Recently, I’ve been working with a psychology student interested in how our brains perceive fractal images in nature (trees, clouds, lanscapes, etc.). I dug up some old PostScript programs which reproduced images from The Algorithmic Beauty of Plants, which describes L-systems and how they are used to model images of plants. (Don’t worry if you don’t have the book or aren’t familiar with L-systems — I’ll tell you everything you need to know.)
To make matters concrete, I changed a few parameters in my program to produce part of a Koch snowflake.
The classical way of creating a Koch snowflake is to begin with the four-segment path at the top, and then replace each of the four segments with a smaller copy of this path. Now replace each of the segments with an even smaller copy, and recurse until the copies are so small, no new detail is added.
Algorithmically, we might represent this as
F +60 F -120 F +60 F,
where “F” represents moving forward, and the numbers represent how much we turn left or right (with the usual convention that positive angles move counter-clockwise). If you start off moving to the right from the red dot, you should be able to follow these instructions and see how the initial iteration is produced.
The recursion comes in as follows: now replace each occurrence of F with a copy of these instructions, yielding
F +60 F -120 F +60 F +60
F +60 F -120 F +60 F -120
F +60 F -120 F +60 F +60
F +60 F -120 F +60 F
If you look carefully, you’ll see four copies of the initial algorithm separated by turning instructions. If F now represents moving forward by 1/3 of the original segment length, when you execute these instructions, you’ll get the second image from the top. Try it! Recursing again gives the third image, and one more level of recursion results in the last image.
Thomas thought this pretty interesting, and proceed to ask what would happen if we changed the angles. This wasn’t hard to do, naturally, since the program was already written. He suggested a steeper climb of about 80 degrees, so I changed the angles to +80 and -140.
Surprise! You’ll easily recognize the first two iterations above, but after five iterations, the image closes up on itself and creates an elegant star-shaped pattern.
I was so intrigued by stumbling upon this symmetry, I decided to explore further over the upcoming weekend. My next experiment was to try +80 and -150.
The results weren’t as symmetrical, but after six levels of recursion, an interesting figure with bilateral symmetry emerged. You can see how close the end point is to the starting point — curious. The figure is oriented so that the starting points (red dots) line up, and the first step is directly to the right.
Another question Thomas posed was what would happen if the lengths of the segments weren’t all the same. This was a natural next step, and so I created an image using angles of +72 and -152 (staying relatively close to what I’d tried before), and using 1 and 0.618 for side lengths, since the pentagonal motifs suggested the golden ratio. Seven iterations produced the following remarkable image.
My purpose in writing about these “fractal” images this week is to illustrate the creative process in doing mathematics. This just happened a few days ago (as I am writing this), and so the process is quite fresh in my mind — a question by a student, some explorations, further experimentation, small steps taken one at a time until something truly wonderful emerges. The purist will note that the star-shaped images are not truly fractals, but since they’re created with an algortihm designed to produce a fractal (the Koch snowflake), I’m taking a liberty here….
This is just a beginning! Why do some parameters result in symmetry? How can you tell? When there is bilateral symmetry, what is the “tilt” angle? Where did the -24.7 come from? Each new image raises new questions — and not always easy to answer.
Two weeks ago, this algorithm was collecting digital dust in a subdirectory on my hard drive. A simple question resurrected it — and resulted in a living, breathing mathematical exploration into an intensely intriguing fractal world. This is how mathematics happens.