A5 polytope
 5-simplex   
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In 5-dimensional geometry, there are 19 uniform polytopes with A5 symmetry. There is one self-dual regular form, the 5-simplex with 6 vertices.
Each can be visualized as symmetric orthographic projections in the Coxeter planes of the A5 Coxeter group and other subgroups.
Graphs[edit]
Symmetric orthographic projections of these 19 polytopes can be made in the A5, A4, A3, A2 Coxeter planes. Ak 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] | ||
| A5 | A4 | A3 | A2 | ||
| 1 |
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{3,3,3,3} 5-simplex (hix) |
| 2 | 
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t1{3,3,3,3} or r{3,3,3,3} Rectified 5-simplex (rix) |
| 3 | 
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t2{3,3,3,3} or 2r{3,3,3,3} Birectified 5-simplex (dot) |
| 4 | 
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t0,1{3,3,3,3} or t{3,3,3,3} Truncated 5-simplex (tix) |
| 5 | 
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t1,2{3,3,3,3} or 2t{3,3,3,3} Bitruncated 5-simplex (bittix) |
| 6 | 
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t0,2{3,3,3,3} or rr{3,3,3,3} Cantellated 5-simplex (sarx) |
| 7 | 
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t1,3{3,3,3,3} or 2rr{3,3,3,3} Bicantellated 5-simplex (sibrid) |
| 8 | 
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t0,3{3,3,3,3} Runcinated 5-simplex (spix) |
| 9 | 
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t0,4{3,3,3,3} or 2r2r{3,3,3,3} Stericated 5-simplex (scad) |
| 10 | 
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t0,1,2{3,3,3,3} or tr{3,3,3,3} Cantitruncated 5-simplex (garx) |
| 11 | 
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t1,2,3{3,3,3,3} or 2tr{3,3,3,3} Bicantitruncated 5-simplex (gibrid) |
| 12 | 
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t0,1,3{3,3,3,3} Runcitruncated 5-simplex (pattix) |
| 13 | 
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t0,2,3{3,3,3,3} Runcicantellated 5-simplex (pirx) |
| 14 | 
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t0,1,4{3,3,3,3} Steritruncated 5-simplex (cappix) |
| 15 | 
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t0,2,4{3,3,3,3} Stericantellated 5-simplex (card) |
| 16 | 
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t0,1,2,3{3,3,3,3} Runcicantitruncated 5-simplex (gippix) |
| 17 | 
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t0,1,2,4{3,3,3,3} Stericantitruncated 5-simplex (cograx) |
| 18 | 
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t0,1,3,4{3,3,3,3} Steriruncitruncated 5-simplex (captid) |
| 19 | 
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t0,1,2,3,4{3,3,3,3} Omnitruncated 5-simplex (gocad) |
| A5 polytopes | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
t0 |
t1 |
t2 |
t0,1 |
t0,2 |
t1,2 |
t0,3 | |||||
t1,3 |
t0,4 |
t0,1,2 |
t0,1,3 |
t0,2,3 |
t1,2,3 |
t0,1,4 | |||||
t0,2,4 |
t0,1,2,3 |
t0,1,2,4 |
t0,1,3,4 |
t0,1,2,3,4 | |||||||
References[edit]
- 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, and 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[edit]
- Klitzing, Richard. "5D uniform polytopes (polytera)".