Turtles, Hats and Spectres: Aperiodic structures on a Rhombic tiling

These notes derive aperiodic monotiles (arXiv:2303.10798) from a set of rhombuses with matching rules. This dual construction is used to simplify the proof of aperiodicity by considering the tiling as a colouring game on a Rhombille tiling. A simple recursive substitution system is then introduced to show the existence of a non-periodic tiling without the need for computer-aided verification. A new cut-and-project style construction linking the Turtle tiling with 1-dimensional Fibonacci words provides a second proof of non-periodicity, and an alternative demonstration that the Turtle can tile the plane. Deforming the Turtle into the Hat tile then provides a third proof for non-periodicity by considering the effect on the lattice underlying the Rhombille tiling. Finally, attention turns to the Spectre tile. In collaboration with Erhard K\"unzel and Yoshiaki Araki, we present two new substitution rules for generating Spectre tilings. This pair of conjugate rules show that the aperiodic monotile tilings can be considered as a 2-dimensional analog to Sturmian words.