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{4,3^{4}}, h_{4}{4,3^{4}}, {3,3}×{3,3} duoprism |
Vertex figure | |
Coxeter group | ×2 = [[3^{1,1},3,3,3^{1,1}]] |
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.
This honeycomb is one of 41 uniform honeycombs 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 |
[3^{1,1},3,3,3^{1,1}] | ×1 | , | |
[[3^{1,1},3,3,3^{1,1}]] | ×2 | , , , | |
<[3^{1,1},3,3,3^{1,1}]> ↔ [3^{1,1},3,3,3,4] |
↔ |
×2 | , , , , , , , ,
, , , , , , , |
<2[3^{1,1},3,3,3^{1,1}]> ↔ [4,3,3,3,3,4] |
↔ |
×4 | , ,
, , , , , , , , , |
[<2[3^{1,1},3,3,3^{1,1}]>] ↔ [[4,3,3,3,3,4]] |
↔ |
×8 | , , ,
, , , |
Regular and uniform honeycombs in 5-space:
Space | Family | / / | ||||
---|---|---|---|---|---|---|
E^{2} | Uniform tiling | 0_{[3]} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |
E^{3} | Uniform convex honeycomb | 0_{[4]} | δ_{4} | hδ_{4} | qδ_{4} | |
E^{4} | Uniform 4-honeycomb | 0_{[5]} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |
E^{5} | Uniform 5-honeycomb | 0_{[6]} | δ_{6} | hδ_{6} | qδ_{6} | |
E^{6} | Uniform 6-honeycomb | 0_{[7]} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |
E^{7} | Uniform 7-honeycomb | 0_{[8]} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |
E^{8} | Uniform 8-honeycomb | 0_{[9]} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |
E^{9} | Uniform 9-honeycomb | 0_{[10]} | δ_{10} | hδ_{10} | qδ_{10} | |
E^{10} | Uniform 10-honeycomb | 0_{[11]} | δ_{11} | hδ_{11} | qδ_{11} | |
E^{n-1} | Uniform (n-1)-honeycomb | 0_{[n]} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |