Fun post at Scientific American explaining how fractal kitties can explain Julia sets:
Julia sets for polynomials of degree two are well-understood, although they’re often fractals rather than simple shapes such as circles. The story gets a lot more complicated as the degree increases because higher-degree polynomials are difficult to factor. (The much-maligned quadratic formula—the reason why we can easily discern the roots of degree two polynomials—is our friend!) A little bit is known about the possible shapes for Julia sets of degree 3 and 4 polynomials, but the shapes of the Julia sets of arbitrary polynomials are not yet understood.
Lindsey is a graduate student in mathematics at Cornell University. Her advisor is John Smillie, but Thurston was an unofficial second advisor, and it was his idea to start this research project. “I was sitting in his house, and he was staring off into space and asked, ‘I wonder if Julia sets can be made into shapes,’” she says. Thurston had been working on understanding the Mandelbrot set better, and looking at the shapes of Julia sets was a related pursuit. The Mandelbrot set, one of the most famous fractals, is closely related to Julia sets of degree two polynomials: imagine the polynomial z2+c, where c can be any complex number. The number c is in the Mandelbrot set if 0 is in the filled Julia set of z2+c.
The math may get hairy at times…but then again, so do the images. Ha!