Quarter 7-cubic honeycomb

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quarter 7-cubic honeycomb
(No image)
Type Uniform 7-honeycomb
Family Quarter hypercubic honeycomb
Schläfli symbol q{4,3,3,3,3,3,4}
Coxeter diagram CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
6-face type h{4,35}, 7-demicube t0 D7.svg
h5{4,35}, 7-demicube t05 D7.svg
{31,1,1}×{3,3} duoprism
Vertex figure
Coxeter group ×2 = [[31,1,3,3,3,31,1]]
Dual
Properties vertex-transitive

In seven-dimensional Euclidean geometry, the quarter 7-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 7-demicubic honeycomb, and a quarter of the vertices of a 7-cube honeycomb. [1] Its facets are 7-demicubes, pentellated 7-demicubes, and {31,1,1}×{3,3} duoprisms.

Contents

This honeycomb is one of 77 uniform honeycombs constructed by the Coxeter group, all but 10 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 77 permutations are listed with its highest extended symmetry, and related and constructions:

D7 honeycombs
Extended
symmetry
Extended
diagram
OrderHoneycombs
[31,1,3,3,3,31,1]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png×1CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png, CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png,
[[31,1,3,3,3,31,1]]CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c3.pngCDel split1.pngCDel nodeab c1-2.png×2CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png, CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png
<[31,1,3,3,3,31,1]>
↔ [31,1,3,3,3,3,4]
CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c5.pngCDel 3.pngCDel node c6.pngCDel split1.pngCDel nodeab c7.png
CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c5.pngCDel 3.pngCDel node c6.pngCDel 3.pngCDel node c7.pngCDel 4.pngCDel node.png
×2...
<<[31,1,3,3,3,31,1]>>
↔ [4,3,3,3,3,3,4]
CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c5.pngCDel split1.pngCDel nodeab c6.png
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c5.pngCDel 3.pngCDel node c6.pngCDel 4.pngCDel node.png
×4...
[<<[31,1,3,3,3,31,1]>>]
↔ [[4,3,3,3,3,3,4]]
CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.png
×8...

See also

Regular and uniform honeycombs in 7-space:

Notes

  1. Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318

References

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 133331
E8 Uniform 8-honeycomb 0[9] δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb 0[10]δ10hδ10qδ10
E10Uniform 10-honeycomb0[11]δ11hδ11qδ11
En−1Uniform (n−1)-honeycomb 0[n] δn hδn qδn 1k22k1k21