Omnitruncated 5-simplex honeycomb

Omnitruncated 5-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Omnitruncated simplectic honeycomb
Schläfli symbol t012345{3[6]}
Coxeter–Dynkin diagram
5-face types t01234{3,3,3,3}
4-face types t0123{3,3,3}
{}×t012{3,3}
{6}×{6}
Cell types t012{3,3}
{4,3}
{}x{6}
Face types {4}
{6}
Vertex figure
Irr. 5-simplex
Symmetry ×12, [6[3[6]]]
Properties vertex-transitive

In five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 5-simplex facets.

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).

A5* lattice

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The A*
5
lattice (also called A6
5
) is the union of six A5 lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex.

                                           = dual of        

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This honeycomb is one of 12 unique uniform honeycombs[1] constructed by the   Coxeter group. The extended symmetry of the hexagonal diagram of the   Coxeter group allows for automorphisms that map diagram nodes (mirrors) on to each other. So the various 12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams:

A5 honeycombs
Hexagon
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycomb diagrams
a1  [3[6]]                  
d2  <[3[6]]>          ×21        1,        ,        ,        ,        
p2  [[3[6]]]          ×22        2,        
i4  [<[3[6]]>]          ×21×22        ,        
d6  <3[3[6]]>          ×61        
r12  [6[3[6]]]          ×12        3

Projection by folding

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The omnitruncated 5-simplex honeycomb can be projected into the 3-dimensional omnitruncated cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same 3-space vertex arrangement:

         
         

See also

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Regular and uniform honeycombs in 5-space:

Notes

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  1. ^ mathworld: Necklace, OEIS sequence A000029 13-1 cases, skipping one with zero marks

References

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  • Norman Johnson Uniform Polytopes, Manuscript (1991)
  • 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 [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Space Family           /   /  
E2 Uniform tiling 0[3] δ3 3 3 Hexagonal
E3 Uniform convex honeycomb 0[4] δ4 4 4
E4 Uniform 4-honeycomb 0[5] δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb 0[6] δ6 6 6
E6 Uniform 6-honeycomb 0[7] δ7 7 7 222
E7 Uniform 7-honeycomb 0[8] δ8 8 8 133331
E8 Uniform 8-honeycomb 0[9] δ9 9 9 152251521
E9 Uniform 9-honeycomb 0[10] δ10 10 10
E10 Uniform 10-honeycomb 0[11] δ11 11 11
En-1 Uniform (n-1)-honeycomb 0[n] δn n n 1k22k1k21