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Fractions, fractals and Thue-Morse sequences, part 2
In part 1 of this series of indeterminate length, I mentioned the Thue-Morse sequence, a certain sequence of zeros and ones, and the counterparts of it for more than two symbols. When I was messing around with Thue-Morse I came across a webpage by Zachary Abel that mentioned something remarkable about them: interpreting them as a turtle-graphics program gives you a thing that somehow converges to the Koch snowflake curve, the archetypical simple fractal. (Abel defines the Thue-Morse sequence as the complement of the way most other people define it, starting with 1 instead of 0, but for this purpose it hardly matters.) This paper by Jun Ma and Judy Holdener proves in some detail that this does in fact converge to the Koch snowflake:
This isn't the textbook way that you usually draw the Koch snowflake with turtle graphics; it's simpler. You don't even need a "right turn" rule. You don't need recursion (except inasmuch as recursion is built into the Thue-Morse sequence, which you could argue it is). You just have the turtle follow the sequence, and turn left 60 degrees when it has a 0, and move forward one unit if it has a 1. The turtle at first seems to be crawling around in drunken squiggles, but if you just let it go, the overall shape of those squiggles turns out to be the Koch snowflake.
And, actually, that's not even the only way to do it. It also works if you interpret the bits the other way around. Or if you tell it to turn 120 degrees instead of 60--the fine details of the path are different but it still converges to the snowflake. Or if you give it a bunch of other ways of interpreting the Thue-Morse sequence as turtle instructions. Abel's page explores all sorts of different cases that work.
Not every interpretation works. If you just put in an arbitrary number for the turn angle, most of them seem to just give a mess. Sometimes, that mess starts to resolve into a directional path on a very large scale and it's possible that you still get the Koch snowflake again, but I haven't been patient enough to tell. Maybe you get something else entirely! Here's 70 degrees:
On the other hand, some you wouldn't think work, do. 75 degrees is sort of a punk rock snowflake:
Sometimes symmetries get you. When I first tried this, it didn't work because I was using the same sequence interpretation I use for the Heighway dragon curve: move forward every time, and turn left if you have a 1 and right if you have a 0. That fails because of the symmetries of the Thue-Morse sequence: every four symbols, the turtle ends up moving in its original direction, and you just get a long line with aperiodic wobbles in it, not very interesting. That's true no matter what turn angle you try.
But if you tweak the rule a little to break the symmetry, so that it turns, say, 120 degrees left for a 1 but only 60 degrees right for a 0... then you get the Koch snowflake again. Only the small-scale squiggles are different.
The Ma and Holdener paper proves that this is indeed the Koch curve for the first rule I mentioned, and for the complementary case where the symbols are interpreted the other way around, and they speculate that there are many other related sequences that have the same property. Kennard, Zaremsky and Holdener then found a whole family of these sequences.
But I don't think there's been a comprehensive study of what happens when you change the turtle interpretation of the series, as Abel did and as I'm doing here. I suspect that in many cases, changing the turtle rule is equivalent on a larger scale to modifying the series instead, in some way that keeps it consistent with Ma and Holdener's conditions.
But there's more...
This isn't the textbook way that you usually draw the Koch snowflake with turtle graphics; it's simpler. You don't even need a "right turn" rule. You don't need recursion (except inasmuch as recursion is built into the Thue-Morse sequence, which you could argue it is). You just have the turtle follow the sequence, and turn left 60 degrees when it has a 0, and move forward one unit if it has a 1. The turtle at first seems to be crawling around in drunken squiggles, but if you just let it go, the overall shape of those squiggles turns out to be the Koch snowflake.
And, actually, that's not even the only way to do it. It also works if you interpret the bits the other way around. Or if you tell it to turn 120 degrees instead of 60--the fine details of the path are different but it still converges to the snowflake. Or if you give it a bunch of other ways of interpreting the Thue-Morse sequence as turtle instructions. Abel's page explores all sorts of different cases that work.
Not every interpretation works. If you just put in an arbitrary number for the turn angle, most of them seem to just give a mess. Sometimes, that mess starts to resolve into a directional path on a very large scale and it's possible that you still get the Koch snowflake again, but I haven't been patient enough to tell. Maybe you get something else entirely! Here's 70 degrees:
On the other hand, some you wouldn't think work, do. 75 degrees is sort of a punk rock snowflake:
Sometimes symmetries get you. When I first tried this, it didn't work because I was using the same sequence interpretation I use for the Heighway dragon curve: move forward every time, and turn left if you have a 1 and right if you have a 0. That fails because of the symmetries of the Thue-Morse sequence: every four symbols, the turtle ends up moving in its original direction, and you just get a long line with aperiodic wobbles in it, not very interesting. That's true no matter what turn angle you try.
But if you tweak the rule a little to break the symmetry, so that it turns, say, 120 degrees left for a 1 but only 60 degrees right for a 0... then you get the Koch snowflake again. Only the small-scale squiggles are different.
The Ma and Holdener paper proves that this is indeed the Koch curve for the first rule I mentioned, and for the complementary case where the symbols are interpreted the other way around, and they speculate that there are many other related sequences that have the same property. Kennard, Zaremsky and Holdener then found a whole family of these sequences.
But I don't think there's been a comprehensive study of what happens when you change the turtle interpretation of the series, as Abel did and as I'm doing here. I suspect that in many cases, changing the turtle rule is equivalent on a larger scale to modifying the series instead, in some way that keeps it consistent with Ma and Holdener's conditions.
But there's more...
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