Mathematicians often rhapsodize about the austere elegance of a well-wrought proof. But math also has a simpler sort of beauty that is perhaps easier to appreciate: It can be used to create objects that are just plain pretty—and fascinating to boot.
That beauty was richly on display at an exhibition of mathematical art at the Joint Mathematics Meetings in San Diego in January, where more than 40 artists showed their creations.
Michael Field, a mathematics professor at the University of Houston, finds artistic inspiration in his work on dynamical systems. A mathematical dynamical system is just any rule that determines how a point moves around a plane. Field uses an equation that takes any point on a piece of paper and moves it to a different spot. Field repeats this process over and over again—around 5 billion times—and keeps track of how often each pixel-sized spot in the plane gets landed on. The more often a pixel gets hit, the deeper the shade Field colors it.
The reason mathematicians are so fascinated by dynamical systems is that very simple equations can produce very complicated behavior. Field has found that such complex behavior can create some beautiful images. For example, the dynamical system he depicts in “Coral Star” does some peculiar things as it gets closer to the center (technically, the equation is discontinuous at the origin). So as you get closer and closer to the center, the image gets more and more complex.
“Even apart from the center, the image has quite a lot of depth to it,” Field says. “It’s a feature of the way it’s colored. I’m not so keen on bright primary colors. The shading makes it more interesting.”
This image has an unusual 35-fold symmetry, and Field created it as a present for his wife on their 35th anniversary.
Robert Bosch, a mathematics professor at Oberlin College in Ohio, took his inspiration from an old, seemingly trivial problem that hides some deep mathematics. Take a loop of string and throw it down on a piece of paper. It can form any shape you like as long as the string never touches or crosses itself. A theorem states that the loop will divide the page into two regions, one inside the loop and one outside.
It is hard to imagine how it could do anything else, and if the loop makes a smoothly curving line, a mathematician would think that is obvious too. But if a line is very, very crinkly, it may not be obvious whether a particular point lies inside or outside the loop. Topologists, the type of mathematicians who study such things have managed to construct many strange, “pathological” mathematical objects with very surprising properties, so they know from experience that you shouldn’t assume a proof is unnecessary in cases like this one. And this problem did turn out to be very difficult to solve: It took about 20 years after mathematicians began working on the problem to find a correct proof.
Bosch created a simple string-loop on a page and colored the resulting region inside the loop red and the region outside black. From afar, the image looks like two interlaced loops—one red and one black—that form a Celtic knot. For more information about his method for creating the image, see “Artful Routes”.
Robert Fathauer, an artist with a mathematical puzzle business in Phoenix, Ariz., found that it doesn’t require fancy mathematics to stumble upon remarkable mathematical patterns. He was playing around with various ways of arranging squares in repeating patterns. He started with a red cube and placed five half-sized orange cubes on its exposed faces. Then he put five smaller yellow cubes on the faces of each of those, and five even smaller greenish cubes on the faces of those, and so on.
“After a few iterations, I noticed that something special was happening with that arrangement,” Fathauer says. The shape was approximating a pyramid, with triangular holes punched out. Even more remarkably, he found that the faces of the pyramid formed the Sierpinski Triangle, one of the earliest fractals ever studied.
Andrew Pike took inspiration from a similar Sierpinski fractal in creating his art. The senior at Oberlin College started with a photograph of the Polish mathematician Waclaw Sierpinski and recreated a version of it with tiles made from the “Sierpinski carpet.” To create a Sierpinski carpet, take a square, divide it in a tic-tac-toe pattern, and take out the middle square. Then draw a tic-tac-toe pattern on each remaining square and knock out the middle squares of those. Continuing forever will create the Sierpinski carpet.
Pike stopped short of continuing forever and instead created tiles with different numbers of iterations of the process. Some of the tiles started white, with the knocked-out squares black, and some of them started black, with the knocked-out squares white. This gave him squares that approximated many gradations of gray.
Then he created a computer program that divided the photograph of Sierpinski into tiny squares, averaged the shades of gray in the picture across each individual square, and selected the Sierpinski tile that was closest in shading. “But it didn’t look good,” Pike says. “The transitions were really rough.”
He couldn’t simply make the tiles smaller, because printers can produce dots that are only so tiny. So he used a technique called “dithering.” He calculated the error—the difference between the shading of the photograph and the shading of the most similar Sierpinski tile—and spread it between the other nearby tiles. This effectively softened the image, removing the awkward transitions between tiles.
“We chose the image of Sierpinski because it was self-referential,” Pike says. Seems appropriate for a technique using self-similar fractals.
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