A New Parameterization for Ford Circles

Abstract

Lester Ford introduced Ford circles in 1938 in order to geometrically understand the approximation of an irrational number by rational numbers. We shall construct Ford circles by a recursive geometric procedure. The Ford circles also turn out to be parameterized by the rational numbers. We introduce a new parameterization of the set of Ford circles in terms of triples of relatively prime integers that satisfy a certain equation. This is interesting because we have developed a better approximation for irrational numbers between 0 and 1 and because our new parameterization generalizes to a higher dimension.