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.