Ford Circles
Every rational number p/q (in lowest terms) sits beneath a tangent circle of radius 1/(2q²). These circles never overlap — they only ever kiss. Hidden in their arrangement is the entire Stern-Brocot tree, a structure that contains every positive rational exactly once.
C(p,q): center (p/q, 1/(2q²)), radius 1/(2q²)
What are Ford circles?
Named after mathematician Lester R. Ford Sr. (1886–1967), Ford circles are a beautiful construction from number theory. For every fraction p/q in lowest terms (where gcd(p, q) = 1), draw a circle tangent to the number line at x = p/q with radius 1/(2q²). The remarkable fact: no two Ford circles ever overlap. Two circles are tangent if and only if their fractions are Farey neighbors — that is, |p′q − pq′| = 1.
The Stern-Brocot tree
The Stern-Brocot tree organizes all positive rationals into a binary search tree. Starting from 0/1 and 1/0, each new fraction is the mediant of its neighbors: (a+c)/(b+d) from a/b and c/d. Every Ford circle touches exactly two larger circles above it, and these two parent fractions have the selected fraction as their mediant. Click any circle to see this relationship.
Farey sequences
The Farey sequence Fn lists all fractions between 0 and 1 with denominator at most n, in order. Consecutive fractions in a Farey sequence are always Farey neighbors, meaning their Ford circles are tangent. As you increase the max denominator slider, you are effectively building up successive Farey sequences.