Ford circles are named after American mathematician Lester R. Ford, Sr, who described them in an article in American Mathematical Monthly in 1938.
The Ford circle associated with the fraction p/q is denoted by C[p/q] or C[p,q]. There is a Ford circle associated with every rational number. In addition, the line y=1 is counted as a Ford circle - it can be thought of as the Ford circle associated with infinity.
Two different Ford circles are either disjoint or tangent to one another. No two Ford circles intersect - even though there is a Ford circle tangent to the x-axis at each point on it with rational co-ordinates. If p/q is between 0 and 1, the Ford circles that are tangent to C[p/q] are precisely those associated with the fractions that are the neighbours of p/q in some Farey sequence.
Ford circles can also be thought of as curves in the complex plane. The modular group of transformations of the complex plane map one Ford circle onto another Ford circle.
By interpreting the upper half of the complex plane as a model of the hyperbolic plane (the Poincaré half-plane model) Ford circles can also be interpreted as a tiling of the hyperbolic plane. Any two Ford circles are congruent in hyperbolic geometry. If C[p/q] and C[r/s] are tangent Ford circles, then the half-circle joining (p/q,0) and (r/s,0) that is perpendicular to the x-axis is a hyperbolic line that also passes through the point where the two circles are tangent to one another.
History
Properties
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