Cyclotruncated 6-simplex honeycomb | |
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(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.
Structure
editIt 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
editThis 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 |
See also
editRegular and uniform honeycombs in 6-space:
Notes
edit- ^ * Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 18-1 cases, skipping one with zero marks
References
edit- 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 [1]
- (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]
Space | Family | / / | ||||
---|---|---|---|---|---|---|
E2 | Uniform tiling | 0[3] | δ3 | hδ3 | qδ3 | Hexagonal |
E3 | Uniform convex honeycomb | 0[4] | δ4 | hδ4 | qδ4 | |
E4 | Uniform 4-honeycomb | 0[5] | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
E5 | Uniform 5-honeycomb | 0[6] | δ6 | hδ6 | qδ6 | |
E6 | Uniform 6-honeycomb | 0[7] | δ7 | hδ7 | qδ7 | 222 |
E7 | Uniform 7-honeycomb | 0[8] | δ8 | hδ8 | qδ8 | 133 • 331 |
E8 | Uniform 8-honeycomb | 0[9] | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
E9 | Uniform 9-honeycomb | 0[10] | δ10 | hδ10 | qδ10 | |
E10 | Uniform 10-honeycomb | 0[11] | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)-honeycomb | 0[n] | δn | hδn | qδn | 1k2 • 2k1 • k21 |