Omnitruncated 5-simplex honeycomb

Five dimensional space-filling tessellation From Wikipedia, the free encyclopedia

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.

Omnitruncated 5-simplex honeycomb
(No image)
TypeUniform honeycomb
FamilyOmnitruncated simplectic honeycomb
Schläfli symbolt012345{3[6]}
Coxeter–Dynkin diagram
5-face typest01234{3,3,3,3}
4-face typest0123{3,3,3}
{}×t012{3,3}
{6}×{6}
Cell typest012{3,3}
{4,3}
{}x{6}
Face types{4}
{6}
Vertex figure
Irr. 5-simplex
Symmetry×12, [6[3[6]]]
Propertiesvertex-transitive

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

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

Summarize
Perspective

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:

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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
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Projection by folding

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:

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See also

Regular and uniform honeycombs in 5-space:

Notes

References

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