A Ford Circle is a circle associated with each fraction p⁄q, in simplest form, from 0 to 1. Each circle has centre (p⁄q, 1⁄2q2), and is tangent to the x-axis (i.e. with radius 1⁄2q2). These circles have the amazing property that they never intersect — not for any pair of fractions. Further, given two fractions, p⁄q and r⁄s, if |ps-qr| = 1 the corresponding two circles touch.
Use the activity below to explore the properties of these circles, and the many patterns they generate.
A Farey Sequence of order n is the sequence of simplest form fractions between 0 and 1 which, when have denominators less than or equal to n, arranged in increasing order. These sequences are useful for generating a set of rational numbers to plot as Ford Circles.