User:Tomruen/List of D5 polytopes
{{See|D5_polytope}}
{{DISPLAYTITLE:{{user other|{{NAMESPACE}}:{{ROOTPAGENAME}}/}}List of D5 polytopes}}
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|+ Orthographic projections in the D5 Coxeter plane |
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In 5-dimensional geometry, there are 23 uniform polytopes with D5 symmetry, 8 are unique, and 15 are shared with the B5 symmetry. There are two special forms, the 5-orthoplex, and 5-demicube with 10 and 16 vertices respectively.
They can be visualized as symmetric orthographic projections in Coxeter planes of the D5 Coxeter group, and other subgroups.
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Graphs
Symmetric orthographic projections of these 8 polytopes can be made in the D5, D4, D3, A3, Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry. The B5 plane is included, with only half the [10] symmetry displayed.
These 8 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.
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!rowspan=3|# !colspan=5|Coxeter plane projections !rowspan=3|Coxeter-Dynkin diagram |
| [10/2]
![8] ![6] ![4] ![4] |
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| B5
!D5 !D4 !D3 !A3 |
| 1
|80px |80px |80px |80px |80px | {{CDD|nodes_10ru|split2|node|3|node|3|node}} (121) |
| 2
|80px |80px |80px |80px |80px | {{CDD|nodes_10ru|split2|node_1|3|node|3|node}} t0,1(121) |
| 3
|80px |80px |80px |80px |80px | {{CDD|nodes_10ru|split2|node|3|node_1|3|node}} t0,2(121) |
| 4
|80px |80px |80px |80px |80px | {{CDD|nodes_10ru|split2|node|3|node|3|node_1}} t0,3(121) |
| 5
|80px |80px |80px |80px |80px | {{CDD|nodes_10ru|split2|node_1|3|node_1|3|node}} t0,1,2(121) |
| 6
|80px |80px |80px |80px |80px | {{CDD|nodes_10ru|split2|node_1|3|node|3|node_1}} t0,1,3(121) |
| 7
|80px |80px |80px |80px |80px | {{CDD|nodes_10ru|split2|node|3|node_1|3|node_1}} t0,2,3(121) |
| 8
|80px |80px |80px |80px |80px | {{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1}} t0,1,2,3(121) |
References
- 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, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html 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
- {{KlitzingPolytopes|polytera.htm|5D|uniform polytopes (polytera)}}
Notes
{{reflist}}
{{Polytopes}}