Ford circle

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Ford circles. A circle rests upon each fraction in lowest terms. The darker circles shown are for the fractions 0/1, 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5 and 4/5. Each circle is tangential to the base line and its neighboring circles. Irreducible fractions with the same denominator have circles of the same size.

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[edit]

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[edit]

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[edit]

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[edit]

  1. ^ a b c d e f Ford, L. R. (1938), "Fractions", The American Mathematical Monthly 45 (9): 586–601, doi:10.2307/2302799, JSTOR 2302799, MR 1524411 .
  2. ^ a b Coxeter, H. S. M. (1968), "The problem of Apollonius", The American Mathematical Monthly 75: 5–15, 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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