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5-simplex honeycomb

## Summary

5-simplex honeycomb
(No image)
Type Uniform 5-honeycomb
Family Simplectic honeycomb
Schläfli symbol {3[6]}
Coxeter diagram
5-face types {34} , t1{34}
t2{34}
4-face types {33} , t1{33}
Cell types {3,3} , t1{3,3}
Face types {3}
Vertex figure t0,4{34}
Coxeter groups ${\displaystyle {\tilde {A}}_{5}}$×2, <[3[6]]>
Properties vertex-transitive

In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. These facet types occur in proportions of 2:2:1 respectively in the whole honeycomb.

## A5 lattice

This vertex arrangement is called the A5 lattice or 5-simplex lattice. The 30 vertices of the stericated 5-simplex vertex figure represent the 30 roots of the ${\displaystyle {\tilde {A}}_{5}}$  Coxeter group.[1] It is the 5-dimensional case of a simplectic honeycomb.

The A2
5
lattice is the union of two A5 lattices:

The A3
5
is the union of three A5 lattices:

.

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

This honeycomb is one of 12 unique uniform honeycombs[2] constructed by the ${\displaystyle {\tilde {A}}_{5}}$  Coxeter group. The extended symmetry of the hexagonal diagram of the ${\displaystyle {\tilde {A}}_{5}}$  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]]         ${\displaystyle {\tilde {A}}_{5}}$
d2  <[3[6]]>         ${\displaystyle {\tilde {A}}_{5}}$ ×21        1,        ,        ,        ,         p2  [[3[6]]]         ${\displaystyle {\tilde {A}}_{5}}$ ×22        2,
i4  [<[3[6]]>]         ${\displaystyle {\tilde {A}}_{5}}$ ×21×22        ,
d6  <3[3[6]]>         ${\displaystyle {\tilde {A}}_{5}}$ ×61
r12  [6[3[6]]]         ${\displaystyle {\tilde {A}}_{5}}$ ×12        3

## Projection by folding

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

Regular and uniform honeycombs in 5-space:

## Notes

1. ^ "The Lattice A5".
2. ^ mathworld: Necklace, OEIS sequence A000029 13-1 cases, skipping one with zero marks

## References

• 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 ${\displaystyle {\tilde {A}}_{n-1}}$  ${\displaystyle {\tilde {C}}_{n-1}}$  ${\displaystyle {\tilde {B}}_{n-1}}$  ${\displaystyle {\tilde {D}}_{n-1}}$  ${\displaystyle {\tilde {G}}_{2}}$  / ${\displaystyle {\tilde {F}}_{4}}$  / ${\displaystyle {\tilde {E}}_{n-1}}$
E2 Uniform tiling {3[3]} δ3 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 4-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 6 6
E6 Uniform 6-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 7-honeycomb {3[8]} δ8 8 8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 9 9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 10 10
E10 Uniform 10-honeycomb {3[11]} δ11 11 11
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k22k1k21