(See the sibling comment with a "10." in it for context.) 4. The Socolar-Taylor tile. This is pretty cool. So, suppose you have an infinite supply of tiles of

(See the sibling comment with a "10." in it for context.) 4. The Socolar-Taylor tile. This is pretty cool. So, suppose you have an infinite supply of tiles of some set of shapes and sizes (trivial simple case: squares of a single size). You might be able to use them to tile the whole plane (e.g., the obvious square tiling, like the pixels of a computer display). The easiest tilings to think about, like this one, are periodic: they repeat themselves. Some others are aperiodic: they don't repeat. In the 1960s, a guy called Wang proved a nice theorem: there is a computer program that can answer the question "here are some tiles; can I tile the plane with them?" if, and only if, every set of tiles that can tile the plane can do so periodically. (This isn't hugely surprising in retrospect; the idea is that you can recognize when there's a tiling if and only if all tilings are effectively finite.) And then Robert Berger found a set of (thousands of) tiles that can tile the plane, but only aperiodically. (I lies slightly: the tilings in the paragraph above are all of a very special kind: the tiles are squares with carefully designed wiggles on their edges that restrict which ones can go next to which others. Equivalently, they're squares with coloured sides and abutting sides' colours have to match.) So then the question arises: Berger did this with 20,000 different kinds of tile, but how few do you need? You have probably heard of "Penrose tilings": Roger Penrose found a set of two tiles (these ones aren't square) that will tile the plane but only aperiodically. So ... can you do it with only one? It turns out you can, and Socolar and Taylor found a tile that does it. You can do this with a hexagon and some restrictions (like the colours on those square tiles earlier) on what can match with what, or you can perturb the hexagon so that the restrictions fall out naturally from the geometry (like putting wiggles in the sides of the squares). Unfortunately, the perturbations required make the Socolar-Taylor tile disconnected -- it's made of multiple bits that move around together! I am inclined to doubt that anything will come of this that's useful elsewhere in mathematics, but it's incredibly cool.