Rational circle tangent to the real line
In mathematics, a Ford circle is a circle in the Euclidean plane, in a family of circles that are all tangent to the formula_1-axis at rational points. For each rational number formula_2, expressed in lowest terms, there is a Ford circle whose center is at the point formula_3 and whose radius is formula_4. It is tangent to the formula_1-axis at its bottom point, formula_6. The two Ford circles for rational numbers formula_2 and formula_8 (both in lowest terms) are tangent circles when formula_9 and otherwise these two circles are disjoint.
History.
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. In the 17th century René Descartes discovered Descartes' theorem, a relationship between the reciprocals of the radii of mutually tangent circles.
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. Given the size of the two outer large circles, what is the size of the small circle between them? The answer is equivalent to a Ford circle:
formula_10
Ford circles are named after the American mathematician Lester R. Ford, Sr., who wrote about them in 1938.
Properties.
The Ford circle associated with the fraction formula_2 is denoted by formula_12 or formula_13 There is a Ford circle associated with every rational number. In addition, the line formula_14 is counted as a Ford circle – it can be thought of as the Ford circle associated with infinity, which is the case formula_15
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. If formula_2 is between 0 and 1, the Ford circles that are tangent to formula_12 can be described variously as
If formula_12 and formula_18 are two tangent Ford circles, then the circle through formula_6 and formula_30 (the x-coordinates of the centers of the Ford circles) and that is perpendicular to the formula_1-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.
Ford circles are a sub-set of the circles in the Apollonian gasket generated by the lines formula_32 and formula_14 and the circle formula_34
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.
Total area of Ford circles.
There is a link between the area of Ford circles, Euler's totient function formula_35 the Riemann zeta function formula_36 and Apéry's constant formula_37 As no two Ford circles intersect, it follows immediately that the total area of the Ford circles
formula_38
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
formula_39
Simplifying this expression gives
formula_40
where the last equality reflects the Dirichlet generating function for Euler's totient function formula_41 Since formula_42 this finally becomes
formula_43
Note that as a matter of convention, the previous calculations excluded the circle of radius formula_44 corresponding to the fraction formula_45. It includes the complete circle for formula_46, half of which lies outside the unit interval, hence the sum is still the fraction of the unit square covered by Ford circles.
Ford spheres (3D).
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 formula_2, the diameter of this sphere should be formula_48 where formula_49 represents the complex conjugate of formula_50. The resulting spheres are tangent for pairs of Gaussian rationals formula_51 and formula_2 with formula_53, and otherwise they do not intersect each other.
References.
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