A long time ago, Industrial Light and Magic was one of the first organizations to try and make plausible digital actors. They worked on it and their focus groups liked everything except the face. It didn’t look real. Even when the modeling fidelity (I have no idea how this was measured) was much higher for the face, the focus groups still gave it two thumbs down. What was up with this?
The most important things in a human’s life are other humans. We devote absurd amounts of our brains to seeing and interpreting other’s faces.
There is another side to this: the man in the moon. Our giant mental face-recognizers see faces in lots of places where there are no faces, like the face on the front of the moon. Otto, the image shown at the top of this article, is the mascot for an online shop one of my students started while Occupy Math was still in Iowa. Most people see Otto as a face, but he’s really five generalized Sierpinski fractals. The same mental phenomenon that made it hard to make digital “actors” makes it easier to generate cartoon or fractal characters with faces. Make something a little like a face and people see a face. But how was the face constructed?
Today’s Occupy Math looks at these easy-to-design fractals.
The triangle above is created by first picking three corners. Starting at one corner, you randomly jump half-way to a corner selected randomly, and color in the point at that location. The selected corner can be the same corner that was picked before. After you’ve picked thousands of random corners, you get the object above. It is called the Sierpinski triangle. Occupy Math has colored the corners red, blue, and green. We have a moving color as well. It starts the same color as the first corner picked. After that, whenever we move toward a corner, the color of that corner is averaged into the moving color. This lets you see a shadow of the averaging process in the colors.
Occupy Math has a fairly bad case of hacker so as soon as the code that made the triangle was running, he tried a square. Bad day! The square fills in, though the same coloring trick lets you see how it fills in. There are red, green, blue, and yellow corners.
This is when Occupy Math had an idea. Instead of picking any corner of the square, based on the corner you chose last, you may only go to:
- That same corner,
- The corner diagonally across from you, or
- The next corner counter-clockwise.
If you do this you get the following picture.
Ideas are the currency of mathematics. The idea of restricting which corner you can jump to next gave Occupy Math a lot of new fractals.
Otto, the face that starts the post, is made of five fractals – each eye, nose, and mouth as well as the outline of his face. All of these fractals are designed by picking some points and making, for each point, a list of which points are next. The points also have colors to average toward when they are picked, like with the triangle’s corners. My student designed a fractal alphabet:
And many other things can be done with this technology, from making the face for my student’s logo to quilt-like fractals and ferns. Visit the gallery. Occupy Math hopes you’ve enjoyed this visit to an idea in the land of mathematics. There are a number of things Occupy Math didn’t mention about these fractals that appear in one of the articles on the fractal taxonomy web site. If you would like a copy of your name in the fractal alphabet, let Occupy Math know and he will see if he can do something. For this request, use email@example.com. If you like these fractals, comment or tweet!
I hope to see you here again,
University of Guelph,
Department of Mathematics and Statistics