September 2, 2015 — that was the day Thomas sent an email to the faculty in the math department asking if anyone was willing to help him learn something about fractals. And that was when it all began.
Now I’ve told this story in various forms over the past year or so on my blog, but I want to begin a thread along a different direction. I’d like to give a brief chronological history of my work with fractals, with an eye toward the fellow mathematician interested in really understanding what is going on.
I’ll be formal enough to state fairly precisely the mathematical nature of what I’m observing, so as to give the curious reader an idea of the nature of the mathematics involved. But I’ll skip the proofs, so that the discussion doesn’t get bogged down in details. And of course provide lots of pictures….
You might want to go back to read my post of Day007 — there is a more detailed discussion there. But the gist of the post is the question posed by Thomas at one of our early meetings: what happens when you change the angles in the algorithm which produces the Koch curve?
What sometimes happens is that the curve actually closes up and exhibits rotational symmetry. As an example, when the recursive routine
F +40 F +60 F +40 F
is implemented, the following sequence of segments is drawn.
Now this doesn’t look recursive at all, but rather iterative. And you certainly will have noticed something curious — some sequences of eight segments, which I call arms, are traversed twice.
All of this behavior can be precisely described — all that’s involved is some fairly elementary number theory. The crux of the analysis revolves around the following way of describing the algorithm: after drawing the nth segment, turn +40 degrees if the highest power of 2 dividing n is even, and turn +60 degrees if it’s odd. Then move forward.
I would learn later about p-adic valuations (this is just a 2-adic valuation), but that’s jumping a little ahead in the story. I’ll just continue on with what I observed.
What is also true is that the kth arm (in this case, the kth sequence of eight segments) is retraced precisely when the highest power of power of 2 dividing k is odd. This implies the following curious fact: the curve in the video is traced over and over again as the recursion deepens, but never periodically. This is because the 2-adic valuation of the positive integers isn’t periodic.
So I’ve written up some results and submitted them to a math journal. What I essentially do is find large families of angle pairs (like +40 and +60) which close up, and I can describe in detail how the curves are drawn, what the symmetry is, etc. I can use this information to create fractal images with a desired symmetry, and I discuss several examples on a page of my mathematical art website.
As one example, for my talks in Europe this past summer, I wanted to create images with 64 segments in each arm and 42-fold symmetry. I also chose to divide the circle in 336 parts — subdivision into degrees is arbitrary. Or another way of looking at it is that I want my angles to be rational multiples of and I’m specifying the denominator of that fraction.
Why 336? First, I needed to make sure 42 divided evenly into 336, since each arm is then 8/336 away from the previous one (although they are not necessarily drawn that way). And second, I wanted there to be enough angle pairs so I’d have some choices. I knew there would be 96 distinct images from the work I’d already done, so it seemed a reasonable choice. Below is one of my favorites. The angles used here are 160 and 168 of the 336 parts the circle is divided into.
Now my drawing routine has the origin at the center of the squares in the previous two images, so that the rotational symmetry is with respect to the origin. If you think about it for a moment, you’ll realize that if both angles are the same rational multiple of you’ll get an image like the following, where one of the vertices of the figure is at the origin, but the center of symmetry is not at the origin.
Of course it could be (and usually is!) the case that the curve does not close — for example, if one angle is a rational multiple of and the other is an irrational multiple of Even when they do close, some appealing results are obtained by cropping the images.
So where does this bring me? I’d like a Theorem like this: For the recursive Koch algorithm described by
F F F F,
the curve closes up precisely when (insert condition on and ), and has the following symmetry (insert description of the symmetry here).
I’m fairly confident I can handle the cases where the center of symmetry is at the origin, but how to address the case when the center of symmetry is not the origin is still baffling. The only cases I’ve found so far are when but that does not preclude the possibility of there being others, of course.
At this point, I was really enjoying creating digital artwork with these Koch-like images, and was also busy working on understanding the underlying mathematics. I was also happy that I could specify certain aspects of the images (like the number of segments in each arm and the symmetry) and find parameters which produced images with these features. By stark contrast, when I began this process, I would just randomly choose angles and hope for the best! I’d consider myself lucky to stumble on something nice….
So it took about a year to move from blindly wandering around in a two-dimensional parameter space (the two angles to be specified) to being able to precisely engineer certain features of fractal images. The coveted “if-and-only-if” Theorem which says it all is still yet to be formulated, but significant progress toward that end has been made.
But then my fractal world explodes by moving from 2-adic to p-adic. That for next time….