[math-fun] Glider in CA on Penrose tiling (Goucher)
It seems to me the essential features of the Penrose tiling are that (a) there are only a finite set of local patterns (b) the Penrose tiling can be got from a higher dimensional lattice by considering a 2D subspace of the higher space, and projecting stuff that is near to that subspace, orthogonally into the subspace. Due to (a) you can hope to devise a CA and write down only a finite number of rules to define it. Due to (b) there is some notion of "direction" and what physicists call "long-range order." (One way in which that is revealed is the Fourier transform of Penrose, which has delta functions like in diffraction from a crystal. Another is "Ammann bars.") Indeed one way Goucher could have worked (although maybe not the way he actually did it) might be to devise a CA on the higher-D lattice whose rules still worked restricted to the Penrose subset alone. If instead we were to use the Voronoi diagram of Poisson-random points in the plane, then I doubt Goucher (or anyone) could make a CA on THAT with gliders that keep going in some direction... due to properties (a) and (b) now failing...
On 8/7/2012 11:53 AM, Warren Smith wrote:
It seems to me the essential features of the Penrose tiling are that
(a) there are only a finite set of local patterns
(b) the Penrose tiling can be got from a higher dimensional lattice by considering a 2D subspace of the higher space, and projecting stuff that is near to that subspace, orthogonally into the subspace.
Due to (a) you can hope to devise a CA and write down only a finite number of rules to define it.
Due to (b) there is some notion of "direction" and what physicists call "long-range order." (One way in which that is revealed is the Fourier transform of Penrose, which has delta functions like in diffraction from a crystal. Another is "Ammann bars.") Indeed one way Goucher could have worked (although maybe not the way he actually did it) might be to devise a CA on the higher-D lattice whose rules still worked restricted to the Penrose subset alone.
If instead we were to use the Voronoi diagram of Poisson-random points in the plane, then I doubt Goucher (or anyone) could make a CA on THAT with gliders that keep going in some direction... due to properties (a) and (b) now failing...
Hmm. If you only required that the 'glider' go arbitrarily far in any given direction - then I think it might be possible. Or you might have to allow rules that considered the path as well as the current neighborhood. Brent Meeker
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Warren Smith