I found these fantastic lecture notes on Penrose’s aperiodic tilings by Alexander F. Ritter, which he wrote for a masterclass at Oxford in 2014.

There is not much systematic well-structured material on aperiodic tilings around, so I thought I would share this for whoever is interested. The material is accessible to every undergrad that has taken a class in real analysis (some nice application of extracting a converging subsequence from a sequence in a compact space, in the proof of the Extension Theorem), and includes many painstaking hand-drawings that are very helpful in following the arguments. Besides being overall well-written, the notes are also good at hammering home some important points – for example, the fact that the Composition and Decomposition operations that one can make on a tiling are unique and thus reversible in the case of tilings by Penrose’s Kites and Darts tiles, which has a cascade of consequences (they prove aperiodicity of the tilings in a few lines, for example) which do not apply to regular periodic tilings for this very reason (non-uniqueness).