Quarter 6-cubic honeycomb
| quarter 6-cubic honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform 6-honeycomb |
| Family | Quarter hypercubic honeycomb |
| Schläfli symbol | q{4,3,3,3,3,4} |
| Coxeter-Dynkin diagram |       =   |
| 5-face type | h{4,34},  h4{4,34},  {3,3}×{3,3} duoprism |
| Vertex figure | |
| Coxeter group |  ×2 = [[3<sup>1,1</sup>,3,3,3<sup>1,1</sup>]] |
| Dual | |
| Properties | vertex-transitive |
In six-dimensional Euclidean geometry, the quarter 6-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 6-demicubic honeycomb, and a quarter of the vertices of a 6-cube honeycomb.[1] Its facets are 6-demicubes, stericated 6-demicubes, and {3,3}×{3,3} duoprisms.
Related honeycombs
This honeycomb is one of 41 uniform honycombs constructed by the Coxeter group, all but 6 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 41 permutations are listed with its highest extended symmetry, and related 
and 
constructions:
| D6 honeycombs | |||
|---|---|---|---|
| Extended symmetry |
Extended diagram |
Order | Honeycombs |
| [31,1,3,3,31,1] |  |
×1 |  , |
| [[31,1,3,3,31,1]] |    |
×2 | , , , |
| <[31,1,3,3,31,1]> ↔ [31,1,3,3,3,4] |
  ↔  |
×2 | , , , , , , , ,
|
| <2[31,1,3,3,31,1]> ↔ [4,3,3,3,3,4] |
   ↔  |
×4 | ,,
|
| [<2[31,1,3,3,31,1]>] ↔ [[4,3,3,3,3,4]] |
↔ |
×8 | , , ,
|
See also
Regular and uniform honeycombs in 5-space:
- 6-cube honeycomb
- 6-demicube honeycomb
- 6-simplex honeycomb
- Truncated 6-simplex honeycomb
- Omnitruncated 6-simplex honeycomb
Notes
- ↑ Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318
References
- 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
- Richard Klitzing, 6D, Euclidean tesselations#6D
| 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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