No, the title of today’s post is not a typo….
About a month ago, a colleague who takes care of the departmental bulletin boards in the hallway approached me and asked if I’d like to create a bulletin board about mathematical art. There was no need to think it over — of course I would!
Well, of course we would, since I immediately recruited Nick to help out. We talked it over, and decided that I would describe Koch-like fractal images on the left third of the board, Nick would discuss fractal trees on the right third, and the middle of the bulletin board would highlight other mathematical art we had created.
I’ll talk more about the specifics in a future post — especially since we’re still working on it! But this weekend I worked on designing a banner for the bulletin board, which is what I want to share with you today.
I really had a lot of fun making this! I decided to create fractals for as many letters of Imagifractalous! as I could, and use isolated letters when I couldn’t. Although I did opt not to use a third fractal “A,” since I already had ideas for four fractal letters in the second line.
The “I”‘s came first. You can see that they’re just relatively ordinary binary trees with small left and right branching angles. I had already incorporated the ability to have the branches in a tree decrease in thickness by a common ratio with each successive level, so it was not difficult to get started.
I did use Mathematica to help me out, though, with the spread of the branches. Instead of doing a lot of tweaking with the branching angles, I just adjusted the aspect ratio (the ratio of the height to the width of the image) of the displayed tree. For example, if the first “I” is displayed with an aspect ratio of 1, here is what it would look like:
I used an aspect ratio of 6 to get the “I” to look just like I wanted.
Next were the “A”‘s. The form of an “A” suggested an iterated function system to me, a type of transformed Sierpinski triangle. Being very familiar with the Sierpinski triangle, it wasn’t too difficult to modify the self-similarity ratios to produce something resembling an “A.” I also like how the first “A” is reminiscent of the Eiffel Tower, which is why I left it black.
I have to admit that discovering the “R” was serendipitous. I was reading a paper about trees with multiple branchings at each node, and decided to try a few random examples to make sure my code worked — it had been some time since I tried to make a tree with more than two branches at each node.
When I saw this, I immediately thought, “R”! I used this image in an earlier draft, but decided I needed to change the color scheme. Unfortunately, I had somehow overwritten the Mathematica notebook with an earlier version and lost the code for the original “R,” but luckily it wasn’t hard to reproduce since I had the original image. I knew I had created the branches only using simple scales and rotations, and could visually estimate the original parameters.
The “C” was a no-brainer — the fractal C-curve! This was fairly straightforward since I had already written the Mathematica code for basic L-systems when I was working with Thomas last year. This fractal is well-known, so it was an easy task to ask the internet for the appropriate recursive routine to generate the C-curve:
+45 F -90 F +45
For the coloring, I used simple linear interpolation from the RGB values of the starting color to the RGB values of the ending color. Of course there are many ways to use color here, but I didn’t want to spend a lot of time playing around. I was pleased enough with the result of something fairly uncomplicated.
For the “T,” it seemed pretty obvious to use a binary tree with branching angles of 90° to the left and right. Notice that the ends of the branches aren’t rounded, like the “I”‘s; you can specify these differences in Mathematica. Here, the branches are emphasized, not the leaves — although I did decide to use small, bright red circles for the leaves for contrast.
The “L” is my favorite letter in the entire banner! Here’s an enlarged version:
This probably took the longest to generate, since I had never made anything quite like it before. My inspiration was the self-similarity of the L-tromino, which may be made up of four smaller copies of itself.
The problem was that this “L” looked too square — I wanted something with a larger aspect ratio, but keeping the same self-similarity as much as possible. Of course exact self-similarity isn’t possible in general, so it took a bit of work to approximate is as closely as I could. I admit the color scheme isn’t too creative, but I liked how the bold, primary colors emphasized the geometry of the fractal.
The “O” was the easiest of the letters — I recalled a Koch-like fractal image I created earlier which looked like a wheel with spokes and which had a lot of empty space in the interior. All I needed to do was change the color scheme from white-on-gray to black-on-white.
Finally, the “S.” This is the fractal S-curve, also known as Heighway’s dragon. It does help to have a working fractal vocabulary — I knew the S-curve existed, so I just asked the internet again…. There are many ways to generate it, but the easiest for me was to recursively producing a string of 0’s and 1’s which told me which way to turn at each step. Easy from there.
So there it is! Took a lot of work, but it was worth it. I’ll take a photo when it’s actually displayed — and update you when the entire bulletin board is finally completed. We’ve only got until the end of the semester, so it won’t be too long….