Ford Circles — Geometry in the Rationals

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Selected Circle

1 2

Decimal 0.5

Circle Properties

Center (0.5, 0.125)

Radius 1/8

Formula 1/(2·4)

Farey Neighbors (Tangent Circles)

Two circles are tangent iff |ad − bc| = 1 for a/b and c/d.

0/1 1/1

Statistics

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circles visible

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tangencies

Ford Circles & Farey Sequences: Named after Lester Ford (1938), each reduced fraction p/q defines a circle centered at (p/q, 1/(2q²)) with radius 1/(2q²). The remarkable property: two circles are tangent exactly when their fractions are Farey neighbors — meaning |ad − bc| = 1 for fractions a/b and c/d. This connects directly to the Stern-Brocot tree: the mediant of two tangent circles' fractions appears where a third circle nestles between them.