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.



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

 

About the activity

The circles never intersect — somehow all rational numbers are sufficiently far from their neighbours. This is an incredible and deep result.

Given two circles, one based on p⁄q and the other on r⁄s, we have already seen that they touch if |ps-qr| = 1. Additionally, if these circles do touch, then the circle corresponding to the mediant fraction  p + r⁄q + s  is tangent to them both.

Problems involving tangent circles, such as Ford Circles, often appear on Sangaku — colourful, traditional Japanese mathematical puzzle tablets hung from the roofs of Shinto and Buddhist shrines during the 18th and 19th centuries.

Related...

Coding Corner

Click here to explore Farey Sequences with computer code.