Ford circle
, expressed in lowest terms, there is a Ford circle whose center is at the point
The two Ford circles for rational numbers
(both in lowest terms) are tangent circles when
[1] Ford circles are a special case of mutually tangent circles; the base line can be thought of as a circle with infinite radius.
Systems of mutually tangent circles were studied by Apollonius of Perga, after whom the problem of Apollonius and the Apollonian gasket are named.
[2] In the 17th century René Descartes discovered Descartes' theorem, a relationship between the reciprocals of the radii of mutually tangent circles.
[2] Ford circles also appear in the Sangaku (geometrical puzzles) of Japanese mathematics.
A typical problem, which is presented on an 1824 tablet in the Gunma Prefecture, covers the relationship of three touching circles with a common tangent.
There is a Ford circle associated with every rational number.
is counted as a Ford circle – it can be thought of as the Ford circle associated with infinity, which is the case
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 coordinates.
(the x-coordinates of the centers of the Ford circles) and that is perpendicular to the
-axis (whose center is on the x-axis) also passes through the point where the two circles are tangent to one another.
The centers of the Ford circles constitute a discrete (and hence countable) subset of the plane, whose closure is the real axis - an uncountable set.
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.
[4] By interpreting the upper half of the complex plane as a model of the hyperbolic plane (the Poincaré half-plane model), Ford circles can be interpreted as horocycles.
In hyperbolic geometry any two horocycles are congruent.
When these horocycles are circumscribed by apeirogons they tile the hyperbolic plane with an order-3 apeirogonal tiling.
There is a link between the area of Ford circles, Euler's totient function
[5] As no two Ford circles intersect, it follows immediately that the total area of the Ford circles is less than 1.
In fact the total area of these Ford circles is given by a convergent sum, which can be evaluated.
From the definition, the area is Simplifying this expression gives where the last equality reflects the Dirichlet generating function for Euler's totient function
this finally becomes Note that as a matter of convention, the previous calculations excluded the circle of radius
, half of which lies outside the unit interval, hence the sum is still the fraction of the unit square covered by Ford circles.
The concept of Ford circles can be generalized from the rational numbers to the Gaussian rationals, giving Ford spheres.
In this construction, the complex numbers are embedded as a plane in a three-dimensional Euclidean space, and for each Gaussian rational point in this plane one constructs a sphere tangent to the plane at that point.
For a Gaussian rational represented in lowest terms as
The resulting spheres are tangent for pairs of Gaussian rationals
