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| [[File:Circumferències de Ford.svg|400px|right|thumb|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 [[tangent]]ial to the base line and its neighboring circles. Irreducible fractions with the same denominator have circles of the same size.]]
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| In [[mathematics]], a '''Ford circle''' is a [[circle]] with [[centre (geometry)|centre]] at <math>(p/q,1/(2q^2))</math> and [[radius]] <math>1/(2q^2),</math> where <math>p/q</math> is an [[irreducible fraction]], i.e. <math>p</math> and <math>q</math> are [[coprime]] [[integer]]s. Each Ford circle is tangent to the horizontal axis <math>y=0,</math> and any two circles are either [[tangent circles|tangent]] or disjoint from each other.<ref name="ford"/>
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| ==History==
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| 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.<ref name="coxeter">{{citation
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| | last = Coxeter | first = H. S. M. | authorlink = Harold Scott MacDonald Coxeter
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| | journal = [[The American Mathematical Monthly]]
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| | mr = 0230204
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| | pages = 5–15
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| | title = The problem of Apollonius
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| | volume = 75
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| | year = 1968}}.</ref> In the 17th century [[René Descartes]] discovered [[Descartes' theorem]], a relationship between the reciprocals of the radii of mutually tangent circles.<ref name="coxeter"/>
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| 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:<ref>{{citation
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| | last1 = Fukagawa | first1 = Hidetosi
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| | last2 = Pedoe | first2 = Dan
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| | isbn = 0-919611-21-4
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| | location = Winnipeg, MB
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| | mr = 1044556
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| | publisher = Charles Babbage Research Centre
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| | title = Japanese temple geometry problems
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| | year = 1989}}.</ref>
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| :<math>\frac{1}{\sqrt{r_\text{middle}}} = \frac{1}{\sqrt{r_\text{left}}} + \frac{1}{\sqrt{r_\text{right}}}.</math>
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| Ford circles are named after the American mathematician [[Lester R. Ford|Lester R. Ford, Sr.]], who wrote about them in 1938.<ref name="ford">{{citation
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| | last = Ford | first = L. R. | authorlink = Lester R. Ford
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| | doi = 10.2307/2302799
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| | issue = 9
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| | journal = [[The American Mathematical Monthly]]
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| | jstor = 2302799
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| | mr = 1524411
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| | pages = 586–601
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| | title = Fractions
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| | volume = 45
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| | year = 1938}}.</ref>
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| ==Properties==
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| The Ford circle associated with the fraction <math>p/q</math> is denoted by <math>C[p/q]</math> or <math>C[p,q].</math> There is a Ford circle associated with every [[rational number]]. In addition, the line <math>y=1</math> is counted as a Ford circle – it can be thought of as the Ford circle associated with [[infinity]], which is the case <math>p=1,q=0.</math>
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| Two different Ford circles are either [[Disjoint sets|disjoint]] or [[tangent]] to one another. No two interiors of Ford circles intersect, even though there is a Ford circle tangent to the [[Cartesian coordinate system|''x''-axis]] at each point on it with [[rational number|rational]] coordinates. If <math>p/q</math> is between 0 and 1, the Ford circles that are tangent to <math>C[p/q]</math> can be described variously as
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| #the circles <math>C[r/s]</math> where <math>|p s-q r|=1,</math><ref name="ford"/>
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| #the circles associated with the fractions <math>r/s</math> that are the neighbours of <math>p/q</math> in some [[Farey sequence]],<ref name="ford"/> or
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| #the circles <math>C[r/s]</math> where <math>r/s</math> is the next larger or the next smaller ancestor to <math>p/q</math> in the [[Stern–Brocot tree]] or where <math>p/q</math> is the next larger or next smaller ancestor to <math>r/s</math>.<ref name="ford"/>
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| Ford circles can also be thought of as curves in the [[complex plane]]. The [[modular group Gamma|modular group]] of transformations of the complex plane maps Ford circles to other Ford circles.<ref name="ford"/>
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| By interpreting the upper half of the complex plane as a model of the [[Hyperbolic geometry|hyperbolic plane]] (the Poincaré half-plane model) Ford circles can also be interpreted as a [[tessellation|tiling]] of the hyperbolic plane by [[horocycle]]s. Any two Ford circles are [[congruence (geometry)|congruent]] in [[hyperbolic geometry]].<ref>{{citation
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| | last = Conway | first = John H. | author-link = John Horton Conway
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| | isbn = 0-88385-030-3
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| | location = Washington, DC
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| | mr = 1478672
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| | pages = 28–33
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| | publisher = Mathematical Association of America
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| | series = Carus Mathematical Monographs
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| | title = The sensual (quadratic) form
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| | volume = 26
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| | year = 1997}}.</ref> If <math>C[p/q]</math> and <math>C[r/s]</math> are tangent Ford circles, then the half-circle joining <math>(p/q,0)</math> and <math>(r/s,0)</math> that is perpendicular to the <math>x</math>-axis is a hyperbolic line that also passes through the point where the two circles are tangent to one another.
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| Ford circles are a sub-set of the circles in the [[Apollonian gasket]] generated by the lines <math>y=0</math> and <math>y=1</math> and the circle <math>C[0/1].</math><ref>{{citation
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| | last1 = Graham | first1 = Ronald L. | author1-link = Ronald Graham
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| | last2 = Lagarias | first2 = Jeffrey C. | author2-link = Jeffrey Lagarias
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| | last3 = Mallows | first3 = Colin L.
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| | last4 = Wilks | first4 = Allan R.
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| | last5 = Yan | first5 = Catherine H.
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| | arxiv = math.NT/0009113
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| | doi = 10.1016/S0022-314X(03)00015-5
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| | issue = 1
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| | journal = Journal of Number Theory
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| | mr = 1971245
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| | pages = 1–45
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| | title = Apollonian circle packings: number theory
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| | volume = 100
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| | year = 2003}}.</ref>
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| ==Total area of Ford circles==
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| There is a link between the area of Ford circles, [[Euler's totient function]] <math>\varphi,</math> the [[Riemann zeta function]] <math>\zeta,</math> and [[Apéry's constant]] <math>\zeta(3).</math><ref>{{citation
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| | last = Marszalek | first = Wieslaw
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| | doi = 10.1007/s00034-012-9392-3
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| | issue = 4
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| | journal = Circuits, Systems and Signal Processing
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| | pages = 1279–1296
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| | title = Circuits with oscillatory hierarchical Farey sequences and fractal properties
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| | volume = 31
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| | year = 2012}}.</ref> As no two Ford circles intersect, it follows immediately that the total area of the Ford circles
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| :<math>\left\{ C[p,q]: 0 \le \frac{p}{q} \le 1 \right\}</math>
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| 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
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| :<math> A = \sum_{q\ge 1} \sum_{ (p, q)=1 \atop 1 \le p < q }\pi \left( \frac{1}{2 q^2} \right)^2.</math>
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| Simplifying this expression gives
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| :<math> A = \frac{\pi}{4} \sum_{q\ge 1} \frac{1}{q^4}
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| \sum_{ (p, q)=1 \atop 1 \le p < q } 1 =
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| \frac{\pi}{4} \sum_{q\ge 1} \frac{\varphi(q)}{q^4} =
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| \frac{\pi}{4} \frac{\zeta(3)}{\zeta(4)},</math>
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| where the last equality reflects the [[Dirichlet generating function]] for [[Euler's totient function]] <math>\varphi(q).</math> Since <math>\zeta(4)=\pi^4/90,</math> this finally becomes
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| :<math> A = \frac{45}{2} \frac{\zeta(3)}{\pi^3}\approx 0.872284041.</math>
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://www.cut-the-knot.org/proofs/fords.shtml Ford's Touching Circles] at [[cut-the-knot]]
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| * {{mathworld|urlname=FordCircle|title=Ford Circle}}
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| {{DEFAULTSORT:Ford Circle}}
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| [[Category:Circles]]
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| [[Category:Fractions]]
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