2-adic numbering for binary tilings
I’ve posted quite a bit about the binary tiling of the hyperbolic plane recently, including what you get when you shrink its vertical edges, a related “nowhere-neat” tessellation, the connection to Smith charts and Escher, a method to 3-color its tiles, a half-flipped variation of the tiling, and its applications in proving that folding origami is hard. But I thought there might be room for one more post, in honor of the Wikipedia binary tiling article’s new Good Article Status. This one is about something I wanted to include in the Wikipedia article, but couldn’t, because I couldn’t find published sources describing it: a way of numbering tiles that encodes their position in a tiling and instantly proves the assertion in the article that there are uncountably many binary tilings.
The basic idea is very simple: in the binary tiling (in its conventional view as a pattern of similar squares or rectangles in the Poincaré half-plane), if you look directly upward from any tile, some of the tiles above it will extend farther to the left, and some of them will extend farther to the right. We will encode this by a binary sequence, where a 1 encodes a tile that extends to the left, and a 0 encodes a tile that extends to the right. (This seems backwards, but there’s a reason for this choice that will become clear soon.) To make it backwards in a different way, I want to write this sequence in right-to-left order, so that for instance the encoding …1100 means that the closest two tiles above it extend to the right, and the next two extend to the left. Here’s a picture with some examples. The label in each square only encodes the tiles that you can see in the picture, but the tiling itself can continue above these squares in multiple different ways, resulting in different continuations of these labels.
