0 Kenneth StephensonThe Approximation of Conformal Structuresvia Circle PackingKenneth StephensonAbstract. This is a pictorial tour and survey of circle packing tech-niques in the approximation of classical conformal objects. It beginswith numerical conformal mapping and the conjecture of Thurstonwhich launched this topic, moves to approximation of more generalanalytic functions, and ends with recent work on the approximation ofconformal tilings and conformal structures.x1 IntroductionA circle packing is a conguration of circles with a specied pattern oftangencies. The regular hexagonal or \penny" packing in the plane |every circle tangent to six others | is certainly familiar to everyone,and the literature contains a smattering of other examples stretchingback to the ancient Greeks. But I oer this as a rst illustration of thetype of packings we will discuss.

S2

Circle Packing 1Circle packing has its beginning as a distinct topic with appli-cations to 3-manifolds in Thurston's Notes [20]. Its connections toanalytic functions, the subject of this survey, can be traced preciselyto a conjecture of Thurston at the 1985 Bieberbach conference con-cerning the use of circle packings to approximate conformal mappingsand to its subsequent proof by Rodin and Sullivan [16].I must strongly emphasis right at the beginning that our topichas almost no contact with the more familiar \sphere packing" stud-ies, which address density and coverage issues | how many ping pongballs t in a boxcar. Look to our rst gure: it is the pattern of tan-gencies and the triangular interstices which are central. These circleshave had to assume widely varied sizes in order to t together in aprescribed pattern | this packing and the source of its combinatoricswill come up later.When I outline the basic notions of circle packing in a moment,you will nd that these objects present an intriguing blend of localgeometric rigidity and global geometric exibility. Mappings betweenone conguration of circles realizing a certain tangency pattern andanother conguration realizing the same pattern turn out to behavein fundamental ways like analytic mappings and have become centralto the theory. Two lines of development have emerged regarding thesemaps. The rst, following the lines of Thurston's 1985 conjecture,concerns their use in the approximation of analytic functions | thatis the subject of this survey. However, the second, which addresses themany analogies between these maps and analytic functions, also playsa role here since discoveries in that theory present us with several newtargets for approximation.This will be a decidedly pictoral survey, and I will rely heavilyon the reader's geometric feeling for analytic functions. We start withtraditional \conformal mapping", where circle packing has little torecommend it, in a numerical sense, over classical methods. However,we move on to more general analytic functions, where it begins toshow some advantages, and nally reach quite new, unbroken groundin the approximation of conformal structures. In recent applicationsto Riemann surfaces and conformal tiling which I survey here, circlepacking may stand as the only technique available.My thanks to the organizers of Computational Methods in Func-tion Theory '97 and to the University of Cyprus for their hospitality.At the end of the paper I point the reader to sources for a circle pack-ing bibliography and for the software package CirclePack used for thenumerical experiments illustrated here.

2 Kenneth Stephenson1.1 Classical Conformal GeometryOur view of analytic functions will be almost exclusively geometric |you will see no power series or integral formulae here, forget complexarithmetic. Think in terms of conformality (angle preserving), curva-ture and metrics, surfaces and coverings, branching and projections,`mappings' not `functions' | the rubber-sheet imagery suggested byFigure 1.f

g

h

D

Figure 1. Geometry of classical analytic functions on the discIn asking you to rely on geometric themes and intuition, it mighthelp to recall a saying which goes to the heart of conformal geometry:namely, analytic functions f map innitesimal circles to innitesimalcircles. Moreover, jf 0j is the stretching of these innitesimal circles.Taking this image seriously and \zooming" in on points z and f(z),one can imagine f mapping a pattern of innitesimal circles in thedomain to the pattern of innitesimal image circles in the range. Ifyou can get your mind around this image, you will not be far from thetruth. In the discrete theory, we simply choose to work with actualcircles!

Circle Packing 31.2 Circle Packing BasicsThe theory began on the sphere with this theorem, proven successivelyby Koebe, Andreev, and Thurston: Let K be an abstract triangulationof a topological sphere. There exists a circle packing P on the Riemannsphere S2 having the combinatorics of K and P is unique up to Mobiustransformations and inversions of S2. The conditions on P are thateach vertex of v 2 K is represented by a circle cv 2 P and that ifvertices v and w are connected by an edge in K, then cv and cw aretangent.The richness of circle packings may be traced to their dual na-tures: combinatoric in the pattern of prescribed tangencies and geo-metric in the realization by actual circles. Until x2.4, our circle pack-ings will lie in the sphere S2, the plane C , or the hyperbolic plane,represented here as D with the Poincare metric. Working with themrequires a rather modest system of bookkeeping: Complex: K denotes an abstract simplicial 2-complex triangulatinga topological surface D. Label: R is a collection fR(v)g of positive numbers associated withvertices v of K. If we are working in a metric space D, these representputative radii. We restrict attention to settings with Riemannian met-rics of constant curvature 0; 1, or 1, described as euclidean, spherical,or hyperbolic, respectively. Angle Sums: A label R determines an angle sum R(v) at eachvertex of K. In particular, for each face hv; u; wi 2 K, let R(v;u;w)denote the angle at cv in a triangle formed by the centers of a mutuallytangent triple hcv; cu; cwi of circles of radii hR(v); R(u); R(w)i; the lawof cosines appropriate to the geometry of R lets us compute . ThenR(v) =Phv;u;wi R(v;u; w); where the sum is over all faces hv; u; wi 2K containing v. Packing: P is a conguration of circles in the metric space D satis-fying the tangency pattern of K. If the radii are listed in the label R,we would write P $ K(R). Carrier: carr(P ) denotes the concrete geometric complex inD formedby connecting the centers of tangent circles of P with geodesic seg-ments; this is simplicially equivalent to the abstract complex K.Figure 2 illustrates these basics with a simple nite complexlabeled K; (b) is a rather generic hyperbolic packing for K in D , (c)a more deliberate euclidean packing with rectangular carrier, (d) isthe maximal packing (in D ) to be discussed shortly. The carriers are

4 Kenneth Stephensonshown for reference. Note that the radii and even the geometry change| it is the combinatorics which these packings share.(a) (b)

(c) (d)

D

D

K

Figure 2. Circle packings for a simple complex KThe surprise, of course, is that for a given complex K thereexist any packings whatsoever, much less a potentially huge variety ofpackings. Computationally, the labels are the key.Denition 1. A label R is a packing label for K if the angle sumR(v) is an integral multiple of 2 for every interior vertex v.This is clearly a necessary condition for R to represent the radii of apacking P , since R(v) = 2n; n 1; simply re ects the fact that thecircles cvi for the neighbors of v must reach precisely n times around cv.It is also a sucient condition if there are no topological obstructions:

Circle Packing 5Lemma 1. If K triangulates a simply connected surface, then a labelR for K represents the radii for a circle packing of K if and only if Ris a packing label.We say little about circle centers because they play a secondaryrole: once a packing label R is found, the circles are laid out in D(S2;C , or D , as appropriate) using a process akin to analytic contin-uation. The resulting circle packing P $ K(R) is essentially unique| that is, it is unique up to a conformal automorphism of D.A bit of terminology: A packing is univalent if its circles havemutually disjoint interiors. Univalence fails locally at a branch circle(or vertex), that is, at an interior circle whose neighbors wrap two ormore times around it (angle sum 4; 6; ). Figure 3 illustrates abranch circle; the same nine circles which wrap once about the centralcircle in Figure 3(a) wrap exactly twice around the smaller centralcircle of Figure 3(b). The branch set for a packing P , denoted br(P ),

(a) (b)Figure 3. Local univalence versus branchingis the set of branch circles, repeated according to multiplicities. If allinterior angle sums are 2, then br(P ) = ; and the packing is calledlocally univalent. Univalence can fail, even without branch circles,when circles from one part of a packing overlap those from another.Packing Variety: Given a complex K, there are numerousresults on the existence and variety of packings for K; we need twomain types. Let's begin with the extreme rigidity displayed by max-imal packings, proven by Koebe in the nite case [14], Beardon and

6 Kenneth StephensonStephenson in the bounded degree case [2], and He and Schramm ingeneral [11]:Theorem 1. Let K triangulate a simply connected surface. Then forD equal to precisely one of the spaces S2;C ; or D , there exists a circlepacking PK for K in D with the property that PK is univalent andhas a carrier lling D. Furthermore, PK is essentially unique.This circle packing PK is called a maximal packing and K is saidto be spherical, parabolic, or hyperbolic as PK lies in S2; C ; or D ,respectively.

(a) (b)

(c) (d)

S2

D D

Figure 4. Maximal PackingsFigure 4 illustrates the four basic situations. A complex K isspherical i it triangulates a topological sphere; in (a) we see the regu-lar pentagonal packing (every circle with ve neighbors). The familiar

Circle Packing 7regular hexagonal or \penny" packing (b) is parabolic, the regular \sep-tagonal" packing (c) is hyperbolic. Computationally, packings such asthat of Figure 4(d) are the most important. Here K triangulates aclosed topological disc (in fact, this is the maximal packing for thecomplex of Figure 2). Note that carr(PK) \lls" D in the sense thatits boundary circles are horocycles (euclidean circles internally tan-gent to @D ) and the boundary vertices have been pushed to the idealboundary. Spherical packings are computed by puncturing at a vertex,packing the rest of the complex in D , then projecting back to S2. In-nite packings are handled as appropriately rescaled limits of packingsin D for nite subcomplexes.In contrast to the rigidity of maximal packings, let's move tothe variety available when we construct packings meeting prescribedboundary value and branching conditions.Let K triangulate a closed topological disc. Let = fb1; ; bkgbe a legal branch set for K; this is a (perhaps empty) list of interiorvertices of K, with possible repetitions. (Necessary/sucient combi-natorial conditions on have been proven by Dubejko [9] and Bowers[4].) @K denotes the boundary vertices of K.Theorem 2. Let K, , and the boundary label g : @K ! (0;1)be given. Then there exists a unique euclidean packing label R for Kwith branch set and satisfying R(w) = g(w); w 2 @K. The sameresult holds in hyperbolic geometry, where g is also allowed to assumethe value +1.In other words, there exists an essentially unique circle packinghaving a prescribed branch set and prescribed boundary radii. In hy-perbolic geometry, innite radii correspond to horocycles. Thus themaximal packing for K is the packing in D having prescribing branchset = ; and prescribed boundary radii g(w) =1; w 2 @K.Don't let this global malleability of circle packings obscure thelocal rigidity implied by the packing condition at each vertex. Thelocal-to-global linkage is moderated by the combinatorics of K.Discrete Analytic Functions: We can attempt to imposesome order on the numerous packings for a given complex K by us-ing a function theory paradigm. Dene discrete analytic functions asmaps between circle packings which preserve tangency and orientation.Various packings of K may now be associated with discrete analyticfunctions f : PK ! P . Compare the univalent, locally univalent,and branched discrete analytic functions of Figure 5 to their classicalmodels in Figure 1.As classical analytic functions map innitesimal circles to in-nitesimal circles, so our discrete versions operate with real circles.

8 Kenneth Stephenson

D

f

g

h

PK

P1

P2

P3

Figure 5. Discrete analytic functions on the discFundamental behavior is surprisingly similar. There is a ratio functionf# dened on vertices of K by f#(v) = radius(f(cv))=radius(cv); thisre ects the amount that f stretches or shrinks cv and is the analogof jf 0j. Hidden in P3 is a branch circle like that of Figure 3(b); thetopological behavior of the mapping is precisely that at a branch pointof an analytic function.Computation: Finally, let me point out that the compu-tational problem centers on nding packing labels. Thurston sug-gested an iterative algorithm which works beautifully in hyperbolic andeuclidean geometry to solve the boundary value problems describedabove. In this survey, we will take the algorithm for granted; see [8]for a description and an ecient implementation.

Circle Packing 9x2 ApproximationLet's proceed now to the principal approximation issues. I will breakmy survey into four topics: the original conformal mapping settingof Thurston's Conjecture, generalizations to other analytic functions,then on to new uses in conformal tiling and conformal structures onsurfaces.2.1 Conformal MappingFigure 6 illustrates Thurston's 1985 conjecture. Given the Jordan do-main , cut out a nite piece of a regular hexagonal packing of circleshaving radii 1=n. Write Pn for this nite circle packing and let Knencode its combinatorics. If PKn is the maximal packing for Kn, thenfn : PKn ! Pn represents for us a discrete conformal mapping fromD to . The illustration captures domains and ranges for three suchmappings (appropriate normalizations are assumed).Thurston conjectured and Rodin and Sullivan proved that thediscrete conformal maps fn converge uniformly on compact subsetsof D to the classical conformal mapping F : D ! as n ! 1.From that convergence one can also show that the ratio functions f#nconverge uniformly on compact sets to jF 0j.I will not go into the proof of Thurston's conjecture, but I wouldlike to highlight the two key ingredients, which reappear in one guise oranother in every setting we consider: namely, renement and distortioncontrol. \Renement" refers to the process of conveniently generatingsequences of successively ner circle packings (that is, more numerousand smaller circles) appropriate to a setting | hexagonal packings withsuccessively smaller radii serve that purpose in Figure 6. \Distortion"concerns the comparison of a face in (the carrier of) one packing withthe corresponding face in another packing of the same complex, nor-mally quantied in terms of quasiconformal distortion. Less distortionof carrier faces converts to \more conformal" mapping behavior. InFigure 7, observe the distortion of various triangles on the right com-pared to the corresponding equilateral triangles on the left. Distortionnear the center is smaller.The important Hexagonal Packing Lemma of Rodin and Sulli-van and its descendants imply, roughly speaking, that in comparingtwo univalent packings the amount of distortion that a given face canmanifest decreases as its combinatorial distance from the boundary in-creases. The Rodin/Sullivan Theorem has been considerably extendedto accommodate more general combinatorics and to quantify the con-vergence. The deepest and most general results are due to Z-X. Heand Oded Schramm [12, 13]; see also [19, 18].

10 Kenneth Stephensonfn

fn+1

fn+2

D

D

D

Ω

Ω

ΩFigure 6. Illustrating Thurston's Conjecture

Circle Packing 11(a) (b)Figure 7. The Hexagonal Packing LemmaNumerical Methods: On the more practical side, for thoseinvolved in numerical conformal mapping circle packing may appear tobe a disappointment: the \packing" process itself | e.g., computingthe radii for PKn | turns out to be rather slow, data sets are bulky, andthe process of laying out the packing itself introduces additional error.Circle packing certainly cannot compete with the classical numericalmethods for speed and accuracy.Circle packing does have some strengths. For instance, it quiteeasily handles multiply connected regions. Figure 8(b) illustrates ageneric 4-connected region and the packing P it cuts from a regularhex packing. Augment the complex underlying P with an \ideal" ver-

f

ΩD

(a) (b)

P

Figure 8. A multiply connected example

12 Kenneth Stephensontex for each hole; that is, a common neighbor for the circles boundingthat hole. The augmented complex is simply connected, and displayingits maximal packing with the circles corresponding to the ideal verticesremoved leaves the \circle domain" on the left in Figure 8. This then is(discretely) conformally equivalent to the original domain . (Z-X. Heand oded Schramm in [11] applied circle packing techniques in a majoradvance on Koebe's Conjecture regarding innitely connected circledomains; their work is far deeper than our simple numerical examplehere.)Prototyping: In the settings of traditional conformal map-ping, circle packing may be helpful at what I will call prototyping.Suppose one plans to apply Schwarz-Christoel to approximate a con-formal map of D to the polygon R of Figure 9(b). One might start

(a) (b)

f

a

b

c

d

a

b

c

d

Figure 9. Prototyping a Schwarz-Christoel Mapwith a discrete conformal map f involving a modest number of cir-cles (just over 5000 in this instance). I've marked only the four mostvisible circles associated with the indicated corners of the polygon;these and the other corner preimages can serve as the initial guessesfor the Schwarz-Christoel parameters. Prototyping could be valu-able in other situations as well, such as the multiply connected settingdiscussed earlier.2.2 Analytic FunctionsClassical analytic function theory involves more than just conformalmaps. So it is, too, with the discrete theory. The numerous paral-lels are not the aim in this survey, but in passing note that the dis-crete theory involves fairly general functions on D , polynomials and a

Circle Packing 13smattering of entire and meromorphic functions on the plane, rationalfunctions on the sphere, and discrete maps on Riemann surfaces. Thetheory has its own Schwarz-Pick, Uniformization, Liouville and Picardtheorems, and its own notions of covering theory, extremal length,harmonic measure, and Brownian motion (random walks). It is ap-proaching something like a \full service" function theory, though thereremain many gaps.I have chosen two examples to illustrate the computational andapproximation side of this more general discrete theory: one involvesa function in the disc algebra, the other a nite Blaschke product.A Disc Algebra Example: Consider the closed curve ofFigure 10, with a nite number of transverse self-intersections. Usingwinding numbers alone, one can verify that is the image of the unitcircle under a disc algebra function (continuous on D and analytic onD ); one can also determine the distribution of branch values (imagesof branch points) among the components of C . (See [15].)

f

F

π

γFigure 10. Approximating a disc algebra function

14 Kenneth StephensonActual construction of a disc algebra solution is another matter;Schwarz-Christoel methods apply, but I'm not aware of implementa-tions. We can construct the analogous discrete solutions by introducingbranch points to our earlier univalent methods.Our aim is construction of a multi-sheeted image packing; weuse a \ball bearing" rather than a hexagonal pattern as our startingmaterial and overlay the region dened by as in the univalent case.However, several \sheets" (up to three deep inside the smallest loop)may be inferred from and the branch structure, so appropriate piecesare cut out and attached as in the upper right of Figure 10. Theresulting packing P has a complex K which is simply connected. Ourdiscrete disc algebra function F results from mapping PK to P with fand projecting with to a packing in C . The pasting resulted in twobranch points, one simple and one of order 2. These are the shadedcircles in PK ; the former has 8 neighbors which map 2-to-1 onto the4 neighbors of its projected image circle, the latter has 12 neighborsmapped 3-to-1 onto the 4 neighbors of its projected image circle.As in the conformal setting, a sequence of rened packings con-structed in this manner will (with suitable normalization) convergeuniformly on compact subsets of the disc to the classical disc algebrasolution. We can routinely generate the necessary renements fromthis initial coarse construction, as we will see later, so this is in fact afairly practical procedure.Finite Blaschke Products: Classical nite Blaschke prod-ucts arise not only in function theory on the disc and its applications,but also in studies of rational functions, iteration theory, and topo-logical covering theory. An n-fold Blaschke product is typically repre-sented as a product of n Mobius factors, one for each zero, multipliedby a unimodular constant. It assumes each value in the disc n times,counting multiplicities, and maps the unit circle n times around itself.Lacking arithmetic, we need a more geometric description for thediscrete construction. The nite Blaschke products are precisely the\proper" analytic maps of the disc to itself, and the argument principlein turn implies that an n-fold Blaschke product has n1 branch points,counting multiplicities. These geometric features are enough for ourconstruction, as I illustrate with a 4-fold example.Take as the domain a maximal packing PK in the unit disc,shown on the left in Figure 11. I have identied three shaded interiorcircles for branching. We repack the complex in hyperbolic geometry,specifying that: (1) boundary circles get innite hyperbolic radii; (2)designated branch circles get angle sum 4 rather than the usual 2.The resulting packing label allows us to lay out the range packing.Displaying all its circles on the right would lead to visual meltdownsince points are covered four times, so I have drawn only the chain ofboundary horocycles and the 3 branch circles (they appear as extremely

Circle Packing 15b

Figure 11. Boundary circles for 4-fold discrete Blaschke productsmall dots towards the middle) in the image packing. A careful surveywould conrm that those boundary circles wrap 4 times around theunit circle. The construction is complete, aside from applying a Mobiustransformation of D for the sake of normalization.To approximate a particular classical 4-fold Blaschke productB, we would rst need to know its branch points. Then we would runthis construction with successively ner complexes Kn, with branchcircles approximating the desired branch points, and with appropriatenormalizations. The sequence fbng of discrete nite Blaschke productswould converge uniformly on compact subsets of D to B.The simplicity of the discrete construction is quite impressive;the repacking process converts the local geometric conditions on bound-ary radii and branch points to the inevitable global behavior. Classicalmethods would have diculty coping with prescribed branch points.On the other hand, constructing a discrete nite Blaschke product withspecied zeros is tough.2.3 Conformal TilingsConformal tilings, introduced by Phil Bowers and me in [7], involvetiles with specied \conformal" rather than euclidean shapes. Thoughclassical objects, they were inspired by experiments with circle packingsand circle packings remain the only avenue for approximating their trueconformal shapes.I'll illustrate with a nite example based on the simple schematicpattern of Figure 12(a). Pasting together seven equilateral trianglesin the prescribed pattern, as suggested in Figure 12(b), yields a topo-logical disc with a piecewise ane structure; there are three \cone"points, each of angle 4=3. There is a standard (and very classical)

16 Kenneth Stephenson(a) (b) (c)Figure 12. Combinatorial, ane, and conformal stucturesprocedure for dening a conformal structure compatible with such anane structure; its atlas involves appropriate power maps z 7! z3=2 atthe cone points (see, e.g., Beardon [1]). We arrive, then, at a Riemannsurface, and if we map it conformally onto an equilateral triangle, car-rying along the markings of the original seven triangles, we arrive atthe triangle T of Figure 12(c).The curves marked on T are uniquely determined by the originalcombinatorics and by what Bowers and I call a re ective structure |namely, any two of the seven triangular pieces having a common edgeare anticonformal re ections of one another across that edge. Howcan one approximate these curves in T ? I am not aware of any classi-cal numerical methods which apply. However, circle packing providesdiscrete curves which do the trick.

(a)

(b)Figure 13. The discrete construction

Circle Packing 17The discrete construction mimics the original. Conceptually, wetake seven copies of the packing of Figure 13(a) and attach them toone another by identifying circles along appropriate edges, just as inFigure 12(b). (In fact, only the resulting combinatorics are impor-tant.) When the results are repacked to form an equilateral carrier,the repacking process forces the circle sizes to adjust to atten out thecone points, as in Figure 13(b).Figure 14 shows the same process carried out with two succes-sively more rened packings; one can see the marked curves approach-ing limit shapes.

stage-2 refinement stage-4 refinementFigure 14. Circle packings at two additional renement levelsI want to take a moment to comment on the renement pro-cess used here. It is called \hex renement" and operates as shownin Figure 15. Hex renement can be applied to any triangulation andoriginal stage-1 stage-2 stage-3Figure 15. Hex renement process

18 Kenneth Stephensonhas several valuable features: New combinatoric data is very easy togenerate. The original vertices of a triangulation remain, with theirdegrees unchanged. New (interior) vertices are degree six. Thismeans that under successive hex renement, one is in good position todeploy the Hex Packing Lemma. \Renement" is actually a combi-natorial process | one must always compute new radii for the renedcomplex. The self-scaling nature of hex renements may provide apriori estimates which will save eort in the repacking computation.Innite Tilings: The processes involved in our simple exam-ple, both the classical construction and its discrete analogue, work withonly minor adjustment in much more general circumstances. We willlook at some innite patterns here and move to patterns on compactsurfaces in the next section.

Figure 16. Four examples of innite conformal tilings of the plane

Circle Packing 19Figure 16 illustrates four innite conformally regular tilings of C .Each tile can be mapped conformally onto a regular euclidean n-gon,corners going to corners (and there are additional conditions whichwe won't go into). These particular patterns are quite special; theyarise from \subdivision rules" being applied by Jim Cannon, WilliamFloyd, and Walter Parry to the study of Thurston's GeometrizationConjecture. (See [7] and references therein.) My thanks to Bill Floydfor the data needed to generate these tilings.What is our interest in tilings? All of these pictures were gener-ated via circle packing | the circles themselves aren't shown, but onlycertain edges of their carriers. Packings bring an experimental com-ponent to this research and they provide approximations (like thesepictures) of the true conformal shapes of the tiles | capabilities nototherwise available.I have space to investigate only one example, so let's look atthe \twisted pentagonal" tiling in the upper right of Figure 16. InFigure 17(a), denote the three dark curves as 0; 1; 2, from innerto outer. These mark central \aggregate tiles" associated with therst three stages of the subdivision rule which generated this pattern.Stare at the picture hard enough and you might guess at some scalingrelationship among the i's. In fact, using 0:55e0:23 , I have

(a) (b)Figure 17. Aggregates in the twisted pentagonal tilingoverlaid 2 2, 1, and 0 in Figure 17(b). Our hunch was right,the vertices i coincide almost perfectly with vertices of i1. Youcan probably anticipate what the curves 3; 4; look like; it appearsthat limn!1(n n) converges to some fractal curve , that is, theaggregate tiles converge to a fractal tile. The same holds for other tiles

20 Kenneth Stephensonin the pattern (not marked), leading to a fractal pentagonal tiling of theplane. Motivated by these very experiments, Walter Parry has usedclassical methods to prove this scaling property, though the precisevalue of is not known.Central issues in conformal tiling include the \type" problem| does a pattern tile the plane or the hyperbolic plane | questionsabout tile shapes, sizes, distortions, and questions about symmetriesand self-similarities within the overall pattern. But this is a new topic,and new phenomena appear in nearly every experiment. Numericalcomputation and display are currently the pivotal tools.2.4 Grothendieck DessinsLet me begin this section by introducing a certain type of Riemann sur-face construction. Suppose S denotes a compact, orientable topologicalsurface and K is a triangulation of S. One can paste euclidean equilat-eral triangles together in the pattern of K to impose a piecewise anestructure on S. Precisely as described in the seven-triangle example ofFigure 13, that ane structure determines a conformal structure on S,making it into a Riemann surface. Surfaces constructed this way willbe called equilateral surfaces.We quote this surprising result of G. Bely [3], for which we needone other bit of terminology. A nonconstant meromorphic function ona compact Riemann surface which branches only over points f0; 1;1gis termed a Bely map.Theorem 3. For a Riemann surface R of positive genus, the followingstatements are equivalent:(a) R is equilateral.(b) R supports a Bely map.(c) There exists a dening equation for R whose coecients liein an algebraic number eld F over the rationals.In other words, in the Teichmuller space Teich(S) for surfacesof genus g > 0 there is a set of points which simultaneously enjoyscombinatorial, function theoretic, and algebraic characterizations. By(a) this set is countable, and by (c) it is dense in Teich(S).Let's turn now to the discrete parallels. We nd that S and Kdetermine a point in Teichmuller space in a second way by this result(see [20, 2]).Theorem 4. Let K be a triangulation of a compact oriented topolog-ical surface S. Then there exists a unique Riemann surface R home-omorphic to S that supports a univalent circle packing PK with thecombinatorics of K.

Circle Packing 21In other words, the combinatorics prescibed for a circle packingwill determine the geometry of the space in which the packing can berealized. (The metric for this packing PK is the intrinsic metric on R;that is, the metric of constant curvature 0 or 1 which R inherits fromits universal covering surface.) When S has positive genus, the surface

Figure 18. \Equilateral" versus \packable" 2-toriR which supports the packing for K is called a packable surface ofTeich(S). The packable surfaces are clearly countable, and by a resultof Bowers and me [6], also dense. With a certain technical conditionthat we may assume for K, one can also dene a discrete meromorphicmap from PK to a packing in S2 which branches only over f0; 1;1g;a discrete Bely map. In other words, we have discrete parallels for theclassical objects described earlier.Dessins: Equilateral surfaces are at the core of a program ini-tiated by Grothendieck to study the absolute Galois group Gal(Q =Q )and to address what is known as the Inverse Galois Problem. Thisamazing theory begins with a \child's drawing" D on S and ends withan algebraic number eld. Brie y, the drawing leads to a triangulationof S, the triangulation leads to an equilateral Riemann surface SD anda Bely map BD on SD, giving the so-called \Bely pair" (SD; BD).Schematically, D ! K ! (SD; BD):

22 Kenneth StephensonOf course this is all quite abstract and in only limited special circum-stances are any concrete details about the surface and the Bely mapknown. (See [17].)On the other hand, the parallel discrete development can becarried out at various renement stages and has the advantage of beingcomputable. The idea is to apply n-stages of hex renement to K,giving a complex K(n), then to construct the associated discrete Belypair (s(n)D ; b(n)D ) consisting of s(n)D 2 Teich(S) and discrete Bely mapb(n)D : PK(n) ! S2. Schematically,D ! K ! K(n) ! (s(n)D ; b(n)D ):Phil Bowers and I have proven the following result [5]:Theorem 5. Let D be a drawing on a compact orientable surface S.Then the discrete Bely pairs (s(n)D ; b(n)D ) converge to the classical Belypair (SD; BD) as n ! 1. In other words, s(n)D converges to SD inthe Teichmmuller metric of Teich(S) and the functions b(n)D convergeuniformly to BD on S.

A B

C

D

CD

A

B

Figure 19. Sample dessins of genus 0 and genus 2As examples, let me discuss the two dessins of Figure 19. Therst, a very simple genus 0 drawing D on the sphere, is shown inFigure 19(a) along with its canonical triangulation, four shaded andfour unshaded faces. Here s(n)D = SD for all n since there is only oneconformal structure on S2. Nevertheless, the convergence of b(n)D toBD allows us to approximate the conformal shapes of the faces of K.The stage n = 3 construction is shown in Figure 20; the packing is the

Circle Packing 23one shown at the beginning of the paper, but the faces of K are markedin its carrier as the shaded and unshaded regions, and the drawing Ditself is shown as the heavy curve. The discrete Bely map b(3)D mapsthis packing to the spherical packing on the right with multiplicity4-to-1, the shaded faces mapping to the shaded hemisphere.S2

(a) (b)

bD(3)

Figure 20. A genus 0 discrete Bely mapOur second example is the dessin D of Figure 19(b); the indi-cated side-pairings show this to be a genus 2 case. I have packed this atthe original \coarse" stage and at a stage-3 hex renement and displaythe results in Figure 21. The packings are laid out in D as fundamen-tal domains for the universal covering maps from D onto s(0)D and s(3)D ,respectively. The heavy curve in each case is D and the faces of K arealternately shaded and unshaded.The packing computations are carried out directly in hyperbolicgeometry with the combinatorics of the 2-torus in tact. The results arethen laid out as a fundamental domain where numerical information,such as side-pairing maps, can be read o. (Note the side-pairingsdesignated on the upper packing.) The fundamental domains convergeas n!1 to the fundamental domain of the universal covering of SD.One thing to note is the surprising accuracy of even the coarse packing| its eight boundary cusps dier by less than 1=500 from those of thestage-3 renement.By the way, have our discrete objects lost their connection tonumber elds? In the original sense, yes; but in fact, number eldsreenter in a new and intriguing way as entries in the associated coveringmaps.

24 Kenneth Stephenson

(a)

(b)

DD

A

B

A B

C

D

C

Figure 21. Fundamental domains for genus 2 dessins

Circle Packing 25Classical Surfaces: Let's wrap up our look at surfaces withtwo classical examples which (in hindsight) are associated with dessins.The quite famous image in Figure 22 is Klein's surface. He was ableto publish the original image more than a century ago because of thesurface's ubiquitous symmetries (an order 168 symmetry group) | thetriangles are all geodesic. The image here, on the other, generated viacircle packing, took approximately two seconds for ten digit accuracystarting from purely combinatorial data.

Figure 22. Klein's surfaceAnother famous classical example is Picard's curve y3 = x41, agenus three surface. It is actually associated with two distinct dessins.One is shown in Figure 23(a); this is a fundamental domain for arst stage coarse approximation and the circles have been left in forreference. This classical surface also has many symmetries, but theedges of its faces are not all geodesic. As far as I know, aside fromcircle packing methods, there would be no way to approximate thesurface or to create these images.

26 Kenneth Stephenson

Figure 23. Picard's surfacex3 Concluding RemarksThe survey examples suggest a general paradigm: given a classicalobject of interest | a function, a Riemann surface, or a conformalshape | proceed as follows: Construct a combinatorial abstraction | that is, a triangulation and(possibly) a branch structure. Endow the triangulation with an appropriate ane structure. Cur-rently we need an \equilateral" model, but certain recent developmentsare expected to allow more general decompositions. Circle pack the result. This is the local-to-global step; nature endowsthe abstraction with a global discrete conformal geometry compatiblewith the local packing conditions. Apply a process of repeated renement and repacking to constructa sequence of discrete conformal objects.

Circle Packing 27If the local geometry is faithfully encoded, then faithful global conse-quences appear to be inevitable and the discrete conformal objectsconverge to their classical counterpart. The methods clearly havebroad applicability, and the equilateral models, hex renement, andthe hexagonal packing lemma t so well together that one can contem-plate a rather comprehensive and fairly automated \analytic engine"(to steal a phrase) for approximating classical surfaces and functionsin a variety of situations.The methods are also geometrically honest: the discrete objectsare generally very faithful to their classical models, so one's intuitionremains in tact; the computations take place within the intrinsic ge-ometry; and the packings provide a natural means of visualization.3.1 Numerical IssuesOf course, the devil is in the details. Standard issues of accuracyand rates of convergence remain almost totally open for circle packingmethods. These may be quite dicult; the original discrete model, thecomputed packing labels, and the laying-out process, all introduce er-rors. A beginning has been made with estimates of the quasiconformaldistortion associated with discrete analytic functions, primarily in thehexagonal setting; for the deepest results, see the work of Z.X. He andOded Schramm [13]. The packing algorithm has some beautiful geom-etry and can be modeled probabilistically, so there is hope of eventualconvergence estimates [18].Nonetheless, we seem to be stuck with the heuristic approachfor now | experiments, checks for internal consistency, and, whenpossible, comparisons to classical examples. In situations where circlepacking is the only available tool, we may be forgiven for using it inexperiments without rst settling the various details of convergence.3.2 Technical IssuesVisit my homepage at www.math.utk.edu/skens for a circle packingbibliography, further packing examples, and directions for acquiringthe software package CirclePack, which runs under X-Windows.I have based my circle packing algorithms on an iterative schemesuggested by Thurston; recent improvements are detailed in [8]. Mod-erate packings, say 5,00010,000 circles, typically require less than aminute; packings of 100,000+ circles are routine, though data sets be-come bulky. Hex renement and \multi-grid" methods open the wayfor parallelization and potentially signicant eciency improvements.See [10] for discriptions of several experiments and [5] for sample tim-ings.

28 Kenneth Stephenson

Figure 24. Packing options: tangency, inversive distances, overlaps3.3 ConclusionCircle packing is a new and, I think, very exciting topic. It comes withsubstantial and interesting problems, new capabilities, some availablefor the rst time, an experimental and visual environment, and manydirections for further development. As merely one example, Figure 24illustrates inversive distance and overlap packings; these generaliza-tions of \tangency" bode even greater exibility for discrete conformalmethods in the future. Moreover, I have left out emerging uses of thisgeometry inspired by circle packing in areas such as graph embeddingand modeling of various physical phenomena.The traditional applications of conformal mapping| areas whichhave motivated some truly beautiful mathematics over the years | areincreasingly falling to numerical PDE and other methods. At the sametime, conformal geometry in the broader sense seems to be growing inimportance, both in theory and applications. We are confronted withsubstantial opportunities for numerical approximation, but with ex-ibility, convenience, visualization, and the intuition gained from real-time experiments taking precedence over traditional speed and preci-sion. Incongruous as it may seem, \circle packing" might provide justthe thing in this new approximation environment.Finally, I admit to loving these pictures, so I want to leave withthis last one: a \dodecahedral" tiling from a division rule of Jim Can-non, data from Bill Floyd. None of the circles actually used in theconstruction are shown, but an intricate pattern of \circles" seems tobe emerging from the edge pattern itself!

Circle Packing 29

Figure 25. Cannon's dodecahedral patternAcknowledgement. The author gratefully acknowledges support ofthe National Science Foundation and the Tennessee Science Alliance.References[1] A. F. Beardon, A primer on Riemann surfaces, London Math. Soc.Lecture Note Series, vol. 78, Cambridge University Press, Cam-bridge, 1984.[2] Alan F. Beardon and Kenneth Stephenson, The uniformization the-orem for circle packings, Indiana Univ. Math. J. 39 (1990), 13831425.[3] G. Bely, On Galois extensions of a maximal cyclotomic eld, Math.USSR Izv. 14 (1979), 247256 (English).[4] Philip L. Bowers, The upper Perron method for labelled complexeswith applications to circle packings, Proc. Camb. Phil. Soc. 114(1993), 321345.[5] Philip L. Bowers and Kenneth Stephenson, Uniformizing dessinsand Bely maps via circle packing, submitted.[6] , Circle packings in surfaces of nite type: An in situ ap-proach with applications to moduli, Topology 32 (1993), 157183.[7] , A \regular" pentagonal tiling of the plane, Conformal Ge-ometry and Dynamics 1 (1997), 5886.

30 Kenneth Stephenson[8] Chuck Collins and Kenneth Stephenson, A circle packing algorithm,preprint.[9] Tomasz Dubejko, Branched circle packings and discrete Blaschkeproducts, Trans. Amer. Math. Soc. 347 (1995), no. 10, 40734103.[10] Tomasz Dubejko and Kenneth Stephenson, Circle packing: Exper-iments in discrete analytic function theory, Experimental Mathe-matics 4 (1995), no. 4, 307348.[11] Zheng-Xu He and Oded Schramm, Fixed points, Koebe uni-formization and circle packings, Ann. of Math. 137 (1993), 369406.[12] , On the convergence of circle packings to the Riemannmap, Invent. Math. 125 (1996), 285305.[13] , The C1-convergence of hexagonal disk packings to theRiemann map, Acta Mathematica, to appear.[14] P. Koebe, Kontaktprobleme der Konformen Abbildung, Ber. Sachs.Akad. Wiss. Leipzig, Math.-Phys. Kl. 88 (1936), 141164.[15] Morris L. Marx, Extensions of normal immersions of s1 into r2,Trans. Amer. Math. Soc. 187 (1974), 309326.[16] Burt Rodin and Dennis Sullivan, The convergence of circle pack-ings to the Riemann mapping, J. Dierential Geometry 26 (1987),349360.[17] L. Schneps (ed.), The Grothendieck theory of dessins d'enfants,London Math. Soc. Lecture Note Series, vol. 200, Cambridge Univ.Press, Cambridge, 1994.[18] Kenneth Stephenson, A probabilistic proof of Thurston's conjec-ture on circle packings, Rendiconti del Seminario Mate. e Fisico diMilano, to appear.[19] , Circle packings in the approximation of conformal map-pings, Bulletin, Amer. Math. Soc. (Research Announcements) 23,no. 2 (1990), 407415.[20] William Thurston, The geometry and topology of 3-manifolds,Princeton University Notes, preprint.Kenneth StephensonDept. of Math.University of TennesseeKnoxville, TN 37996U.S.A.kens@math.utk.edu