An Algorithm for Domino Tilings of a Simply Connected Region, Height Functions & Conway's Tiling Group

Rough notes from a presentation I gave on domino tilings and height functions. A fascinating example of when very intuitive and natural questions about tilings can be solved using ideas and tools from algebraic topology and geometric/combinatorial group theory.

As a way to visualise height functions and the domino group better, I also made a modular sculpture in brass, sycamore and pear woods with some 3d-printed parts. The sculpture allows one to go from one tiling to another using their lift in the Cayley graph of the domino group and visualise sequences of flips from that perspective, while seeing the projected tiling underneath! Here are a few pictures.