[math-fun] Hexagonal analogue of Langton's ant
A few decades ago I played with a variant of Langton's ant in which the sites that the bug visits are hexagons rather than squares. I called it the "bee" (as opposed to "ant") model, and I could have sworn I wrote up something about it, but my efforts to dig up anything have been so far unavailing. Even if I didn't in fact ever publish anything about it, I'm sure others had the same idea, and perhaps they published something. If so, can any of you provide any links? Just as in the ant case (described in http://arxiv.org/pdf/math/9501233v1.pdf), one can show that for certain combinations of the initial state of the universe and the rule-string of the bee, the state of the universe will exhibit bilateral symmetry infinitely often. However, the proof of the "fundamental theorem of myrmecology" (the path of an ant is always unbounded) does not transfer over to melittology. (Maybe the boundedness claim is even false; I can't remember if I ever found a counterexample.) I studied a very simple combination of initial-state and rule-string that demonstrably gives states of the universe with bilateral symmetry infinitely often, and such that the path of the bee appears to be unbounded, but I was never able to prove the latter assertion; I looked in vain for a combinatorially-defined "arrow of time" for this system. Since a couple of decades have passed, maybe this problem has been solved while I was thinking about others things! Jim Propp
The most well-known hexagonal analogues of Langton's ant are Paterson's Worms. Over the years, the worms have been gradually 'solved' (that is to say, it has been proved whether or not the worm halts), with a single unsolved exception: the fabled 'Worm 1042015', which sadly was not solved on 01/04/2015 (although that would have made an *excellent* April Fool's joke). http://www.mathpuzzle.com/MAA/01-Paterson%27s%20Worms/mathgames_10_24_03.htm... Many of the later worms were solved by Benjamin Chaffin and Tom Rokicki. Sincerely, Adam P. Goucher
Sent: Thursday, May 21, 2015 at 11:41 PM From: "James Propp" <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Hexagonal analogue of Langton's ant
A few decades ago I played with a variant of Langton's ant in which the sites that the bug visits are hexagons rather than squares. I called it the "bee" (as opposed to "ant") model, and I could have sworn I wrote up something about it, but my efforts to dig up anything have been so far unavailing.
Even if I didn't in fact ever publish anything about it, I'm sure others had the same idea, and perhaps they published something. If so, can any of you provide any links?
Just as in the ant case (described in http://arxiv.org/pdf/math/9501233v1.pdf), one can show that for certain combinations of the initial state of the universe and the rule-string of the bee, the state of the universe will exhibit bilateral symmetry infinitely often.
However, the proof of the "fundamental theorem of myrmecology" (the path of an ant is always unbounded) does not transfer over to melittology. (Maybe the boundedness claim is even false; I can't remember if I ever found a counterexample.)
I studied a very simple combination of initial-state and rule-string that demonstrably gives states of the universe with bilateral symmetry infinitely often, and such that the path of the bee appears to be unbounded, but I was never able to prove the latter assertion; I looked in vain for a combinatorially-defined "arrow of time" for this system.
Since a couple of decades have passed, maybe this problem has been solved while I was thinking about others things!
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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James Propp