Cyclotruncated 6-simplex honeycomb
| Cyclotruncated 6-simplex honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform honeycomb |
| Family | Cyclotruncated simplectic honeycomb |
| Schläfli symbol | t0,1{3[7]} |
| Coxeter diagram |      |
| 6-face types | {35}  t{35}  2t{35}  3t{35}  |
| Vertex figure | Elongated 5-simplex antiprism |
| Symmetry |  ×2, [[3[7]]] |
| Properties | vertex-transitive |
In six-dimensional Euclidean geometry, the cyclotruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.
It can be constructed by seven sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-simplex honeycomb divisions on each hyperplane.
Related polytopes and honeycombs
This honeycomb is one of 17 unique uniform honeycombs[1] constructed by
the Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:
A6 honeycombs
| Heptagon symmetry |
Extended symmetry |
Extended diagram |
Extended group |
Honeycombs |
|---|---|---|---|---|
| a1 | [3[7]] |   |
|
|
| i2 | [[3[7]]] |     |
×2 |
|
| r14 | [7[3[7]]] |   |
×14 |
Template:Hidden A6
See also
Regular and uniform honeycombs in 6-space:
- 6-cubic honeycomb
- 6-demicubic honeycomb
- 6-simplex honeycomb
- Omnitruncated 6-simplex honeycomb
- 222 honeycomb
Notes
- ↑
- Weisstein, Eric W., "Necklace", MathWorld., A000029 18-1 cases, skipping one with zero marks
References
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- 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
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
| Fundamental convex regular and uniform honeycombs in dimensions 2–10 | |||||
|---|---|---|---|---|---|
| Family |  |
 |
 |
 |
 /  /  |
| Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
| Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
| Uniform 5-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
| Uniform 6-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
| Uniform 7-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
| Uniform 8-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
| Uniform 9-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| Uniform n-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |
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