  A staircase is the set of points in Z^2 below a given rational line in the
plane that have Manhattan Distance less than 1 to the line. Staircases are
closely related to Beatty and Sturmian sequences of rational numbers.
Connecting the geometry and the number theoretic concepts, we obtain three
equivalent characterizations of Sturmian sequences of rational numbers, as well
as a new proof of Barvinok's Theorem in dimension two, a recursion formula for
Dedekind-Carlitz polynomials and a partially new proof of White's
characterization of empty lattice tetrahedra. Our main tool is a recursive
description of staircases in the spirit of the Euclidean Algorithm.
