[Small 'Garden of Eden' patterns?] - A New Kind of Science: The NKS ForumA New Kind of Science: The NKS Forum
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Small 'Garden of Eden' patterns?
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Posted by: Tony Smith
I've lately been testing my pattern recognition abilities on the overall flow of patterns in my favourite irreversible cellular automata and become curious as to why nobody seems to have identified any reasonably small "Garden of Eden" patterns. Apparently the smallest found to date has 81 live cells which fit in a 13x12 rectangle, and which boringly evolves to a block and a blinker in 7 generations.
Certainly any such pattern can be trivially extended to construct unbounded series of larger gardens of eden, but only finding one unique out of 2^156 possibilities at even that size strikes me as unlikely. Of course proving that a pattern is a garden of eden is non-trivial, so exhaustively searching is even less of an option. Yet when looked at the other way around, starting with small enough random patterns you most often finish up at some common stable or cyclic descendant, quite often via one of a similarly small and unevently distributed evolutionary tracks, so there could be some expectation that gardens of eden are relatively numerous.
In some ways a starting configuration is a bit like a raindrop falling on a landmass. It's path to the sea will most likely be via one of the major river systems, but some drops will finish up in the most minor of streams. So I guess it might be possible that garden of eden patterns really only correspond to the ridgelines and thus a vanishingly small fraction of the pattern landscape. But that doesn't really feel right to me, especially in view of the results I've had from exhaustive searches in much smaller configurations of Trapper which has garden of eden configurations exceeding 50% for trap width 14 and increasing for larger trap widths.
On 435ff of the NKS book, Wolfram focuses on reversible cellular automata, but in my first post here I discussed how my work with Trapper suggested we might not need the hypothesised universal simple program to be reversible to account for the reversible laws of basic physics. The purpose of this post is not to try to push that argument any further, especially not before I can finally get back to properly presenting my Trapper results, but just to ask whether anybody has any better ideas for finding garden of eden patterns in Life and other rules.
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