Acoustic guitar
Rectified 24-cell | ||
Schlegel diagram 8 of 24 cuboctahedral cells shown | ||
Type | Uniform polychoron | |
Schläfli symbol | r{3,4,3} rr{3,3,4} r{31,1,1} | |
Coxeter-Dynkin diagrams | Template:CDD Template:CDD Template:CDD or Template:CDD | |
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 is a uniform 4-dimensional polytope (or uniform polychoron), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by reducing the icositetrachoron's cells to cubes or cuboctahedra.
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 √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
Template:24-cell 4-cube Coxeter plane graphs
Stereographic projection | |
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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 | Template:CDD | Template:CDD | Template:CDD |
Facets | 3: Template:CDD 2: Template:CDD |
2,2: Template:CDD 2: Template:CDD |
1,1,1: Template:CDD 2: Template:CDD |
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
The rectified 24-cell can also be derived as a cantellated 16-cell: Template: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, editied 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)
- Template:PolyCell
- Template:KlitzingPolytopes