Sketch · 2026-09-23
Ford circles.
Every reduced fraction p over q gets a circle. Its radius is one over twice q squared. Its centre sits directly above p over q on the number line, at that same height. Two of these circles never overlap. They only ever touch, and when they touch, their fractions differ by exactly one over q times s. That is the whole rule, and the packing it produces is dense enough to hide the rationals inside it.
Drag left or right to slide the number line. Scroll or pinch to zoom. Circles for larger denominators appear as you zoom in, because a circle of radius one over two q squared shrinks fast. Hover over any circle to read its fraction and the neighbours it touches.
Note
Lester Ford introduced these circles in 1938 to give a geometric picture of Diophantine approximation. Two fractions p/q and r/s in lowest terms are called Farey neighbours when p s minus q r equals plus or minus one. Ford's theorem says that exactly those pairs correspond to circles that are tangent, and every other pair of circles is disjoint. Neighbours kiss, strangers stay apart.
The mediant of two neighbours p/q and r/s is (p+r)/(q+s), and it drops in between them as a new tangent circle sitting on top of the two originals. Doing this over and over walks the Stern-Brocot tree, and every reduced fraction shows up exactly once. That is why the packing looks self-similar under zoom: any gap between two touching circles is again a gap between two touching circles.
The picture also gives Hurwitz's theorem a physical intuition. To approximate an irrational number x by rationals, aim a vertical ray down at x and read off which circles it pierces. The deeper the ray goes into a circle, the better that fraction approximates x. The golden ratio is the number that the ray hits hardest: no rational approximates it more efficiently than the ratio of consecutive Fibonacci numbers.
Sources
- Ford, L. R. (1938). Fractions. The American Mathematical Monthly 45(9): 586-601.
- Hardy, G. H., Wright, E. M. (1979). An Introduction to the Theory of Numbers, 5th ed. Oxford University Press, chapters III and XI.
- Conway, J. H., Guy, R. K. (1996). The Book of Numbers. Copernicus, pages 152-156.