I had a link to an paper proving some basic-sounding theorems about the bee. At the moment, I can't find it among my links about Langton's Ant. The alternative to the path being unbounded is that the bug cycles; there are "initial" states producing very short cycles for the bee. (Scare quotes because such a state is just one of the states in the loop.) The bee I'm talking about has a hexagonal cell with one bit of state: turn left or right and toggle the state, as with Langton's Ant. I believe Patterson's worm has a lattice where each edge has a bit of state, and once eaten, an edge can't be traveled again (and so it's sometimes possible for the worm to paint itself into a corner and halt). --Steve
Date: Thu, 21 May 2015 18:41:36 -0400 From: James Propp <jamespropp@gmail.com>
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
------------------------------ Date: Fri, 22 May 2015 04:24:01 +0200 From: "Adam P. Goucher" <apgoucher@gmx.com>
The most well-known hexagonal analogues of Langton's ant are Paterson's Worms.