Ford Circles & Farey Fractions
Every reduced fraction p/q can be drawn as a Ford circle of radius 1/(2q²) resting on the number line at p/q, and two Ford circles never overlap — they are tangent exactly when the fractions are neighbours in a Farey sequence. This tool builds the fractions between 0 and 1 by taking mediants (the Stern–Brocot / Farey construction), draws their Ford circles, and lets you raise the maximum denominator to watch the gaps fill with ever smaller kissing circles.
Runs 100% in your browser — simulations are computed locally on your device.
Read the full guide to this tool
Notes
- The mediant of a/b and c/d is (a+c)/(b+d); it always lands between them and is already in lowest terms here.
- The Ford circle for p/q has radius 1/(2q²), so larger denominators give smaller circles.
- Two Ford circles are tangent exactly when |ps − rq| = 1 — the neighbour condition in a Farey sequence.
- Runs 100% in your browser — simulations are computed locally on your device.
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