The Markov machine model is a simple model of computation which is equivalent in power to the Turing machine model, and gives the same time complexity classes modulo P. Typomatic is a cute, colorful, and bizarrely addictive tool for watching Markov machines at work. I’ve had great luck, and great fun, using Typomatic as a hands-on way of introducing middle through high school students to the idea of computability.

These interactive slides use 1d voter models to illustrate basic features of neutral theory—a vision of how genetic and ecological diversity can emerge even without selective pressure. The slides show how questions about the persistence and spatial organization of lineages can be rephrased, in these models, as questions about random walks.

Slice a triply periodic wooden sculpture along an irrational plane. If you ink the cut surface and press it against a page, the pattern you print will be quasiperiodic. Patterns like these help physicists see how metals conduct electricity in strong magnetic fields. These interactive slides explore the printing process described above and sketch its connection to the physics of conductivity.

You can think of a translation structure as a pinstripe suit for a Riemann surface. These drawings show variations on a suit for a twice-punctured surface of genus two, sewn from six bolts of colored cloth. They placed in UT Austin's Visualizing Science contest for 2015.

The long and short radii a and b of an ellipse lie on the hyperbola a² – b² = f², where f is half the distance between the foci.

At a conference, Ren Yi asked whether you can modify the midpoint map to get irregular polygons as attracting fixed points. For the affine midpoint map, this sketch shows that the answer is clearly yes, but I don’t know how to prove it. I also don't know which polygons you can get.

In which π takes a unexpected interest in horticulture.

In which π takes an unexpected interest in games of chance.