Guillotine Partitions and the Hipparchus Operad

If you dissect a square into n similar rectangles, what proportions can these rectangles have? Folks on Mathstodon figured this out for n ≤ 7, and I blogged about it here recently. But I was left feeling that some deeper structure governed this problem.

Various people on Mathstodon, including Steven Stanicki, David Eppstein and Rahul Narain, convinced me of the importance of a certain class of dissections called ‘guillotine partitions’. I started suspecting that these were connected to an op...