Ford Circles

Lester R. Ford presented these circles and proved their tangency and non-overlap properties in his 1938 paper “Fractions”, published in The American Mathematical Monthly.

Each reduced fraction \(\frac{a}{b}\) defines a Ford circle with radius \(r=\frac{1}{2b^2}\) and center \((\frac{a}{b},r)\).

Any two distinct Ford circles on the same side of a single number line are either tangent or disjoint: they never overlap! In fact, two Ford circles corresponding to \(\frac{a}{b}\) and \(\frac{c}{d}\) are tangent exactly when

\[ |ad-bc|=1. \]

When this holds, \(\frac{a}{b}\) and \(\frac{c}{d}\) are called adjacent fractions, and their Ford circles are tangent at

\[ \left(\frac{ab+cd}{b^2+d^2},\;\frac{1}{b^2+d^2}\right). \]

I think that’s pretty cool!

Note: these coordinates are not derived in the tangency proof in Ford’s original paper, but they are offered as an exercise in Allen Hatcher’s (Cornell professor!!!!) Topology of Numbers, Chapter 1, Exercise 4, page 32. Check out my proof here!

The homepage background repeats this construction with some slight edits. Read about the specific modifications.

Inspired by Jacob Rus’s Ford Circles.