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Ford circle
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{{short description|Rational circle tangent to the real line}} [[File:Ford_circles_colour.svg|upright=1.35|thumb|Ford circles for {{math|''p''/''q''}} with {{mvar|q}} from 1 to 20. Circles with {{math|''q'' ≤ 10}} are labelled as {{math|{{sfrac|''p''|''q''}}}} and color-coded according to {{mvar|q}}. Each circle is [[tangent]] 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]] in the [[Euclidean plane]], in a family of circles that are all tangent to the <math>x</math>-axis at [[rational number|rational]] points. For each rational number <math>p/q</math>, expressed in lowest terms, there is a Ford circle whose [[center (geometry)|center]] is at the point <math>(p/q,1/(2q^2))</math> and whose radius is <math>1/(2q^2)</math>. It is tangent to the <math>x</math>-axis at its bottom point, <math>(p/q,0)</math>. The two Ford circles for rational numbers <math>p/q</math> and <math>r/s</math> (both in lowest terms) are [[tangent circles]] when <math>|ps-qr|=1</math> and otherwise these two circles are disjoint.<ref name="ford"/> ==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.<ref name="coxeter">{{citation | last = Coxeter | first = H. S. M. | authorlink = Harold Scott MacDonald Coxeter | journal = [[The American Mathematical Monthly]] | mr = 0230204 | pages = 5–15 | title = The problem of Apollonius | volume = 75 | year = 1968 | issue = 1 | doi=10.2307/2315097| jstor = 2315097 }}.</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"/> 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 | last1 = Fukagawa | first1 = Hidetosi | last2 = Pedoe | first2 = Dan | isbn = 0-919611-21-4 | location = Winnipeg, MB | mr = 1044556 | publisher = Charles Babbage Research Centre | title = Japanese temple geometry problems | year = 1989}}.</ref> :<math>\frac{1}{\sqrt{r_\text{middle}}} = \frac{1}{\sqrt{r_\text{left}}} + \frac{1}{\sqrt{r_\text{right}}}.</math> 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 | last = Ford | first = L. R. | authorlink = Lester R. Ford | doi = 10.2307/2302799 | issue = 9 | journal = [[The American Mathematical Monthly]] | jstor = 2302799 | mr = 1524411 | pages = 586–601 | title = Fractions | volume = 45 | year = 1938}}.</ref> ==Properties== [[File:Comparison_Ford_circles_Farey_diagram.svg|thumb|upright=1.35|link={{filepath:comparison_Ford_circles_Farey_diagram.svg}}|Comparison of Ford circles and a Farey diagram with circular arcs for ''n'' from 1 to 9. Note that each arc intersects its corresponding circles at right angles. In [{{filepath:comparison_Ford_circles_Farey_diagram.svg}} {{nowrap|the SVG image,}}] hover over a circle or curve to highlight it and its terms.]] 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> 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 # the circles <math>C[r/s]</math> where <math>|p s-q r|=1,</math><ref name="ford"/> # the circles associated with the fractions <math>r/s</math> that are the neighbors of <math>p/q</math> in some [[Farey sequence]],<ref name="ford"/> or # 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"/> If <math>C[p/q]</math> and <math>C[r/s]</math> are two tangent Ford circles, then the circle through <math>(p/q,0)</math> and <math>(r/s,0)</math> (the x-coordinates of the centers of the Ford circles) and that is perpendicular to the <math>x</math>-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 Gamma|modular group]] of transformations of the complex plane maps Ford circles to other Ford circles.<ref name="ford"/> 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 | last1 = Graham | first1 = Ronald L. | author1-link = Ronald Graham | last2 = Lagarias | first2 = Jeffrey C. | author2-link = Jeffrey Lagarias | last3 = Mallows | first3 = Colin L. | last4 = Wilks | first4 = Allan R. | last5 = Yan | first5 = Catherine H. | arxiv = math.NT/0009113 | doi = 10.1016/S0022-314X(03)00015-5 | issue = 1 | journal = Journal of Number Theory | mr = 1971245 | pages = 1–45 | title = Apollonian circle packings: number theory | volume = 100 | year = 2003| s2cid = 16607718 }}.</ref> 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 be interpreted as [[horocycle]]s. In [[hyperbolic geometry]] any two horocycles are [[congruence (geometry)|congruent]]. When these [[horocycle]]s are [[tangential polygon|circumscribed]] by [[apeirogon]]s they [[tessellation|tile]] the hyperbolic plane with an [[order-3 apeirogonal tiling]]. <!-- ref to check <ref>{{citation | last = Conway | first = John H. | author-link = John Horton Conway | isbn = 0-88385-030-3 | location = Washington, DC | mr = 1478672 | pages = 28–33 | publisher = Mathematical Association of America | series = Carus Mathematical Monographs | title = The sensual (quadratic) form | volume = 26 | year = 1997}}.</ref> --> ==Total area of Ford circles== 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 | last = Marszalek | first = Wieslaw | doi = 10.1007/s00034-012-9392-3 | issue = 4 | journal = Circuits, Systems and Signal Processing | pages = 1279–1296 | title = Circuits with oscillatory hierarchical Farey sequences and fractal properties | volume = 31 | year = 2012| s2cid = 5447881 }}.</ref> As no two Ford circles intersect, it follows immediately that the total area of the Ford circles :<math>\left\{ C[p,q]: 0 < \frac{p}{q} \le 1 \right\}</math> 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 :<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> Simplifying this expression gives :<math> 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)},</math> 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 :<math> A = \frac{45}{2} \frac{\zeta(3)}{\pi^3}\approx 0.872284041.</math> Note that as a matter of convention, the previous calculations excluded the circle of radius <math>\frac{1}{2}</math> corresponding to the fraction <math>\frac{0}{1}</math>. It includes the complete circle for <math>\frac{1}{1}</math>, 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) == [[File:Ford-Kugeln.png|thumb|Ford spheres above the complex domain]] The concept of Ford circles can be generalized from the rational numbers to the [[Gaussian rational]]s, giving Ford spheres. In this construction, the [[complex number]]s 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 <math>p/q</math>, the diameter of this sphere should be <math>1/2q\bar q</math> where <math>\bar q</math> represents the [[complex conjugate]] of <math>q</math>. The resulting spheres are [[tangent]] for pairs of Gaussian rationals <math>P/Q</math> and <math>p/q</math> with <math>|Pq-pQ|=1</math>, and otherwise they do not intersect each other.<ref>{{citation|title=Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning|first=Clifford A.|last=Pickover|authorlink=Clifford A. Pickover|publisher=Oxford University Press|year=2001|isbn=9780195348002|contribution=Chapter 103. Beauty and Gaussian Rational Numbers|pages=243–246|url=https://books.google.com/books?id=52N0JJBspM0C&pg=PA243}}.</ref><ref>{{citation|year=2015|arxiv=1503.00813|title=Ford Circles and Spheres|first=Sam|last=Northshield|bibcode=2015arXiv150300813N}}.</ref> ==See also== * [[Apollonian gasket]] – a fractal with infinite mutually tangential circles in a circle instead of on a line * [[Steiner chain]] * [[Pappus chain]] ==References== {{reflist}} ==External links== * [http://www.cut-the-knot.org/proofs/fords.shtml Ford's Touching Circles] at [[cut-the-knot]] * {{mathworld|urlname=FordCircle|title=Ford Circle}} * {{cite web|last1=Bonahon|first1=Francis|authorlink=Francis Bonahon|title=Funny Fractions and Ford Circles|date=9 June 2015 |url=https://www.youtube.com/watch?v=0hlvhQZIOQw |archive-url=https://ghostarchive.org/varchive/youtube/20211221/0hlvhQZIOQw |archive-date=2021-12-21 |url-status=live|publisher=[[Brady Haran]]|accessdate=9 June 2015|format=YouTube video}}{{cbignore}} {{DEFAULTSORT:Ford Circle}} [[Category:Circle packing]] [[Category:Fractions (mathematics)]]
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