Английская Википедия:A4 polytope
| Файл:4-simplex t0.svg 5-cell Шаблон:CDD |
In 4-dimensional geometry, there are 9 uniform polytopes with A4 symmetry. There is one self-dual regular form, the 5-cell with 5 vertices.
Symmetry
A4 symmetry, or [3,3,3] is order 120, with Conway quaternion notation +1/60[I×Шаблон:Overline].21. Its abstract structure is the symmetric group S5. Three forms with symmetric Coxeter diagrams have extended symmetry, [[3,3,3]] of order 240, and Conway notation ±1/60[I×Шаблон:Overline].2, and abstract structure S5×C2.
Visualizations
Each can be visualized as symmetric orthographic projections in Coxeter planes of the A4 Coxeter group, and other subgroups. Three Coxeter plane 2D projections are given, for the A4, A3, A2 Coxeter groups, showing symmetry order 5,4,3, and doubled on even Ak orders to 10,4,6 for symmetric Coxeter diagrams.
The 3D picture are drawn as Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at position 0 are shown solid.
| # | Name | Coxeter diagram and Schläfli symbols |
Coxeter plane graphs | Schlegel diagram | Net | ||
|---|---|---|---|---|---|---|---|
| A4 [[5]] = [10] |
A3 [4] |
A2 [[3]] = [6] |
Tetrahedron centered | ||||
| 5 | *runcinated 5-cell | Шаблон:CDD t0,3{3,3,3} |
Файл:4-simplex t03.svg | Файл:4-simplex t03 A3.svg | Файл:4-simplex t03 A2.svg | Файл:Schlegel half-solid runcinated 5-cell.png | Файл:Small prismatodecachoron net.png |
| 6 | *bitruncated 5-cell decachoron |
Шаблон:CDD 2t{3,3,3} |
Файл:4-simplex t12.svg | Файл:4-simplex t12 A3.svg | Файл:4-simplex t12 A2.svg | Файл:Schlegel half-solid bitruncated 5-cell.png | Файл:Decachoron net.png |
| 9 | *omnitruncated 5-cell | Шаблон:CDD t0,1,2,3{3,3,3} |
Файл:4-simplex t0123.svg | Файл:4-simplex t0123 A3.svg | Файл:4-simplex t0123 A2.svg | Файл:Schlegel half-solid omnitruncated 5-cell.png | Файл:Great prismatodecachoron net.png |
Coordinates
The coordinates of uniform 4-polytopes with pentachoric symmetry can be generated as permutations of simple integers in 5-space, all in hyperplanes with normal vector (1,1,1,1,1). The A4 Coxeter group is palindromic, so repeated polytopes exist in pairs of dual configurations. There are 3 symmetric positions, and 6 pairs making the total 15 permutations of one or more rings. All 15 are listed here in order of binary arithmetic for clarity of the coordinate generation from the rings in each corresponding Coxeter diagram.
The number of vertices can be deduced here from the permutations of the number of coordinates, peaking at 5 factorial for the omnitruncated form with 5 unique coordinate values.
| # | Base point | Name (symmetric name) |
Coxeter diagram | Vertices | |
|---|---|---|---|---|---|
| 1 | (0, 0, 0, 0, 1) (1, 1, 1, 1, 0) |
5-cell Trirectified 5-cell |
Шаблон:CDD Шаблон:CDD |
5 | 5!/(4!) |
| 2 | (0, 0, 0, 1, 1) (1, 1, 1, 0, 0) |
Rectified 5-cell Birectified 5-cell |
Шаблон:CDD Шаблон:CDD |
10 | 5!/(3!2!) |
| 3 | (0, 0, 0, 1, 2) (2, 2, 2, 1, 0) |
Truncated 5-cell Tritruncated 5-cell |
Шаблон:CDD Шаблон:CDD |
20 | 5!/(3!) |
| 5 | (0, 1, 1, 1, 2) | Runcinated 5-cell | Шаблон:CDD | 20 | 5!/(3!) |
| 4 | (0, 0, 1, 1, 2) (2, 2, 1, 1, 0) |
Cantellated 5-cell Bicantellated 5-cell |
Шаблон:CDD Шаблон:CDD |
30 | 5!/(2!2!) |
| 6 | (0, 0, 1, 2, 2) | Bitruncated 5-cell | Шаблон:CDD | 30 | 5!/(2!2!) |
| 7 | (0, 0, 1, 2, 3) (3, 3, 2, 1, 0) |
Cantitruncated 5-cell Bicantitruncated 5-cell |
Шаблон:CDD Шаблон:CDD |
60 | 5!/2! |
| 8 | (0, 1, 1, 2, 3) (3, 2, 2, 1, 0) |
Runcitruncated 5-cell Runcicantellated 5-cell |
Шаблон:CDD Шаблон:CDD |
60 | 5!/2! |
| 9 | (0, 1, 2, 3, 4) | Omnitruncated 5-cell | Шаблон:CDD | 120 | 5! |
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
