6simplex honeycomb
In sixdimensional Euclidean geometry, the 6simplex honeycomb is a spacefilling tessellation (or honeycomb). The tessellation fills space by 6simplex, rectified 6simplex, and birectified 6simplex facets. These facet types occur in proportions of 1:1:1 respectively in the whole honeycomb.
6simplex honeycomb  

(No image)  
Type  Uniform 6honeycomb 
Family  Simplectic honeycomb 
Schläfli symbol  {3^{[7]}} 
Coxeter diagram  
6face types  {3^{5}} t_{2}{3^{5}} 
5face types  {3^{4}} t_{2}{3^{4}} 
4face types  {3^{3}} 
Cell types  {3,3} 
Face types  {3} 
Vertex figure  t_{0,5}{3^{5}} 
Symmetry  ×2, [[3^{[7]}]] 
Properties  vertextransitive 
A6 lattice
This vertex arrangement is called the A6 lattice or 6simplex lattice. The 42 vertices of the expanded 6simplex vertex figure represent the 42 roots of the Coxeter group.[1] It is the 6dimensional case of a simplectic honeycomb. Around each vertex figure are 126 facets: 7+7 6simplex, 21+21 rectified 6simplex, 35+35 birectified 6simplex, with the count distribution from the 8th row of Pascal's triangle.
The A^{*}
_{6} lattice (also called A^{7}
_{6}) is the union of seven A_{6} lattices, and has the vertex arrangement of the dual to the omnitruncated 6simplex honeycomb, and therefore the Voronoi cell of this lattice is the omnitruncated 6simplex.
Related polytopes and honeycombs
This honeycomb is one of 17 unique uniform honeycombs[2] constructed by the Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:
A6 honeycombs  

Heptagon symmetry 
Extended symmetry 
Extended diagram 
Extended group 
Honeycombs 
a1  [3^{[7]}] 
 
i2  [[3^{[7]}]]  ×2 
 
r14  [7[3^{[7]}]]  ×14 

Projection by folding
The 6simplex honeycomb can be projected into the 3dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
See also
Regular and uniform honeycombs in 6space:
Notes
 http://www2.research.att.com/~njas/lattices/A6.html

 Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 181 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, WileyInterscience Publication, 1995, ISBN 9780471010036
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10] (1.9 Uniform spacefillings)
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
Fundamental convex regular and uniform honeycombs in dimensions 29  

Space  Family  / /  
E^{2}  Uniform tiling  {3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  {3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  {3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  {3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  {3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  {3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
E^{8}  Uniform 8honeycomb  {3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
E^{9}  Uniform 9honeycomb  {3^{[10]}}  δ_{10}  hδ_{10}  qδ_{10}  
E^{n1}  Uniform (n1)honeycomb  {3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 