A new geometric approach to Sturmian words

We introduce a new geometric approach to Sturmian words by means of a mapping that associates certain lines in the n x n -grid and sets of finite Sturmian words of length n. Using this mapping, we give new proofs of the formulas enumerating the finite Sturmian words and the palindromic finite Sturmian words of a given length. We also give a new proof for the well-known result that a factor of a Sturmian word has precisely two return words.

💡 Research Summary

The paper presents a novel geometric framework for studying finite Sturmian words, establishing a one‑to‑one correspondence between certain lines drawn in an n × n integer grid and binary strings of length n that are precisely the finite Sturmian words. The authors begin by recalling the defining properties of Sturmian sequences—balancedness and minimal factor complexity n + 1—and note that most existing proofs rely on combinatorial or number‑theoretic constructions such as cutting sequences of irrational rotations.
The core construction is as follows. For a fixed n, consider all straight lines that start at the origin (0,0) and end at the opposite corner (n,n) with slope k∈