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For an irrational number $0<\alpha <1$, the characteristic Sturmian word of slope $\alpha$ is the binary sequenceEach term is zero or one because the arguments differ by less than one. This fixes an intercept convention: it is the lower mechanical word with intercept $\alpha$.

For $\alpha =(3-\sqrt{5})/2$, the beginning is $0100101001\dots$, the Fibonacci word. Like every Sturmian word, it has exactly $n+1$ distinct factors of length $n$. Its relation to continued fractions makes its recursive block structure reflect the partial quotients of its slope.

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.

Fix an irrational number $\alpha =[0;a_1,a_2,\dots ]$ and let $q_k$ be its convergent denominators, with $q_{-1}=0,q_0=1$. Ostrowski numeration writes each nonnegative integer uniquely as a finite sum $N={\sum }_{k\ge 0}{b}_{k+1}{q}_k$, subject to $0\le b_1<a_1$, $0\le b_{k+1}\le a_{k+1}$ for $k\ge 1$, and $b_{k+1}=a_{k+1}\Rightarrow b_k=0$.

The digit restrictions remove the ambiguities caused by $q_{k+1}=a_{k+1}{q}_k+q_{k-1}$. For $\alpha =(\sqrt{5}-1)/2$, all partial quotients equal one, so this becomes the representation by nonconsecutive Fibonacci numbers, with the duplicate initial weight removed. The numeration is useful for linking integer expansions to irrational rotations and Sturmian words.