One detail-level problem with the 4-cell neighborhood is: Consider the options for a 0 cell with three 0 neighbors and one 1 neighbor: If the successor state is 0, then no pattern can grow beyond its bounding box, preventing replication and unbounded computations. If the successor state is 1, most patterns grow at the speed of light and fill space with 1s. There are several ways to get around this problem, but they all involve complications: More states, or more history, or varying the geometry, etc. Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] on behalf of Adam P. Goucher [apgoucher@gmx.com] Sent: Tuesday, December 17, 2013 9:22 AM To: math-fun Subject: [EXTERNAL] Re: [math-fun] "life" generalized CAs classified ala Wolfram?
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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun