Omnitruncated 8-simplex honeycomb

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Omnitruncated 8-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Omnitruncated simplectic honeycomb
Schläfli symbol {3[9]}
Coxeter–Dynkin diagrams CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png
7-face types t01234567{3,3,3,3,3,3,3}
Vertex figure Omnitruncated 8-simplex honeycomb verf.png
Irr. 8-simplex
Symmetry ×18, [9[3[9]]]
Properties vertex-transitive

In eight-dimensional Euclidean geometry, the omnitruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 8-simplex facets.

Contents

The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).

A*
8
lattice

The A*
8
lattice (also called A9
8
) is the union of nine A8 lattices, and has the vertex arrangement of the dual honeycomb to the omnitruncated 8-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 8-simplex

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes 01lr.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lr.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01lr.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lr.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01lr.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 10l.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 01l.png = dual of CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png.

This honeycomb is one of 45 unique uniform honeycombs [1] constructed by the Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:

A8 honeycombs
Enneagon
symmetry
Symmetry Extended
diagram
Extended
group
Honeycombs
a1[3[9]]CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png

CDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lr.pngCDel 3ab.pngCDel branch.pngCDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 10l.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 10l.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lr.pngCDel 3ab.pngCDel branch.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 10l.pngCDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lr.pngCDel 3ab.pngCDel branch 01l.pngCDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.png

CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 10l.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lr.pngCDel 3ab.pngCDel branch 01l.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lr.pngCDel 3ab.pngCDel branch 10l.pngCDel node.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 10l.pngCDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 10l.png

i2[[3[9]]]CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel 3ab.pngCDel nodeab c3.pngCDel 3ab.pngCDel nodeab c4.pngCDel 3ab.pngCDel branch c5.png×2

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png 1 CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.png 2

CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.png

CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png

CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png

CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png

CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png

CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png

i6[3[3[9]]]CDel node c3.pngCDel split1.pngCDel nodeab c1.pngCDel 3ab.pngCDel nodeab c1.pngCDel 3ab.pngCDel nodeab c3.pngCDel 3ab.pngCDel branch c1.png×6CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.png
r18[9[3[9]]]CDel node c1.pngCDel split1.pngCDel nodeab c1.pngCDel 3ab.pngCDel nodeab c1.pngCDel 3ab.pngCDel nodeab c1.pngCDel 3ab.pngCDel branch c1.png×18CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png 3

See also

Regular and uniform honeycombs in 8-space:

Notes

    • Weisstein, Eric W. "Necklace". MathWorld ., OEIS sequenceA000029 46-1 cases, skipping one with zero marks

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