Mechanical word

For slope $0\le \alpha \le 1$ and intercept $\rho$, the lower mechanical word has letters $w_n=\lfloor (n+1)\alpha +\rho \rfloor -\lfloor n\alpha +\rho \rfloor$. These letters are zero or one. Replacing floors by ceilings defines the upper mechanical word. An irrational slope gives a Sturmian word.

The floor-based mechanical word $w_n=\lfloor (n+1)\alpha +\rho \rfloor -\lfloor n\alpha +\rho \rfloor$.