Английская Википедия:B4 polytope
| Файл:4-cube t0.svg Tesseract Шаблон:CDD |
Файл:4-cube t3.svg 16-cell Шаблон:CDD |
In 4-dimensional geometry, there are 15 uniform 4-polytopes with B4 symmetry. There are two regular forms, the tesseract and 16-cell, with 16 and 8 vertices respectively.
Visualizations
They can be visualized as symmetric orthographic projections in Coxeter planes of the B5 Coxeter group, and other subgroups.
Symmetric orthographic projections of these 32 polytopes can be made in the B5, B4, B3, B2, A3, Coxeter planes. Ak has [k+1] symmetry, and Bk has [2k] symmetry.
These 32 polytopes are each shown in these 5 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
The pictures are drawn as Schlegel diagram perspective projections, centered on the cell at pos. 3, with a consistent orientation, and the 16 cells at position 0 are shown solid, alternately colored.
| # | Name | Coxeter plane projections | Schlegel diagrams |
Net | |||||
|---|---|---|---|---|---|---|---|---|---|
| F4 [12] |
B4 [8] |
B3 [6] |
B2 [4] |
A3 [4] |
Cube centered |
Tetrahedron centered | |||
| 13 | *rectified 16-cell (Same as 24-cell) Шаблон:CDD = Шаблон:CDD r{3,3,4} = {3,4,3} |
Файл:24-cell t3 F4.svg | Файл:24-cell t0 B4.svg | Файл:24-cell t3 B3.svg | Файл:24-cell t3 B2.svg | Файл:24-cell t0 B2.svg | Файл:Schlegel half-solid rectified 16-cell.png | Файл:24-cell net.png | |
| 14 | *cantellated 16-cell (Same as rectified 24-cell) Шаблон:CDD = Шаблон:CDD rr{3,3,4} = r{3,4,3} |
Файл:24-cell t2 F4.svg | Файл:24-cell t1 B4.svg | Файл:24-cell t2 B3.svg | Файл:24-cell t2 B2.svg | Файл:24-cell t1 B2.svg | Файл:Schlegel half-solid cantellated 16-cell.png | Файл:Rectified icositetrachoron net.png | |
| 15 | *cantitruncated 16-cell (Same as truncated 24-cell) Шаблон:CDD = Шаблон:CDD tr{3,3,4} = t{3,4,3} |
Файл:24-cell t23 F4.svg | Файл:4-cube t123.svg | Файл:24-cell t23 B3.svg | Файл:4-cube t123 B2.svg | Файл:24-cell t01 B2.svg | Файл:Schlegel half-solid cantitruncated 16-cell.png | Файл:Truncated icositetrachoron net.png | |
| # | Name | Coxeter plane projections | Schlegel diagrams |
Net | |||||
|---|---|---|---|---|---|---|---|---|---|
| F4 [12] |
B4 [8] |
B3 [6] |
B2 [4] |
A3 [4] |
Cube centered |
Tetrahedron centered | |||
| 16 | alternated cantitruncated 16-cell (Same as the snub 24-cell) Шаблон:CDD = Шаблон:CDD sr{3,3,4} = s{3,4,3} |
Файл:24-cell h01 F4.svg | Файл:24-cell h01 B4.svg | Файл:24-cell h01 B3.svg | Файл:24-cell h01 B2.svg | Файл:Schlegel half-solid alternated cantitruncated 16-cell.png | Файл:Snub disicositetrachoron net.png | ||
Coordinates
The tesseractic family of 4-polytopes are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform 4-polytopes. All coordinates correspond with uniform 4-polytopes of edge length 2.
| # | Base point | Name | Coxeter diagram | Vertices | |
|---|---|---|---|---|---|
| 3 | (0,0,0,1)Шаблон:Radic | 16-cell | Шаблон:CDD | 8 | 24-34!/3! |
| 1 | (1,1,1,1) | Tesseract | Шаблон:CDD | 16 | 244!/4! |
| 13 | (0,0,1,1)Шаблон:Radic | Rectified 16-cell (24-cell) | Шаблон:CDD | 24 | 24-24!/(2!2!) |
| 2 | (0,1,1,1)Шаблон:Radic | Rectified tesseract | Шаблон:CDD | 32 | 244!/(3!2!) |
| 8 | (0,0,1,2)Шаблон:Radic | Truncated 16-cell | Шаблон:CDD | 48 | 24-24!/2! |
| 6 | (1,1,1,1) + (0,0,0,1)Шаблон:Radic | Runcinated tesseract | Шаблон:CDD | 64 | 244!/3! |
| 4 | (1,1,1,1) + (0,1,1,1)Шаблон:Radic | Truncated tesseract | Шаблон:CDD | 64 | 244!/3! |
| 14 | (0,1,1,2)Шаблон:Radic | Cantellated 16-cell (rectified 24-cell) | Шаблон:CDD | 96 | 244!/(2!2!) |
| 7 | (0,1,2,2)Шаблон:Radic | Bitruncated 16-cell | Шаблон:CDD | 96 | 244!/(2!2!) |
| 5 | (1,1,1,1) + (0,0,1,1)Шаблон:Radic | Cantellated tesseract | Шаблон:CDD | 96 | 244!/(2!2!) |
| 15 | (0,1,2,3)Шаблон:Radic | cantitruncated 16-cell (truncated 24-cell) | Шаблон:CDD | 192 | 244!/2! |
| 11 | (1,1,1,1) + (0,0,1,2)Шаблон:Radic | Runcitruncated 16-cell | Шаблон:CDD | 192 | 244!/2! |
| 10 | (1,1,1,1) + (0,1,1,2)Шаблон:Radic | Runcitruncated tesseract | Шаблон:CDD | 192 | 244!/2! |
| 9 | (1,1,1,1) + (0,1,2,2)Шаблон:Radic | Cantitruncated tesseract | Шаблон:CDD | 192 | 244!/2! |
| 12 | (1,1,1,1) + (0,1,2,3)Шаблон:Radic | Omnitruncated 16-cell | Шаблон:CDD | 384 | 244! |
References
- J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Шаблон:ISBN (Chapter 26)
- H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- 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, Шаблон:ISBN Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
- (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]
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
External links
- Шаблон:KlitzingPolytopes
- Uniform, convex polytopes in four dimensions:, Marco Möller Шаблон:In lang
- Шаблон:PolyCell
