A Ford Circle is a circle associated with each fraction pq, in simplest form, from 0 to 1. Each circle has centre (pq, 12q2), and is tangent to the x-axis (i.e. with radius 12q2). These circles have the amazing property that they never intersect — not for any pair of fractions. Further, given two fractions, pq and rs, 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.

Ford Circles
Linking algebra and geometry
Enter a fraction to add its associated Ford Circle to the chart, or use the Farey buttons to generate a sequence of circles.
Click a circle to see its associated rational number. Click and hold to magnify.

Look for patterns in the "Farey" Circles, or try adding a sequence of fractions that follow a pattern of their own, and see if there is an associated pattern in the circles.

Enter a fraction