Circle packings in the plane date back over 2000 years and have attracted the attention of princesses and Nobel Prize winners alike. We give a brief history and describe some recent developments

Pollicott, Mark

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     warwick.ac.uk/lib-publications      Manuscript version: Published Version The version presented in WRAP is the published version (Version of Record).  Persistent WRAP URL: http://wrap.warwick.ac.uk/106292                             How to cite: The repository item page linked to above, will contain details on accessing citation guidance from the publisher.  Copyright and reuse: The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions.   Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.  Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.  Publisher’s statement: Please refer to the repository item page, publisher’s statement section, for further information.  For more information, please contact the WRAP Team at: wrap@warwick.ac.uk  FEATURE 1Appolonius circle countingMARK POLLICOTTAbstract. Circle packings in the plane date back over 2000 years and have attracted the attention ofprincesses and Nobel prize winners alike. We give a brief history and describe some recent developments.Appolonius and his circlesThe famous geometer Apollonius of Perga (circa 240-190 BC) made many important contributions to math-ematics, perhaps the best known nowadays being thecoining of the modern terms ellipse, hyperbola andparabola. Apollonius also established the followingbasic result.Theorem 1 (Apollonius). Given three tangent circlesin the plane there are precisely two more circles eachof which mutually tangent to the original three.Figure 1: The three initialcircles (or radii r1, r2, r3, say)and the new Apolloniancircles (of radii r0, r4, say)There is a simplemodern proof ofthis theorem usingMöbius maps [7]. Bytransforming to in-nity one of thepoints where twoof the original cir-cles touch the imagecircles are straightlines. The new cir-cles are then sim-ply translates of thethird circle.Descartes and Princess Elizabeth of BohemiaPrincess Elizabeth (1618-1680) was the daughter ofKing Frederick of Bohemia, whose ill-fated reignlasted a mere 1 year and 4 days. Her superior educa-tion included instruction from the great french math-ematician and philosopher Rene Descartes (1596-1650). Included in their correspondence was a formularelating the radii r0, r4 of the two additional circlesgiven by Apollonius’ theorem to the radii r1, r2 andr3 of the original three circles.Theorem 2 (Descartes-Princess Elizabeth). The twopossible solutions ξ to the quadratic equation1ξ2+1r 21+1r 22+1r 23=(1ξ+1r1+1r2+1r3)2correspond to r0 and −r4.Sadly, when the princess petitioned Queen Chris-tine of Sweden for support in recovering her landsthe only consequence was that Descartes went toStockholm, dying there of pneumonia due to having5:00am audiences in a drafty palace. Thwarted in herattempts to recover her family’s lands in Bohemia,the princess retreated to a nunnery.Frederick SoddyThe formula was rediscovered two centuries laterby Frederick Soddy (1877-1956), and for this reasonthe circles are sometimes called Soddy circles. Soddyis probably more famous for having won the NobelPrize for Chemistry in 1921, and having introducedthe terms isotopes and chain reaction.In 1936 he published the same theorem in the formof a poem in the journal Nature entitled The KissPrecise [8].Part of “The kiss precise”Four circles to the kissing come.The smaller are the benter.The bend is just the inverse ofThe distance from the center.Though their intrigue left Euclid dumbThere’s now no need for rule of thumb.Since zero bend’s a dead straight lineAnd concave bends have minus sign,The sum of the squares of all four bendsIs half the square of their sum.2 FEATURECounting innitely many circlesWe can continue to add circles ad innitum by itera-tively applying Theorem 1 to every triples of mutuallytangent circles we see. Let (rn)∞n=0 enumerate theirradii.Example 1. If we start with 1r0 = 3,1r1= 5, 1r2 =1r3=8 then we can order the reciprocals of all the newradii. In the example above this becomes(1/rn) = 5, 8, 8, 12, 12, 20, 20, 21, 29, 29, 32, 32, · · ·5881212202021292932324444454848535353535656686868687777777777929293939393101101104104108108116116117117117120120120120125125128128132132140140141141149149149149156156165168168173173173173173173173173176176188188189189192192197197197197200200204204213213213213213213221221221221221224224224224228228236236237237245245245245252252260260261261264264269269276276284284285288288293293293293293293293293296296308308308308308308312312312312317317317317317317332332332332333333336336336336341341348348356356357357357357357357357365365365365368368368368368368372372380380380380381381389389389389389389392392392392396396404404405405413413413413413413413413420420420420437437437437437452452452452453453453453456456456456461461461461477477477477480480485485485485488488492492501501501501509509512512525528528533533533533533533536536536 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2: Innitely many circlesfrom the Apolloniusconstruction.The sequence ofradii tends to inn-ity (or equivalentlythe sequence ofradii (rn) tends tozero) since the to-tal area of thedisjoint disks en-closed by the cir-cles pi∑r 2n < +∞which is boundedby the area insidethe outer circle.Why do we getnatural numbers?It is an interesting property of the identity that ifr0, r1, r2, r3 are reciprocals of natural numbers thenso is r4 since1r0+1r4= 2(1r1+1r2+1r0+1r3)by the formula of Descartes-Princess Elizabeth. Pro-ceeding iteratively, we see that rn is the reciprocalof a natural numbers for all n ≥ 0.It is a natural question to ask how quickly such thesequence (1/rn) grows.Kontorovich-OhWe denote by N ( ) = #{n : rn >  } the nite num-ber of circles with radii greater than  > 0. We havealready observed that N ( ) → +∞ as  → 0. Thiswas considerably strengthened by A. Kontorovichand H. Oh in the following theorem.Theorem 3 (Kontorovich-Oh). There exist C > 0 andδ > 0 such that the number N ( ) ∼ −δ as  → 0,i.e.,lim→0N ( )−δ= C .The sequence looks a bit like ...One might compare this with a similar lookingresult for prime numbers2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, . . .For any x > 0 let pi(x) denote the nite num-ber of primes numbers less than x . Sincethere are innitely many primes we see thatpi(x) tends to innity as x tends to innity. Amore rened estimate comes from the follow-ing famous theorem proved in 1896.Theorem 4 (Prime Number Theorem). Thefunction pi(x) is asymptotic to xlog x as x → +∞,i.e., limx→+∞ pi(x)x/log x = 1.One might (rightly) imagine that these tworesults might be proved in a similar way.The value δWe can consider the compact setA in the plane givenby the closure of the countable union of circles A.Figure 3: A fractal set A in theplaneThe value of δ isequal to the Haus-dor Dimension ofthis set A (i.e.,thenatural notion of“size” for fractalsets). Although noexplicit expressionis known for δ ithas been numeri-cally estimated tobe δ = 1 ·30568 . . .[4].Any initial conguration of circles in the plane can bemapped onto any others by a Möbius map, which willalso map the corresponding circles in each packing.Since the Hausdor Dimension is preserved by anyLipschitz map we deduce that the same value of δoccurs for any initial choice of r1, r2 and r3.FEATURE 3A pinch of complex analysisRather than describe the original proof in [5] it ispossible to give a proof of the counting theorem forradii of circles in the Apollonian circle packing whichis analogous to that of Hadamard’s famous proof ofthe Prime Number Theorem [6]. We can consider thecomplex functionη(s ) =∑nr snwhere the summation is over all the radii of circlesin the Apollonian circle packing A. This converges toa non-zero analytic function for Re (s ) > δ and ananalytic function with the following properties:(1) η(s ) has a simple pole at s = δ; and(2) η(s ) has no zeros or poles on the lineRe (s ) = δIn particular, we can writeη(s ) = Cs − δ + ψ(s )where ψ(s ) is analytic in a neighbourhood of Re (s ) ≥δ. We can writeη(s ) =∫ ∞0r sdN (r )and then the properties (1) and (2) above are enoughto ensure that by the classical Ikehara-Wiener taube-rian theorem we have that N (r ) ∼ Cr −δ as r → 0[3].Comparison with the Prime Num-ber TheoremThe original proof of the Prime Number Theo-rem used the Riemann Zeta functionζ(s ) =∞∑n=11nswhich converges to a non-zero analytic func-tion for Re (s ) > 1 and the analytic extensionhas the following properties:(1) ζ(s ) has a simple pole at s = 1; and(2) ζ(s ) has no zeros or poles on the lineRe (s ) = 1.The asymptotic formula for the pi(x) thenfollows from an application of the classicalIkehara-Wiener tauberian theorem.Phillip BeecroftThere remains the non-minor detail of proving theproperties (1) and (2) for η(s ).A useful dynamical approach to generating circlepackings was discovered by Philip Beecroft (1818-1862), a school teacher from Hyde, in Greater Manch-ester, who was the son of a miller and lived quietlywith his two sisters. His approach was based on in-versions in four complementary circles, a result hepublished in the pleasantly named Lady’s and Gen-tleman’s diary from 1842 [1], [2].Theorem 5 (Beecroft). Given 4 mutually tangent cir-cles C1,C2,C3,C4 we can associate 4 new mutuallytangent "dual" circles K1,K2,K3,K4 passing throughthe points Ci ∩C j (1 ≤ i < j ≤ 4).K2K1K3K4C4C2C3C1The 4 original circles C1,C2,C3,C4 and their 4 dual circlesK1, k2,K3,K4In the course of his work, Beecroft too rediscoveredthe formula of Descartes and Princess Elizabeth.Apollonian groupsConsidering the circles to lie in the complex plane, foreach of the dual circles Ki , with center ci and radiusri , say, we can dene a Möbius map Ti (z ) = r2(z−c )|z−c |(i = 1, 2, 3, 4). These maps generate a group G calledthe Apollonian group.4 FEATUREThe group G gives a systematic way to study thecircles in A. The images gCi of the four initial circlesCi under elements g ∈ G correspond to the circlesin A, with a few trivial exceptions. In particular, wecan writeη(s ) =4∑i=1∑g ∈Γr (gCi ).where r (·) just denotes the radius of a circle.Of course, one might ask what we have gainedthrough writing the function η(s ) in this way. Infact the group is closely related to an example of aKleinian group and the function η(s ) is essentiallywhat is called a Poincaré series. There are dynamicalmethods available to establish (1) and (2).REFERENCE[1] P. Beecroft, The Concordent Circles, The Lady’sand Gentleman’s Diary, The Company of Station-ers, London (1843)[2] H. Coxeter An Absolute Property of Four Mutu-ally Tangent Circles. In: Prékopa A., Molnár E. (eds)Non-Euclidean Geometries. Mathematics and ItsApplications, vol 581. Springer, Boston, MA, 2006.[3] F. Ellison and W. Ellison, Prime Numbers, Wiley,New York, 1985.[4] C. McMullen, Hausdor dimension and confor-mal dynamics III: Computation of dimension, Amer.J. Math. 120 (1998) 691–721.[5] A. Kontorovich and H. Oh, Apollonian circlepackings and closed horospheres on hyperbolic3-manifolds, J. Amer. Math. Soc. 24 (2011) 603–648.[6] M. Pollicott and M. Urbaski, AsymptoticCounting in Conformal Dynamical Systems,arXiv:1704.06896.[7] P. Sarnak, Integral Apollonian Packings, Amer.Math. Monthly 118 (2011) 291–306.[8] F. Soddy, The Kiss Precise, Nature, 137 (1936)1021.Mark PollicottMark Pollicott is a profes-sor of mathematics atWarwick University. Hismain research interestsare in dynamical sys-tems and ergodic theory.In his spare time Marklikes to wrestle alligators.Paintings of Descartes and the PrincessThe painting of Gerrit van Honthorst, Triumph of the Winter Queen: Allegory of the Just depicts QueenElizabeth of Bohemia (seated in the chariot) with her late husband and son looking on. Amoungst theaccompanying children, Princess Elizabeth is dressed in blue. The Queen was the granddaughter ofJames I of England, and had Gunpowder plot of 1605 succedeed she should have been made queen ofEngland (at the age of 9 years).In the painting by Louis-Michel Dumesnil, Rene Descartes conducts a demonstration at the court of QueenChristina of Sweden, Descartes appears in the foreground to the right.

Appolonius circle counting

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Appolonius circle counting

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