Aperiodic Geometry, Roger Penrose, Oxford University

If I were to prove the fact that Penrose tiles (with matching rules!) only allow for non-periodic tilings, I’d start with substitution rules, inflation and deflation and the up-down generation of tilings. Given a valid tiling, you can replace its tiles with smaller tiles from the same set. That’s deflation. This you can use to show that you can make tilings covering arbitrary areas. But, in this context more interestingly, you can also perform the opposite direction, which is inflation. So if a tiling fills the whole plane, then you can locally replace combinations of tiles with larger tiles. This you can prove locally, by showing that any finite combination of tiles which can not be composed in this way also cannot lead to an infinite tiling because somewhere something doesn’t match up. Composition is even unique, which again can be shown locally. Once you have the fact that every tile is part of an inflated version in a unique way, you can use this to label each tile. A small-triangl