Aperiodic tile sets from Sturmian lattices

We give an explicit algorithm to construct aperiodic tile sets based on Sturmian words of quadratic slopes. The method works for any quadratic irrational slope, and we can produce an aperiodic tile set whose underlying scaling constant is a unit of any real quadratic field. There are two key ingredients in our construction. The first one is the ``Sturmian lattices'', an interesting grid structure generated by Sturmian words that emerged in an aperiodic monotile called Smith Turtle. The second is the bounded displacement equivalence of Delone sets, which plays a central role in this construction. A classification of Sturmian lattices and complete proofs are given in the full version.