Description by Cruz Godar (first-year graduate student), 2020, Digital animations rotating on the screen
Finite Subdivisions: This one isn’t too complicated! Draw a polygon and connect its center of mass to every vertex, splitting it into triangles. Do the same thing for every triangle, and repeat until you have a beautiful, crystal-like spiderweb. This is called barycentric subdivision, and while it’s only one kind of finite subdivision, it’s one of the most picturesque.
Abelian Sandpiles: Draw a grid and start dropping sand grains in the center square. As soon as you get to four grains, the pile topples, spilling one grain into each of the four surrounding squares. By the time you’ve dropped fifteen grains in the center, you’ll have five piles of three grains each. The next grain topples the center pile, and then each of the surrounding four topple too. Four of those grains end up back in the center, where they topple again, and then the system is stable again. If we start with a massive number of grains in the center, the pattern that’s eventually produced is surprisingly complex. Amazingly, when we have more than one pile that will topple at the same time, the order we topple them has no effect on the final outcome, and because of this, the model is called Abelian.
Julia Sets: It’s hard to find fractals more famous than the Mandelbrot and Julia sets. By iterating the function f(z) = z^2 + c on points z in the complex plane, some points stay bounded (colored black) and others don’t (colored more brightly the closer they come to staying bounded). By varying c, we can animate through an entire family of fractals, all of which share a beautiful resemblance.
Juliabulbs: The holy grail of 3D fractals, Juliabulbs are a landmark accomplishment. Translating the Mandelbrot set to three dimensions is impossible to do canonically, since there is no appropriate 3-dimensional number system analogous to the complex numbers. Nevertheless, Daniel White and Paul Nylander managed to use spherical coordinates to produce something spectacular in its own right, showing just what 3D fractals could be and igniting the search for more.
Newton’s Method: Commonly taught in beginning calculus courses, Newton’s Method is an algorithm to find a root of a function given a starting guess. Here, that function is a polynomial with eight varying roots, and we try every complex number as a starting guess, coloring a point based on the root it eventually approaches. Remarkably, the boundary between roots isn’t a smooth curve, but an infinitely detailed fractal.
Quasi-Fuchsian Groups: A Möbius transformation is a relatively simple function defined on the complex numbers. By choosing two transformations and applying them repeatedly in different orders to a single point, we reveal a folded and detailed boundary. Despite its complexity, these fractals are still just curves, and it’s examples like them that make results about generic curves so difficult to prove.
Wilson’s Algorithm: Ever wanted to make your own maze? If you’ve tried, you might find it’s not as easy as you might think. Draw just one long path, and it’s too easy; draw lots of small ones and you’ll often run into the same problem. If even a small maze is tricky to make, a massive one seems downright impossible — and yet pick up any puzzle book for children and you’ll see pages upon pages of them. How do they do it? Enter Wilson’s Algorithm. It’s a method that makes maze-making as easy as following a recipe, and what’s more, it produces a truly random one — every possible maze of a given size has the same chance of being created. We draw the maze one random path at a time, and then color it by filling from the center outward: points colored pink may look close to those colored red, but their path to one another through the maze is actually as long as possible.