Roger Penrose is well known for his collaborations with Stephen Hawking studying black holes. Perhaps being a cosmologist made him interested in the work of Kepler. And perhaps that led him to thinking about tilings with pentagons. In any case, he ended up discovering a some remarkable things about tiling. In particular, he discovered Penrose tiles.
Unlike most tilings, Penrose tiles have rules for how edges can be matched. It is not enough that edges have the same length. The rules can be captured by adding tabs to the edges that guarantee that the tiles are joined according to the rules. And remarkably, a tiling that follows these rules is aperiodic, which means it is not a repeating pattern as you’ve seen with regular polygons. Below is an example of Penrose P3 tiles.
Amazingly, Penrose’s tilings ended up being useful in explaining physical phenomena that was discovered after Penrose discovered the tiles. You can read more about applications to crystallography, as well as the notion of inflation in the article Penrose Tiles talk across the Miles.