1|4|node}} {{CDD|node_1|3|node|split1|nodes_11}} or {{CDD|node|splitsplit1|branch3_11|node_1}} | |
| 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 1152 B4 [3,3,4], order 384 D4 [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 {{radic|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
{{24-cell_4-cube_Coxeter_plane_graphs|t1|100|t2}}| 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 | node|3|node_1|4|node|3|node}} | node|4|node_1|3|node|3|node_1}} | nodes_11|split2|node|3|node_1}} |
| Facets | 3: {{CDD>node|3|node_1|4|node}} 2: {{CDD|node_1|4|node|3|node}} | 2,2: {{CDD>node_1|3|node|3|node_1}} 2: {{CDD|node|4|node_1|2|node_1}} | 1,1,1: {{CDD>node_1|3|node|3|node_1}} 2: {{CDD|node_1|2|node_1|2|node_1}} |
| Vertex figure |
Alternate names
- Rectified 24-cell, Cantellated 16-cell (Norman Johnson)
- Rectified icositetrachoron (Acronym rico) (George Olshevsky, Jonathan Bowers)
- Cantellated hexadecachoron
- Disicositetrachoron
- Amboicositetrachoron (Neil Sloane & John Horton Conway)
Related uniform polytopes
{{Demitesseract family}}{{24-cell_family}}The rectified 24-cell can also be derived as a cantellated 16-cell:
{{Tesseract family}}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)
- {{PolyCell | urlname = section2.html| title = 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 23}}
- {{PolyCell | urlname = section3.html| title = 3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 23}}
- {{PolyCell | urlname = section7.html| title = 7. Uniform polychora derived from glomeric tetrahedron B4 - Model 23 }}
- {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|o3x4o3o - rico}}
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