In Imagifractalous! 1, I talked about varying parameters in the usual algorithm for creating the Koch curve to produce a variety of images, and casually mentioned p-adic valuations. What happened was about a year after I began exploring these interesting images, I did a Google search (as one periodically does when researching new and exciting topics), and stumbled upon the paper Arithmetic Self-Similarity of Infinite Sequences.
It was there that I recognized the sequence of 0’s and 1’s I’d been using all along, and this sequence was called a 2-adic valuation (mod 2).
Here’s the definition: the p-adic valuation is the sequence such that the nth term in the sequence is the highest power of p which divides n. So the 2-adic valuation begins
0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, ….
Of course all terms with odd n are 0, and for the even n, you just look at the highest power of 2 dividing that even number. And you can take this sequence (mod 2), or mod anything you like.
I naturally did what any right-thinking mathematician would do — looked up p-adic sequences in the The On-Line Encyclopedia of Integer Sequences. Now it’s not that p-adic valuations are all that obscure, it’s just I had never encountered them before.
Of course there it was: A096268. And if you read through the comments, you’ll see one which states that this sequence can be used to determine the angle you need to turn to create the Koch snowflake.
I wasn’t particularly discouraged — this sort of thing happens all the time when exploring new mathematical ideas. You just get used to it. But something unexpected happened.
I starting experimenting with other p-adic valuations mod 2 (since I had two choices of angles), and found similar interesting behavior. Not only that, p didn’t have to be prime — any integer would do.
But most papers about p-adic valuations assumed p was prime. Why is that? If is a p-adic valuation and p is prime, it’s not hard to show that
an often-used property in developing a theory of p-adic valuations. But it only takes a moment to see that
Bad news for the p-adic theorists, but the fractal images couldn’t seem to care whether p was prime or not….
So I didn’t plunge into researching p-adic valuations, since I needed a treatment which included p composite, which didn’t seem to be out there.
But here’s the neat part. Most of the work I’d done to prove something already known — that a 2-adic valuation (mod 2) could be used to produce a Koch curve — could be used to study generic p. So I was able to make fairly good progress in a relatively short amount of time, since I’d already thought about it a lot before. I suspect it would have taken me quite a bit longer if I’d just casually read about the 2-adic result rather than prove it myself.
The progress made was similar to that of the 2-adic case — number of segments in each arm in the symmetric case, how many solutions given a particular order of symmetry, and so forth.
Now my fractal brain was truly revved up! I enjoyed creating images using various p-adic valuations — the variety seemed another dimension of endless. So I started brainstorming about ways to even further diversify my repertoire of fractal images.
The first two weeks of November were particularly fruitful. Two ideas seemed to coalesce. The first revolved around an old unexplored question: what happened when you didn’t only change the angles in the algorithm to produce the Koch snowflake, but you also changed the lengths?
Of course this seemed to make the parameter space impossibly large, but I was in an adventurous mood, and the only thing at stake was a few moments with Mathematica generating uninteresting images.
But what I found was that if an image closed up with a particular symmetry, then as long as the sequence of edge lengths was appropriately periodic, the image with different edge lengths also closed up with the same order of symmetry!
This was truly mind-boggling at first. But after looking at lots of images and diving into the algorithm, it’s not all that improbable. You can observe that in the image above, segments of the same length occur in six different orientations which are 60 degrees apart, and so will ultimately “cancel out” in any final vector sum and take you right back to the origin.
Now I don’t have the precise characterization of “appropriately periodic” as yet, but I know it’s out there. Just a matter of time.
The second big idea at the beginning of November involved skimming through the paper on arithmetic self-similarity mentioned above. Some results discussed adding one sequence to another, and so I wondered: what if you added two p-adic sequences before taking their sum (mod 2)?
Well, preliminary results were promising only when the p-adic valuations involved were of the same power of some p, like in the image above (which also involves different edge lengths).
These ideas are only in a very preliminary stage — it’s the perennial problem of, well, waiting….. It may look liking adding 2-adic and 3-adic valuations doesn’t get you anywhere, but maybe that’s just because you need so many more iterations to see what’s actually happening. So there’s lots more to explore here.
So as the parameter space gets larger — mutliple different lengths, adding several different p-adic valuations — the variety becomes infinitely diverse, and the analysis becomes that much more involved.
But that makes it all the more intriguing. And it will be all the more rewarding when I finally figure everything out….