Ford circle

Ford circles for q from 1 to 20. Circles with q ≤ 10 are labelled as p/q and colour-coded according to q. Each circle is tangential to the base line and its neighboring circles. Irreducible fractions with the same denominator have circles of the same size. (Click for larger version)

In mathematics, a Ford circle is a circle with centre at (p/q,1/(2q^2)) and radius 1/(2q^2), where p/q is an irreducible fraction, i.e. p and q are coprime integers. Each Ford circle is tangent to the horizontal axis y=0, and any two circles are either tangent or disjoint from each other.[1]

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.[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. 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:[3]

\frac{1}{\sqrt{r_\text{middle}}} = \frac{1}{\sqrt{r_\text{left}}} + \frac{1}{\sqrt{r_\text{right}}}.

Ford circles are named after the American mathematician Lester R. Ford, Sr., who wrote about them in 1938.[1]

Properties

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 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 p/q is between 0 and 1, the Ford circles that are tangent to C[p/q] can be described variously as

  1. the circles C[r/s] where |p s-q r|=1,[1]
  2. the circles associated with the fractions r/s that are the neighbours of p/q in some Farey sequence,[1] or
  3. the circles C[r/s] 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.[1]

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.[1]

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.[4] 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.

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[0/1].[5]

Total area of Ford circles

There is a link between the area of Ford circles, Euler's totient function \varphi, the Riemann zeta function \zeta, and Apéry's constant \zeta(3).[6] As no two Ford circles intersect, it follows immediately that the total area of the Ford circles

\left\{ C[p,q]: 0 \le \frac{p}{q} \le 1 \right\}

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

A = \sum_{q\ge 1} \sum_{ (p, q)=1 \atop 1 \le p < q }\pi \left( \frac{1}{2 q^2} \right)^2.

Simplifying this expression gives

A = \frac{\pi}{4} \sum_{q\ge 1} \frac{1}{q^4} \sum_{ (p, q)=1 \atop 1 \le p < q } 1 = \frac{\pi}{4} \sum_{q\ge 1} \frac{\varphi(q)}{q^4} = \frac{\pi}{4} \frac{\zeta(3)}{\zeta(4)},

where the last equality reflects the Dirichlet generating function for Euler's totient function \varphi(q). Since \zeta(4)=\pi^4/90, this finally becomes

A = \frac{45}{2} \frac{\zeta(3)}{\pi^3}\approx 0.872284041.

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 Ford, L. R. (1938), "Fractions", The American Mathematical Monthly 45 (9): 586–601, doi:10.2307/2302799, JSTOR 2302799, MR 1524411.
  2. ↑ 2.0 2.1 Coxeter, H. S. M. (1968), "The problem of Apollonius", The American Mathematical Monthly 75: 5–15, doi:10.2307/2315097, MR 0230204.
  3. ↑ Fukagawa, Hidetosi; Pedoe, Dan (1989), Japanese temple geometry problems, Winnipeg, MB: Charles Babbage Research Centre, ISBN 0-919611-21-4, MR 1044556.
  4. ↑ Conway, John H. (1997), The sensual (quadratic) form, Carus Mathematical Monographs 26, Washington, DC: Mathematical Association of America, pp. 28–33, ISBN 0-88385-030-3, MR 1478672.
  5. ↑ Graham, Ronald L.; Lagarias, Jeffrey C.; Mallows, Colin L.; Wilks, Allan R.; Yan, Catherine H. (2003), "Apollonian circle packings: number theory", Journal of Number Theory 100 (1): 1–45, arXiv:math.NT/0009113, doi:10.1016/S0022-314X(03)00015-5, MR 1971245.
  6. ↑ Marszalek, Wieslaw (2012), "Circuits with oscillatory hierarchical Farey sequences and fractal properties", Circuits, Systems and Signal Processing 31 (4): 1279–1296, doi:10.1007/s00034-012-9392-3.

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