Ford Circle - Properties

Properties

The Ford circle associated with the fraction p/q is denoted by C or C. 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, which is the case p = 1, q = 0.

Two different Ford circles are either disjoint or tangent to one another. No two interiors of 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 can be described variously as

1. the circles C where |ps − qr| = 1,
2. the circles associated with the fractions r/s that are the neighbours of p/q in some Farey sequence, or
3. the circles C where r/s is the next larger or the next smaller ancestor to p/q in the Stern–Brocot tree or where p/q is the next larger or next smaller ancestor to r/s.

Ford circles can also be thought of as curves in the complex plane. The modular group of transformations of the complex plane maps Ford circles to other Ford circles.

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 by horocycles. Any two Ford circles are congruent in hyperbolic geometry. If C and C 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.

Ford circles are a sub-set of the circles in the Apollonian gasket generated by the lines y = 0 and y = 1 and the circle C.