quarter 5-cubic honeycomb | |
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(No image) | |
Type | Uniform 5-honeycomb |
Family | Quarter hypercubic honeycomb |
Schläfli symbol | q{4,3,3,3,4} |
Coxeter-Dynkin diagram | = |
5-face type | h{4,3^{3}}, h_{4}{4,3^{3}}, |
Vertex figure | Rectified 5-cell antiprism or Stretched birectified 5-simplex |
Coxeter group | ×2 = [[3^{1,1},3,3^{1,1}]] |
Dual | |
Properties | vertex-transitive |
In five-dimensional Euclidean geometry, the quarter 5-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 5-demicubic honeycomb, and a quarter of the vertices of a 5-cube honeycomb.^{[1]} Its facets are 5-demicubes and runcinated 5-demicubes.
This honeycomb is one of 20 uniform honeycombs constructed by the Coxeter group, all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 20 permutations are listed with its highest extended symmetry relation:
D5 honeycombs | |||
---|---|---|---|
Extended symmetry |
Extended diagram |
Extended group |
Honeycombs |
[3^{1,1},3,3^{1,1}] | |||
<[3^{1,1},3,3^{1,1}]> ↔ [3^{1,1},3,3,4] |
↔ |
×2_{1} = | , , ,
, , , |
[[3^{1,1},3,3^{1,1}]] | ×2_{2} | , | |
<2[3^{1,1},3,3^{1,1}]> ↔ [4,3,3,3,4] |
↔ |
×4_{1} = | , , , , , |
[<2[3^{1,1},3,3^{1,1}]>] ↔ [[4,3,3,3,4]] |
↔ |
×8 = ×2 | , , |
Regular and uniform honeycombs in 5-space:
Space | Family | / / | ||||
---|---|---|---|---|---|---|
E^{2} | Uniform tiling | {3^{[3]}} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |
E^{3} | Uniform convex honeycomb | {3^{[4]}} | δ_{4} | hδ_{4} | qδ_{4} | |
E^{4} | Uniform 4-honeycomb | {3^{[5]}} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |
E^{5} | Uniform 5-honeycomb | {3^{[6]}} | δ_{6} | hδ_{6} | qδ_{6} | |
E^{6} | Uniform 6-honeycomb | {3^{[7]}} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |
E^{7} | Uniform 7-honeycomb | {3^{[8]}} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |
E^{8} | Uniform 8-honeycomb | {3^{[9]}} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |
E^{9} | Uniform 9-honeycomb | {3^{[10]}} | δ_{10} | hδ_{10} | qδ_{10} | |
E^{10} | Uniform 10-honeycomb | {3^{[11]}} | δ_{11} | hδ_{11} | qδ_{11} | |
E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |