But Adam, the duals A_n* of the A_n latti are exceptionally nice (without being exceptional). Their Voronoi cells have the property that if two of them intersect, it's always along an (n-1)-face. --Dan On 2013-12-17, at 8:22 AM, Adam P. Goucher wrote:
Marc LeBrun wrote: Alternatively we might start with the simpler 4 cell neighborhood, thus only 2^10 cases.
--the checkerboard neighborhood graph is bipartite, so this probably not a good idea.
A cell is connected to itself, so I fail to see what the problem is.
Anyway, I seem to recall that Edgar F. Codd already looked at all of these 1024 rules and found that none were interesting.
Using the equilateral triangle lattice, 6 neighbors, might be better, though.
Yes, the hexagonal lattice A_2 is awesome. The next time that a single lattice simultaneously wins the packing, covering and quantising problems is the Leech lattice in 24 dimensions (rather unfairly, A_8* beats E_8 in terms of covering).
Sincerely,
Adam P. Goucher
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