A_{5} polytope
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5-simplex |
In 5-dimensional geometry, there are 19 uniform polytopes with A_{5} symmetry. There is one self-dual regular form, the 5-simplex with 6 vertices.
Each can be visualized as symmetric orthographic projections in Coxeter planes of the A_{5} Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 19 polytopes can be made in the A_{5}, A_{4}, A_{3}, A_{2} Coxeter planes. A_{k} graphs have [k+1] symmetry. For even k and symmetrically nodea_1ed-diagrams, symmetry doubles to [2(k+1)].
These 19 polytopes are each shown in these 4 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
# | Coxeter plane graphs | Coxeter-Dynkin diagram Schläfli symbol Name | |||
---|---|---|---|---|---|
[6] | [5] | [4] | [3] | ||
A_{5} | A_{4} | A_{3} | A_{2} | ||
1 | {3,3,3,3} 5-simplex (hix) | ||||
2 | t_{1}{3,3,3,3} or r{3,3,3,3} Rectified 5-simplex (rix) | ||||
3 | t_{2}{3,3,3,3} or 2r{3,3,3,3} Birectified 5-simplex (dot) | ||||
4 | t_{0,1}{3,3,3,3} or t{3,3,3,3} Truncated 5-simplex (tix) | ||||
5 | t_{1,2}{3,3,3,3} or 2t{3,3,3,3} Bitruncated 5-simplex (bittix) | ||||
6 | t_{0,2}{3,3,3,3} or rr{3,3,3,3} Cantellated 5-simplex (sarx) | ||||
7 | t_{1,3}{3,3,3,3} or 2rr{3,3,3,3} Bicantellated 5-simplex (sibrid) | ||||
8 | t_{0,3}{3,3,3,3} Runcinated 5-simplex (spix) | ||||
9 | t_{0,4}{3,3,3,3} or 2r2r{3,3,3,3} Stericated 5-simplex (scad) | ||||
10 | t_{0,1,2}{3,3,3,3} or tr{3,3,3,3} Cantitruncated 5-simplex (garx) | ||||
11 | t_{1,2,3}{3,3,3,3} or 2tr{3,3,3,3} Bicantitruncated 5-simplex (gibrid) | ||||
12 | t_{0,1,3}{3,3,3,3} Runcitruncated 5-simplex (pattix) | ||||
13 | t_{0,2,3}{3,3,3,3} Runcicantellated 5-simplex (pirx) | ||||
14 | t_{0,1,4}{3,3,3,3} Steritruncated 5-simplex (cappix) | ||||
15 | t_{0,2,4}{3,3,3,3} Stericantellated 5-simplex (card) | ||||
16 | t_{0,1,2,3}{3,3,3,3} Runcicantitruncated 5-simplex (gippix) | ||||
17 | t_{0,1,2,4}{3,3,3,3} Stericantitruncated 5-simplex (cograx) | ||||
18 | t_{0,1,3,4}{3,3,3,3} Steriruncitruncated 5-simplex (captid) | ||||
19 | t_{0,1,2,3,4}{3,3,3,3} Omnitruncated 5-simplex (gocad) |
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 ^{[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]
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
External links
- Klitzing, Richard. "5D uniform polytopes (polytera)". https://bendwavy.org/klitzing/dimensions/polytera.htm.
Notes
Original source: https://en.wikipedia.org/wiki/A5 polytope.
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