A6 polytope
 6-simplex    |
In 6-dimensional geometry, there are 35 uniform polytopes with A6 symmetry. There is one self-dual regular form, the 6-simplex with 7 vertices.
Each can be visualized as symmetric orthographic projections in Coxeter planes of the A6 Coxeter group, and other subgroups.
Graphs
[edit]Symmetric orthographic projections of these 35 polytopes can be made in the A6, A5, A4, A3, A2 Coxeter planes. Ak graphs have [k+1] symmetry. For even k and symmetric ringed diagrams, symmetry doubles to [2(k+1)].
These 35 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.
| # | A6 [7] | A5 [6] | A4 [5] | A3 [4] | A2 [3] | Coxeter-Dynkin diagram Schläfli symbol Name |
|---|---|---|---|---|---|---|
| 1 | |  |  |  |  |    t0{3,3,3,3,3} 6-simplex Heptapeton (hop) |
| 2 |  |  |  |  |  | t1{3,3,3,3,3} Rectified 6-simplex Rectified heptapeton (ril) |
| 3 |  |  |  |  |  | t0,1{3,3,3,3,3} Truncated 6-simplex Truncated heptapeton (til) |
| 4 |  |  |  |  |  | t2{3,3,3,3,3} Birectified 6-simplex Birectified heptapeton (bril) |
| 5 |  |  |  |  |  | t0,2{3,3,3,3,3} Cantellated 6-simplex Small rhombated heptapeton (sril) |
| 6 |  |  |  |  |  | t1,2{3,3,3,3,3} Bitruncated 6-simplex Bitruncated heptapeton (batal) |
| 7 |  |  |  |  |  | t0,1,2{3,3,3,3,3} Cantitruncated 6-simplex Great rhombated heptapeton (gril) |
| 8 |  |  |  |  |  | t0,3{3,3,3,3,3} Runcinated 6-simplex Small prismated heptapeton (spil) |
| 9 |  |  |  |  |  | t1,3{3,3,3,3,3} Bicantellated 6-simplex Small birhombated heptapeton (sabril) |
| 10 |  |  |  |  |  | t0,1,3{3,3,3,3,3} Runcitruncated 6-simplex Prismatotruncated heptapeton (patal) |
| 11 |  |  |  |  |  | t2,3{3,3,3,3,3} Tritruncated 6-simplex Tetradecapeton (fe) |
| 12 |  |  |  |  |  | t0,2,3{3,3,3,3,3} Runcicantellated 6-simplex Prismatorhombated heptapeton (pril) |
| 13 |  |  |  |  |  | t1,2,3{3,3,3,3,3} Bicantitruncated 6-simplex Great birhombated heptapeton (gabril) |
| 14 |  |  |  |  |  | t0,1,2,3{3,3,3,3,3} Runcicantitruncated 6-simplex Great prismated heptapeton (gapil) |
| 15 |  |  |  |  |  | t0,4{3,3,3,3,3} Stericated 6-simplex Small cellated heptapeton (scal) |
| 16 |  |  |  |  |  | t1,4{3,3,3,3,3} Biruncinated 6-simplex Small biprismato-tetradecapeton (sibpof) |
| 17 |  |  |  |  |  | t0,1,4{3,3,3,3,3} Steritruncated 6-simplex cellitruncated heptapeton (catal) |
| 18 |  |  |  |  |  | t0,2,4{3,3,3,3,3} Stericantellated 6-simplex Cellirhombated heptapeton (cral) |
| 19 |  |  |  |  |  | t1,2,4{3,3,3,3,3} Biruncitruncated 6-simplex Biprismatorhombated heptapeton (bapril) |
| 20 |  |  |  |  |  | t0,1,2,4{3,3,3,3,3} Stericantitruncated 6-simplex Celligreatorhombated heptapeton (cagral) |
| 21 |  |  |  |  |  | t0,3,4{3,3,3,3,3} Steriruncinated 6-simplex Celliprismated heptapeton (copal) |
| 22 |  |  |  |  |  | t0,1,3,4{3,3,3,3,3} Steriruncitruncated 6-simplex celliprismatotruncated heptapeton (captal) |
| 23 |  |  |  |  |  | t0,2,3,4{3,3,3,3,3} Steriruncicantellated 6-simplex celliprismatorhombated heptapeton (copril) |
| 24 |  |  |  |  |  | t1,2,3,4{3,3,3,3,3} Biruncicantitruncated 6-simplex Great biprismato-tetradecapeton (gibpof) |
| 25 |  |  |  |  |  | t0,1,2,3,4{3,3,3,3,3} Steriruncicantitruncated 6-simplex Great cellated heptapeton (gacal) |
| 26 |  |  |  |  |  | t0,5{3,3,3,3,3} Pentellated 6-simplex Small teri-tetradecapeton (staf) |
| 27 |  |  |  |  |  | t0,1,5{3,3,3,3,3} Pentitruncated 6-simplex Tericellated heptapeton (tocal) |
| 28 |  |  |  |  |  | t0,2,5{3,3,3,3,3} Penticantellated 6-simplex Teriprismated heptapeton (tapal) |
| 29 |  |  |  |  |  | t0,1,2,5{3,3,3,3,3} Penticantitruncated 6-simplex Terigreatorhombated heptapeton (togral) |
| 30 |  |  |  |  |  | t0,1,3,5{3,3,3,3,3} Pentiruncitruncated 6-simplex Tericellirhombated heptapeton (tocral) |
| 31 |  |  |  |  |  | t0,2,3,5{3,3,3,3,3} Pentiruncicantellated 6-simplex Teriprismatorhombi-tetradecapeton (taporf) |
| 32 |  |  |  |  |  | t0,1,2,3,5{3,3,3,3,3} Pentiruncicantitruncated 6-simplex Terigreatoprismated heptapeton (tagopal) |
| 33 |  |  |  |  |  | t0,1,4,5{3,3,3,3,3} Pentisteritruncated 6-simplex tericellitrunki-tetradecapeton (tactaf) |
| 34 |  |  |  |  |  | t0,1,2,4,5{3,3,3,3,3} Pentistericantitruncated 6-simplex tericelligreatorhombated heptapeton (tacogral) |
| 35 |  |  |  |  |  | t0,1,2,3,4,5{3,3,3,3,3} Omnitruncated 6-simplex Great teri-tetradecapeton (gotaf) |
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, 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. "6D uniform polytopes (polypeta)".