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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, which is the case p = 1, q = 0.
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.
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.
- http://www.josleys.com/creatures41.htm - art and graphics based on Ford circles