Circle Packing: A Mathematical Tale - University of Georgiamath.uga.edu/~saarh/Notices_article.pdf · Circle Packing: A Mathematical Tale Kenneth Stephenson 1376 NOTICESOFTHEAMS VOLUME50,

Circle Packing: AMathematical TaleKenneth Stephenson

1376 NOTICES OF THE AMS VOLUME 50, NUMBER 11

The circle is arguably the most studied ob-ject in mathematics, yet I am here to tellthe tale of circle packing, a topic whichis likely to be new to most readers. Thesepackings are configurations of circles

satisfying preassigned patterns of tangency, and wewill be concerned here with their creation, manip-ulation, and interpretation. Lest we get off on thewrong foot, I should caution that this is NOT two-dimensional “sphere” packing: rather than beingfixed in size, our circles must adjust their radii intightly choreographed ways if they hope to fit together in a specified pattern.

In posing this as a mathematical tale, I am ask-ing the reader for some latitude. From a tale youexpect truth without all the details; you know thatthe storyteller will be playing with the plot and tim-ing; you let pictures carry part of the story. We allhope for deep insights, but perhaps sometimes asimple story with a few new twists is enough—mayyou enjoy this tale in that spirit. Readers who wishto dig into the details can consult the “Reader’sGuide” at the end.

Once Upon a Time …From wagon wheel to mythical symbol, predatinghistory, perfect form to ancient geometers, com-panion to π , the circle is perhaps the most celebratedobject in mathematics.

There is indeed a long tradition behind ourstory. Who can date the most familiar of circlepackings, the “penny-packing” seen in the back-ground of Figure 1? Even the “apollonian gasket”(a) has a history stretching across more than twomillennia, from the time of Apollonius of Perga tothe latest research on limit sets. And circles werenever far from the classical solids, as suggested bythe sphere caged by a dodecahedron in (b). Equallyancient is the αρβηλoς or “shoemaker’s knife” in(c), and it is amazing that the Greeks had alreadyproved that the nth circle cn has its center ndiameters from the base. This same result can befound, beautifully illustrated, in sangaku, woodentemple carvings from seventeenth-century Japan.In comparatively recent times, Descartes estab-lished his Circle Theorem for “quads” like that in(d), showing that the bends bj (reciprocal radii) offour mutually tangent circles are related by(b1 + b2 + b3 + b4)2 = 2(b2

1 + b22 + b2

3 + b24) . Nobel

laureate F. Soddy was so taken by this result thathe rendered it in verse: The Kiss Precise (1936).With such a long and illustrious history, is it surprising or is it inevitable that a new idea aboutcircles should come along?

Birth of an IdeaOne can debate whether we see many truly newideas in mathematics these days. With such a richhistory, everything has antecedents—who is to say,for example, what was in the lost books of Apollo-nius and others? Nonetheless, some topics havefairly well-defined starting points.

Kenneth Stephenson is professor of mathematics at the Uni-versity of Tennessee, Knoxville. His email address [email protected].

The author gratefully acknowledges support of the NationalScience Foundation, DMS-0101324.

DECEMBER 2003 NOTICES OF THE AMS 1377

Our story traces its origin to William Thurston’sfamous Notes. In constructing 3-manifolds, Thurstonproves that associated with any triangulation of asphere is a “circle packing”, that is, a configurationof circles which are tangent with one another in thepattern of the triangulation. Moreover, this packingis unique up to Möbius transformations and inver-sions of the sphere. This is a remarkable fact, for thepattern of tangencies—which can be arbitrarily in-tricate—is purely abstract, yet the circle packing su-perimposes on that pattern a rigid geometry. This isa main theme running through our story, that circlepacking provides a bridge between the combinatoricon the one hand and the geometric on the other.

Although known in the topological communitythrough the Notes, circle packings reached a sur-prising new audience when Thurston spoke at the1985 Purdue conference celebrating de Branges’sproof of the Bieberbach Conjecture. Thurston hadrecognized in the rigidity of circle packings some-thing like the rigidity shown by analytic functions,and in a talk entitled “A finite Riemann mappingtheorem” he illustrated with a scheme for con-structing conformal maps based on circle packings.He made an explicit conjecture, in fact, that his “fi-nite” maps would converge, under refinement, toa classical conformal map, the type his Purdue au-dience knew well. As if that weren’t enough,Thurston even threw in an iterative numericalscheme for computing these finite Riemann map-pings in practice, with pictures to back it all up.

So this was the situation for your storyteller as helistened to Thurston’s Purdue talk: a most surpris-ing theorem and beautiful pictures about patterns ofcircles, an algorithm for actually computing them,and a conjectured connection to a favorite topic, an-alytic function theory. This storyteller was hooked!

As for antecedents, Thurston found that his the-orem on packings of the sphere followed from priorwork by E. Andreev on reflection groups, and someyears later Reiner Kuhnau pointed out a 1936 proofby P. Koebe, so I refer to it here as the K-A-T (Koebe-Andreev-Thurston) Theorem. Nonetheless, for ourpurposes the new idea was born at Purdue in 1985,and our tale can begin.

Internal DevelopmentOnce a topic is launched and begins to attract a com-munity, it also begins to develop an internal ecology:special language, key examples and theorems, centralthemes, and—with luck—a few gems to amaze theuninitiated.

The main players in our story, circles, are wellknown to us all, and we work in familiar geomet-ric spaces: the sphere P, the euclidean plane C, andthe hyperbolic plane as represented by the unit discD. Working with configurations of circles, how-ever, will require a modest bit of bookkeeping, so

bear with me while I introduce the essentials neededto follow the story.

• Complex: The tangency patterns forcircle packings are encoded as abstractsimplicial 2-complexes K; we assume Kis (i.e., triangulates) an oriented topo-logical surface.

• Packing: A circle packing P for K is aconfiguration of circles such that foreach vertex v ∈ K there is a corre-sponding circle cv , for each edge〈v, u〉 ∈ K the circles cv and cu are (ex-ternally) tangent, and for each posi-tively oriented face 〈v, u,w〉 ∈ K themutually tangent triple of circles〈cv, cu, cw〉 is positively oriented.

• Label: A label R for K is a collectionof putative radii, with R(v) denoting thelabel for vertex v .

Look to Figure 2 for a very simple first example.Here K is a closed topological disc and P is a euclidean circle packing for K. I show the carrierof the packing in dashed lines to aid in matchingcircles to their vertices in K; there are 9 interior and8 boundary circles. Of course the question is howto find such packings, and the key is the label Rof radii—knowing K, the tangencies, and R, thesizes, it is a fairly simple matter to lay out the circles themselves. In particular, circle centers playa secondary role. The computational effort in

(a) (b)

(d)(c)

C0

C1

C2C3

Figure 1. A long tradition.


circle packing lies mainly in computing labels. It isin these computations that circle packing directlyconfronts geometry and the local-to-global themeplays out. Here, very briefly, is what is involved.

• Flower: A circle cv and the circles tan-gent to it are called a flower. The or-dered chain cv1 , · · · , cvk of tangent cir-cles, the petals, is closed when v is aninterior vertex of K.

• Angle Sum: The angle sum θR(v) forvertex v, given label R, is the sum of theangles at cv in the triangles formed bythe triples 〈cv, cvj , cvj+1〉 in its flower.Angle sums are computed via the appropriate law of cosines; in the euclidean case, for example,

where the sum is over all faces con-taining v .

• Packing Condition The flower of aninterior vertex v can be realized as anactual geometric flower of circles withlabels from R if and only if θR(v) = 2πnfor some integer n ≥ 1.

It is clear that circles trying to form a packingfor K must tightly choreograph their radii. Thepacking condition at interior vertices is necessary,so a label R is called a packing label if θR(v) is a multiple of 2π for every interior v ∈ K. When K issimply connected, this and a monodromy argu-ment yield a corresponding packing P , and the labels are, in fact, radii. When K is multiply con-nected, however, the local packing condition aloneis not enough, and global obstructions become thefocus. Here are a last few pieces of the ecology.

• Miscellany: A packing is univalent ifits circles have mutually disjoint inte-riors. A branch circle cv in a packing Pis an interior circle whose angle sum is2πn for integer n ≥ 2; that is, its petalswrap n times around it. A packing P isbranched if it has one or more branchcircles; otherwise it is locally univalent.(Caution: Global univalence is assumedfor all circle packings in some parts ofthe literature, but not here.) Möbiustransformations map packings to pack-ings; a packing is said to be essentiallyunique with some property if it isunique up to such transformations. Inthe disc, a horocycle is a circle inter-nally tangent to the unit circle and maybe treated as a circle of infinite hyper-bolic radius.

You are now ready for the internal art of circlepacking. Someone hands you a complex K. Do thereexist any circle packings for K? How many? In whichgeometry? Can they be computed in practice? Whatare their properties? What do they look like?

Let’s begin by explicating certain extremal pack-ings shown in Figure 3. The spherical packing in (a)illustrates the K-A-T Theorem using the combina-torics of the soccer ball. However, our developmentreally starts in the hyperbolic plane; Figure 3(b) illustrates the key theorem (the outer circle representsthe boundary of D).

Key Theorem. Let K be a closed disc. There existsan essentially unique circle packing PK for K in Dthat is univalent and whose boundary circles arehorocycles.

The proof involves induction on the number ofvertices in K and simple geometric monotonici-ties, culminating in a result which deserves its ownstatement. Here RK is the hyperbolic packing label

θR(v) =∑

〈v,u,w〉arccos

((R(v)+ R(u))2 + (R(v)+ R(w ))2 − (R(u)+ R(w ))2

2(R(v)+ R(u))(R(v)+ R(w ))

),

K P

Figure 2. Compare packing P to its complex K.


for the packing PK , so this result justifies the adjective “maximal” that I will attach to these extremal packings.

Discrete Schwarz Lemma [DSL]. Let K be a closeddisc and R any hyperbolic packing label for K.Then R(v) ≤ RK(v) for every vertex v of K; equal-ity for any interior vertex v implies R ≡ RK .

Our Key Theorem is easily equivalent to the K-A-T Theorem, but its formulation and proof setthe tone for the whole topic. The next step, for example, is to extend the Key Theorem to open discsK by exhausting with closed discs Kj ↑ K . When weapply the monotonicity of the DSL to the maximallabels Rj for these nested complexes, a funda-mental dichotomy emerges:

as j →∞

either Rj (v) ↓ r (v) > 0 ∀v ∈ Kor Rj (v) ↓ 0 ∀v ∈ K.

In the former case, a geometric diagonalization ar-gument produces a univalent hyperbolic packingPK for K, its label being maximal among hyperboliclabels as in the DSL. In the latter case, the maximalpackings of the Kj may be treated as euclideanand rescaled, after which geometric diagonalizationproduces a euclidean univalent circle packing PKfor K. Archetypes for the dichotomy are the max-imal packings for the constant 6- and 7-degreecomplexes, the well-known penny-packing, and theheptagonal packing of Figure 3(c), respectively. Wecan now summarize the simply connected cases.

Discrete Riemann Mapping Theorem [DRMT]. IfK is a simply connected surface, then there existsan essentially unique, locally finite, univalent cir-cle packing PK for K in one and only one of thegeometries P, C, or D. The complex K is termedspherical, parabolic, or hyperbolic, respectively,and PK is called its maximal packing.

For complexes Kwhich are not simply connnected,topological arguments provide an infinite universalcovering complex K. By the DRMT, K is parabolic orhyperbolic (the sphere covers only itself) and has amaximal packing P in G (i.e., C or D, respectively).Essential uniqueness of P implies existence of a dis-crete group Λ of conformal automorphisms of Gunder which P is invariant. G/Λ defines a Riemannsurface S, and the projection π : G → S carries themetric of G to the intrinsic metric of constant cur-vature on S, euclidean or hyperbolic, as the case maybe. Some quiet reflection and arrow-chasing showsthat π (P) defines a univalent circle packing PK forK in the intrinsic metric on S. In other words, wehave found our maximal circle packing PK for K.

Discrete Uniformization Theorem [DUT]. Let K bea triangulation of an oriented surface S . Then thereexists a conformal structure on S such that the

resulting Riemann surface S supports a circle pack-ing PK for K in its intrinsic metric, with PK univa-lent and locally finite. The Riemann surface S isunique up to conformal equivalence, and PK isunique up to conformal automorphisms of S .

The DUT is illustrated in Figure 3(d) for a torus hav-ing just 10 vertices. I have marked a fundamentaldomain and its images under the covering group.The 10 darkened circles form the torus when youuse the dashed circles for side-pairings.

This theorem completes the existence/unique-ness picture for extremal univalent circle pack-ings. It is quite remarkable that every complex hascircle packings. From the pure combinatorics of Kone gets not only the circle packing but even thegeometry in which it must live! This highlights cen-tral internal themes of the topic:

combinatorics ↔ geometrylocal packing condition ↔ global structure

You can see that the DUT opens a wealth of ques-tions. It is known that the “packable” Riemann surfaces, those supporting some circle packing,are dense in Teichmüller space, but they have yetto be characterized, and the connections betweenK and the differential geometry of S remain largelyunknown.

Extremal packings only scratch the surface; ingeneral a complex K will have a huge variety of additional circle packings if we are allowed to manipulate boundary values and/or branching.When K possesses a boundary, it has been provedthat given any hyperbolic (respectively euclidean) labels for the boundary vertices of K, there exists aunique locally univalent hyperbolic (respectively

(a) (b)

(c) (d)

Figure 3. Maximal circle packing sampler.


euclidean) packing P having those labels as itsboundary radii. In a similar vein, necessary andsufficient conditions for finite sets of branch cir-cles have been established in many cases. The mostI can do here is illustrate a variety of packings fora single complex. I’ve done this in Figure 4, usinga K with good visual cues. The maximal packing isat the top; then below it, left to right, are a univa-lent but nonmaximal hyperbolic packing, euclideanand spherical packings with prescribed boundaryangle sums, and…the last one? At the heart of thislast owl is a single branch circle, one whose petalcircles wrap twice around it. We will refer back tothese images later.

Pretty as the pictures are, the real gems in thistopic are the elementary geometric and monoto-nicity arguments. Challenge yourself with some ofthese:

• Distinct circles can intersect in at most twopoints! Amazingly, Z-X. He and Oded Schrammproved that this is the key to the uniqueness forparabolic maximal packings.

• In hyperbolic geometry the central circle in aflower with n petals has hyperbolic radius no largerthan − log(sin(π/n)) .

• The important Rodin/Sullivan Ring Lemma: forn ≥ 3 there exists a constant cn > 0 such that in anyclosed univalent flower of circles having n petals,no petal can have radius smaller than cn times thatof the center. By Descartes’s Circle Theorem, thebest constants are all reciprocal integers, beginningwith c3 = 1, c4 = 1/4, c5 = 1/12. I’ll let the readercompute c6.

• Figures 5(a) and (b) show hexagonal spirals. The first, created by Coxeter from the “quad” shownin Figure 1(d), is linked to the golden ratio. The second, along with a whole 2-parameter family ofothers, results from an observation of Peter Doyle:for any parameters a, b > 0 , a chain of six circleswith successive radii a, b, b/a,1/a,1/b, a/b willclose up precisely around a circle of radius 1 toform a 6-flower.

• The spherical packing of Figure 5(c), whichhas the same complex as the packing in Figure 3(a),was generated using 〈2,3,5〉 “Schwarz” triangles;if you look closely at one of the twelve shaded cir-cles, you will see that its five neighbors wrap twicearound it.

• And what of Figure 5(d), the snowflake? Some-times a pretty picture is just a pretty picture.

The internals of the topic that I have outlinedhere are wonderfully pure, clean, and accessible,and those who prefer their geometry unadulteratedshould know that pictures and computers are notnecessary for the theory. On the other hand, thepictures certainly add to the topic, and the fact thatthese packings are essentially computable begsthe question of numerical algorithms, which are an-other source of packing pleasure. There are many,many open questions; let me wrap up with one ofmy favorites: He and Schramm proved that if K isan open disc which packs D, i.e., is hyperbolic,then it can in fact pack ANY simply connectedproper subdomain Ω ⊂ C. Consider the combina-torics behind the packing of Figure 3(c) on page1379, for example. Can you imagine your favoritehorribly pathological domain Ω filled—every nookand cranny—with a univalent packing in whichevery circle has seven neighbors? How does onecompute such packings?

(a)

(b) (c)

(d)

Figure 4. Owl manipulations.

(a) (b)

(c) (d)

Figure 5. Geometric gems.


Dust Off the TheoryWhatever internal richness a new topic develops, thedrive of mathematics is towards the broader view.Where does it fit in the grand scheme? What are theanalogies, links, organizing principles, applications?What does this topic need and what can it offer?

The tools we have used so far—basic geometry andtrigonometry, surface topology, covering theory—are hardly surprising in the context, but perhaps youwondered about references to the Schwarz Lemma,Riemann Mapping Theorem, and Uniformization.The claim is, quite frankly, that one can look to ana-lytic function theory as a model for organizing circlepackings. Let’s jump right in.

Definition. A discrete analytic function is a mapf : Q → P between circle packings which preservestangency and orientation.

The study of circle packings P for Kmay now beposed as the study of the discrete analytic functionsf : PK → P; namely, for each vertex v of K definef (Cv ) = cv, where Cv and cv are the circles for v inPK and P, respectively. For first examples look toFigure 4, where the maximal packing is the commondomain for four discrete analytic functions, fa, fb,fc, fd , mapping to the packings (a), (b), (c), (d), respectively.

With this definition we immediately inherit a won-derful nomenclature: fa from Figure 4 is a discrete analytic self-map of D, and fb is a discreteconformal (Riemann) mapping; the discrete analyticfunction from the packing of Figure 3(a) to that ofFigure 5(c) is a discrete rational function with twelvesimple branch points. A map from the penny-pack-ing to the Doyle spiral of Figure 5(a) or (b) is a dis-crete entire function, in fact, a discrete exponentialmap. Among my favorite examples are the discrete proper analytic self-mappings of D, thediscrete finite Blaschke products. And there aremany others: discrete disc algebra functions; dis-crete versions of sine and cosine; a full family of discrete polynomials; and, when K is compact,discrete meromorphic functions, though these arequite challenging and the theory is just in its infancy.

The whole panoply of function-theory machineryalso opens to us. In particular, the names attached tothe theorems in the last section make perfect sense.One comes to recognize the DSL as the hyperboliccontraction principle; analytic self-maps of the hyperbolic plane are hyperbolic contractions. Likeits classical counterpart, it plays a central role on the way to DRMT and DUT. The hyperbolic/parabolicdichotomy for infinite complexes is just the classi-cal “type” problem and yields, for example, the Discrete Liouville Theorem: a parabolic complex Kcan have no bounded circle packing. Other notionsfrom the classical theory also enter: discrete versionsof extremal length, harmonic measure, random walks,maximum principles, analytic continuation, cover-

ing theory, and the list goes on. And if you are look-ing for a derivative, there is a natural analogue for its modulus in the sharp function, defined at circleCv by f #(Cv ) = radius (f (Cv ))/radius (Cv ).

Ultimately, we find in circle packing a remark-ably comprehensive analogue of classical analyticfunctions and the associated theory. And the parallels are not stretches; they almost formulatethemselves, just as the Thurston finite Riemannmapping of Figure 6 is so clearly a discrete con-formal map. The inevitable question, of course:Does the topic provide more than analogy? morethan nomenclature?

This is where Thurston’s startling conjectureenters our tale, for he saw in the rigidity of circlepackings a direct link to conformality. Accordingto Thurston, if one cookie-cuts a region Ω, as in Fig-ure 6, using increasingly fine hexagonal packings,one obtains discrete mappings which converge tothe classical Riemann mapping F : D → Ω . Theconjecture was soon proved by Burt Rodin andDennis Sullivan in the seminal paper of this topic.Their result has now been extended to more gen-eral (i.e., nonhexagonal) and multiply connectedcomplexes, to all three classical geometries, and tononunivalent and branched packings. The Thurstonmodel holds: given a class of functions, formulatethe discrete (i.e., circle packing) analogues, createinstances with increasingly fine combinatorics, appropriately normalized, then watch as the dis-crete versions converge to their classical models.Of course there are details, for example, geomet-ric finiteness conditions on valence and branching,but putting these aside we have this

• Metatheorem: Discrete analytic func-tions converge under refinement to theirclassical analytic counterparts.

Thus discrete Blaschke products converge toBlaschke products, discrete polynomials to poly-nomials, discrete rational functions to rationalfunctions, and so forth. I cannot show these in static pictures, but we can nonetheless capture theintuition quite succinctly: A classical analytic func-tion is said to “map infinitesimal circles to

ΩPK

Figure 6. A Thurston “finite” Riemann mapping.


infinitesimal circles”; a discrete analytic functiondoes the same, but with real circles.

Fortunately for our pictorial tale we can bring thissame intuition to bear in a more directly geometricway. A Riemann surface is one having a conformalstructure, which is, loosely speaking, a consistentway to measure angles. A conformal map betweenRiemann surfaces is one which preserves this mea-surement (magnitude and orientation). (When thingsare appropriately formulated, conformal maps arejust analytic maps and vice versa.) The discussion inthe box above illustrates construction and refine-ment of discrete conformal maps from a Riemann

surface to plane rectangles. A formal statement of thelimit behavior is somewhat involved, but we have

• Metatheorem: Discrete conformalmappings converge under refinement totheir classical counterparts.

Any lingering doubts the reader may have aboutconnections with analyticity should be put to restby the fact that the K-A-T Theorem actually impliesthe Riemann Mapping Theorem for plane domains.The last critical piece involves elementary (thoughby no means easy) geometric arguments of He andSchramm which replace the original quasiconfor-mal methods in Rodin/Sullivan.

cK

fS

n0 n4

p1p0

This is a toy problem in discrete conformal geometry; your materials are shown above. S is a piecewise affine (p.w.a.)surface in 3-space constructed out of ten equilateral triangles; the cartoon K shows how these are pasted togetherand designates four boundary vertices as “corners”. S is not “flat”; it has two interior cone points with cone anglesπ and 3π. The p.w.a. structure on S brings with it a canonical conformal structure, so S is in fact a simply connected Riemann surface. By the Riemann Mapping Theorem there exists an essentially unique conformal mapF : S → R , where R is a plane rectangle and F maps the corner vertices of S to the corners of R. Question: Whatare the shapes of the ten faces when they are mapped to R?

Here is the parallel discrete construction. Mark each equilateral face of S with arcs of circles as in the trianglelabelled “n = 0”. These arcs piece together in S to define an in situ packing Q0. By using prescribed boundary anglesums, Q0 can be computationally flattened to the packing P0 shown below with rectangular carrier and the designated vertices as corners. In our terminology, f0 : Q0 −→ P0 is a discrete conformal mapping, and it carries the ten faces of S to ten triangles in the plane.

Clearly, this “coarse” packing cannot capture the conformal subtleties of S . Therefore, we use a simple “hex-refine” process which respects the p.w.a. structure of S : namely, break each equilateral face into four equilateralfaces half its size. Applying n such refinement steps gives an in situ circle packing Qn in S , and flattening Qn toget a rectangular packing Pn yields a refined discrete conformal mapping fn : Qn −→ Pn. With four stages of re-finement, for example, each face of S looks like the triangle labelled “n = 4”. The packing P4 is shown below withthe ten faces outlined.

As with Thurston’s conjecture for plane regions, it can be proven that the discrete mappings fn converge onS to the classical mapping F . In other words, as you watch successive image packings Pn , you are seeing the tenfaces converge to their true conformal shapes.


That Special SomethingEvery topic exhibits, at least to its adherents, somespecial character that sets it apart, even as it findsits place in the larger scheme. Will it make a lastingimprint on mathematics? Does it represent a para-digm shift? The true believer always holds out hope.

The synergy among mathematics, computation,and visualization that began with Thurston’s 1985talk infuses circle packing with an experimentalcharacter that I believe is unique in mathematics.Let me speak in the context of CirclePack, a graph-ically based software package for creating, manip-ulating, storing, and displaying packings interac-tively on the computer screen. CirclePack handlesarbitrary triangulated surfaces, simply or multiplyconnected, with or without boundary, in any of thethree geometries. Packings range from 4 to1,600,000 circles (the current record in one of BillFloyd’s tilings); those up to roughly 10,000 circlesnow qualify as “routine”, since packing times of afew seconds give an interactive feel. Multiply con-nected packings are manipulated in their intrinsicmetrics and are displayed in the standard geo-metric spaces as fundamental regions with associated side-pairings, as with the torus ofFigure 3(d). Nearly all the images in this paper comedirectly from CirclePack and are typical of what oneviews on-screen during live experiments.

It is true that nearly every topic has come underthe influence of computing in one way or another,even if only in sharing its algorithmic philosophy.What distinguishes circle packing is the depth ofthe interactions among the mathematics, compu-tations, and visualization—the central results haveinvolved all three.

• Mathematics: This is discrete complex analy-sis, so it touches not only function theory but alsopotential theory and brownian motion, Möbius andconformal geometry, number theory, Fuchsian andKleinian groups, Riemann surfaces and Teichmüllertheory, not to mention applications. This is coremathematics, and the key geometric tools are here:topology, boundary conditions, group actions,branching, and, of course, conformality, in the formof the packing condition. And these tools are nottied to preconceived roles. You want to double acomplex across a boundary? slit two surfaces andpaste them together? mix boundary conditions?try some fractional branching? puncture a torus or carry out a Dehn twist? Go ahead! You may missfamiliar tools—no complex arithmetic, no power series, no functional composition—but much ofcomplex analysis is fundamentally geometric, andyou can see it in action.

• Computation: Space prevents me from givingthe numerics of circle packing its due. Thurston’salgorithm works directly on the geometry, manip-ulating the distribution of curvature among thecircles. In the computations of the ten-triangle

pattern in the box on page 1382, for example, thereis a remarkably stationary flow in the computations: comparing Figure 7 to the packing P4 there, youcan almost see the curvature streaming from pointsof excess to points of shortage. There is also amarkov model of Thurston’s algorithm, plus thereare alternative algorithms by Colin de Verdière andby Bobenko and Springborn. Every improvement inalgorithms seems to be associated with new geo-metric insights; curvature flow, for example, has aclassical interpretation and may aid in parallelizingpacking computations. Perhaps the main open ques-tion concerns a provable algorithm that works di-rectly in spherical geometry; at this time spherical

Figure 7. Curvature flow.

(a) (b)

(c) (d)

Figure 8. “If it is triangulated, circle pack it!”


packing is done in the hyperbolic plane and pro-jected to the sphere.

• Visualization: It is the images and interac-tions with them that really set circle packing apart.You come to anticipate the unexpected in everyexperiment: some surprise symmetry or monoto-nicity, a classical behavior borne out, a new twistbegging explanation, the odd outright mystery. Afew examples are shown in Figure 8.

• The “cat’s eye” of (a), mere play with edge-flips(Whitehead moves), led to lattice dislocation graphs used by physics colleagues in studies of 2Dquenching.

• Thurston’s version of K-A-T actually involvedcircle patterns with prescribed overlaps between cir-cles, tangency being just one option. In CirclePackone can specify not only such overlaps but also inversive distance packings, raising some chal-lenging new existence and uniqueness questions;(b) is our owl with randomly prescribed inversivedistances.

• Use of overlaps in “square grid” packings wasinitiated by Oded Schramm and picked up by thoseinvolved with integrable systems. Alexander

Bobenko and his collaborators haveextended these ideas into the study ofdiscrete minimal surfaces and relatedtopics; one of their gorgeous imagesis shown in (c).

• Circle packing powers the visu-alization of millions of knot and linkprojections in Thistlethwaite and

Hoste’s KnotScape program; here the (3,7) torusknot is shown (with the circles that give it shape).Circle packing has been used to find graph sepa-rators, to generate grids, and to study Whiteheadmoves, hence the motto of Figure 8.

I would argue that CirclePack is to a geometerwhat a moderately well-equipped laboratory is toan organic chemist (only safer). The potential foropen-ended experiments is unique, and yet themachinery is accessible to people at all levels; whoknows, a few experiments and you or your studentsmight be hooked!

New Kid on the BlockThe new topic has linked itself to a rich classical veinwhich it has exploited shamelessly: definitions, the-orems, examples, philosophy. But former colleaguesare beginning to feel used—time for the new kid tostep up and contribute.

Much as I would love to tour discrete functiontheory, it is probably better to stick with a single,more directly geometric example. 2D tiling is wellknown for its mixture of combinatorics and geom-etry, and there is a new theme called “conformal”tiling which grew directly out of circle packing ex-periments . One reverses tradition by starting withthe combinatorics and asking, With what tile shapesand in which geometry can these combinatorics berealized? Let me recount the story of the “twistedpentagonal” subdivision tiling, one of many con-ceived by Cannon, Floyd, and Parry in ongoingwork on Thurston’s Geometrization Conjecture;my thanks to Bill Floyd for the combinatorial data.

We begin with the cartoon in Figure 9, whichshows a rule for breaking one pentagon into fiveby adding edges and vertices. Your task, startingwith one pentagon, is to repeatedly apply this sub-division rule: at the first stage you get 5 pentagons,then these are subdivided into 25, and these into125, etc. The combinatorics quickly get out of hand,and circle packing is brought in initially just to getuseful embeddings: at each stage, a barycenteradded to each pentagon gives a triangulation whichcan then be circle-packed, giving shape to the pentagons. The first three stages are shown inFigure 9 with their circles.

The pictures are an immediate help, for after afew additional stages you come to realize that the“subdivision” rule can be replaced by a corre-sponding “expansion” rule. There is, in fact, an es-sentially unique infinite combinatorial tiling T

Figure 9. Three stages of subdivision.

Figure 10. An infinite, subdivision-invariant pattern T.


which is subdivision-invariant: i.e., if you simulta-neously subdivide all the pentagons, the result isagain combinatorially equivalent to T. This T issuggested by Figure 10.

If you caught the spirit of our ten-triangle ex-ample in the box on page 1382, you might try thesame treatment here. Pasting regular euclideanpentagons together in the pattern of T yields asimply connected p.w.a. Riemann surface S . Rie-mann himself would have known that there is a con-formal homeomorphism f from S to one of C orD. The images of the faces under f form a so-calledconformal tiling T with the combinatorics of T. Isthis tiling parabolic or hyperbolic? (i.e., does it liein C or in D?) What are the shapes of its tiles? Doesthe pattern have internal structure? Before circlepacking, there was no way to approach such ques-tions, so they weren’t asked!

Now we have a method. As you might have sus-pected, Figure 10 was created using circle packing;it is a rough approximation of T by “coarse” circlepackings like those of Figure 9. Your first instinctmight be to improve conformal fidelity via refine-ment, as we did in the box on page 1382. The keyexperiments, however, turn out to be of quite adifferent nature. Stare at T for a moment—per-haps let your eyes defocus. The longer you look, themore certain you become of some large-scale pat-tern. Let me help you pull it out. As T is invariantunder subdivision, so must it be invariant under ag-gregation (unsubdividing). On the left in Figure 11are the outlines of the first four stages of aggre-gation; that is, each of the outlined aggregates iscombinatorially equivalent to a subdivision of itspredecessor.

Do you see a hint of a pattern now? A little workin PostScript to dilate, rotate, and overlay the out-lines leads to the picture on the right in Figure 11.The scale factor turns out to be roughly the samefrom one stage to the next, suggesting that the tilingis parabolic. More surprising, you find that the corners of the outline from one stage seem to lineup with corners of the next: each edge at one stageis replaced by a zig-zag of three edges at the next.Motivated by these very images, Cannon, Floyd,Parry, and Rick Kenyon have confirmed all theseobservations. In fact, a wealth of mathematics con-verges in these images: This tiling turns out to beassociated with one of Grothendieck’s dessins d’en-fants on the sphere. Hence there exists a rationalfunction with algebraic coefficients whose iteratesencode the subdivision rule. That iteration gives anassociated Königsfunction k, an entity right out ofnineteenth-century function theory, and T is justthe cell decomposition of C defined by k−1([0,1]).And with the scaling confirmed, renormalization(as started on the right in Figure 11) suggests a limit tiling in the pattern of T which would haveperfect scaling, fractal pentagonal tiles, and a

fractal subdivision rule. It is clear that our discreteexperiments have faithfully captured some truthin classical conformal geometry.

There are many other combinatoric situationswhere similar experiments can be run. At the risk

Figure 11. Outline, scale, rotate, and overlay the aggregates.

Figure 12. Circle packing for conformal structure.


of leaving some mysteries for the reader, I haveshown a few in Figure 12. The two images at the topare straight out of classical function theory. Onthe left, circles (not shown) recreate the “Klein”surface in an image made famous over 100 yearsago in Fricke/Klein. On the right is the more illu-sive “Picard” surface; we can now approach this andother surfaces whose triangulations are not geo-desic. If you are into number theory, the topicknown as dessins d’enfants, children’s drawings,alluded to earlier, tightly binds combinatorics,meromorphic functions, and number fields. Itsstructures are equilateral, like the ten-triangle ex-ample in the box on page 1382, so the topic is a nat-ural for discrete experimentation; the middle im-ages in Figure 12 show genus 0 and genus 2 dessinexamples, respectively. Likewise, the notion of con-formal welding used in complex analysis and inthe study of 3-manifolds is now an experimental re-ality. The bottom images in Figure 12 show a toyexample, where two owls have been welded to-gether, along with a more serious example pro-duced by George (Brock) Williams.

Note how the tables have turned. We are nowstarting with combinatorics. Circle packing then imposes a discrete conformal structure, that is, ageometry manifesting key characteristics of con-formality—notions such as ‘type’, extremal length,moduli of rings, harmonic measure, and curvature.Then one can search for, perhaps even prove, parallel classical results. In summarizing the intu-ition, it seems only fair to let the discrete side take the lead: A discrete conformal structure on a surface is determined by a triangulation; a classi-

cal conformal structure is determined in the sameway, but with “infinitesimal” triangles.

I cannot leave this section without mentioninga point of closure in the theory provided by resultsof He and Schramm. They have made a major advance on a classical conjecture concerned withthe conformal mapping of infinitely connected regions, the so-called Kreisnormierungsproblem,by applying methods which they developed in circle packing. And exactly whose conjecture wasthis? None other than P. Koebe himself: he provedthe finitely connected case and then applied hisclassical methods to establish the K-A-T Theorem!

Reaching OutIt is an article of mathematical faith that every topicwill find connections to the wider world—eventually.For some, that isn’t enough. For some it is real-timeexchange between the mathematics and the appli-cations that is the measure of a topic.

The important roles complex analysis tradition-ally played in the physical sciences—electrostatics,fluid flow, airfoil design, residue computations—are largely gone, replaced by numerical partial differential equations or symbolic packages. But the core of complex analysis is too fundamental togo missing for long. Surfaces embedded in three-space are becoming pervasive in new areas of science, image analysis, and computer visualization,and conformal geometry is all about such surfaces.With new tools to (faithfully) access conformality,perhaps complex analysis has new roles to play.

I would like to illustrate briefly with brain-flattening work that has garnered recent exposureoutside of mathematics. The work is being carriedon by an NSF-sponsored Focused Research Group:Chuck Collins and the author (Tennessee); PhilBowers, Monica Hurdal, and De Witt Sumners(Florida State); and neuroscientist David Rottenberg(Minnesota).

The first image in Figure 13 shows the type of3D data which is becoming routinely availablethrough noninvasive techniques such as MRI (magnetic resonance imaging), in this case, onehemisphere of a human cerebrum. Our mental processing occurs largely in the cortex, the thin layer of neurons (grey matter) on the brain surface.Neuroscientists wishing to apply surface-basedtechniques need to map the cortex to a flat do-main—hence the topic of “brain-flattening”. As youcan see, the cortex is an extremely convoluted surface (the shading here reflects the mean cur-vature), and it is well known that there can existno flat map which preserves its areas or surface distances. However, by the 150-year-old RiemannMapping Theorem there does exist a conformalflat map. Using standard techniques, one can produce a triangulation which approximates thecortical surface from the volumetric data. Figure 13

Figure 13. A 3D cortical hemisphere and three flat maps.


illustrates three discrete conformal flat maps based on such a triangulation (180,000 vertices):clockwise from upper right are spherical, hyper-bolic, and euclidean flat maps.

It is not our goal to discuss the potential scien-tific value in these maps (though I will mention thatour neuroscience colleagues have a surprising affinity for the hyperbolic maps; perhaps theseare reminiscent of the view in a microscope). How-ever, there are some points that do bear on ourstory.

• First, one does not have to believe that con-formality per se has any relevance to an applicationto exploit its amazing richness—existence anduniqueness first, then companion notions such asextremal length and harmonic measure.

• Second, approximation of true conformalitymay be superfluous if its companion structures appear faithfully at coarse stages, as seems oftento be the case with circle packing.

• Finally, the structures take precedence overtechnique; circle-packing experiments can con-tribute to a topic even if other methods replace itin practice. What would be nice to hear at the endof a neuro consult is, “You know, we need to hireanother conformal geometer.”

ConclusionOf course, a mathematical topic itself never con-cludes; the tradition is “definition-theorem-proof-publication” as new contributions add to the line. A mathematical tale, on the other hand, must haveclosure, and the storyteller is allowed to put somepersonal spin on the story (if not a moral).

I have related this tale in the belief that it hassome touch of universality to it. We are drawn tomathematics for a variety of reasons: the clarity ofelementary geometry, the discipline of computation,the challenge of richly layered theory and deepquestions, the beauty of images, or the pleasuresof teaching and applying the results. I feel that I haveseen all these in circle packing, and perhaps youhave glimpsed parallels with your own favoritetopic.

Personally, it has been a pleasure to watch an oldfriend, complex function theory, emerge in a formwith so much appeal: new theory, new applica-tions, stunning visuals, an exciting experimentalslant. For me circle packing is quantum complexanalysis, classical in the limit. The discrete resultsand their proofs are pure mathematics, the pic-tures and software being not only unnecessary,but for some, unwanted. Yet the experimentationand visualization, the very programming itself, areat the research frontier here. In this regard, circlepacking illustrates the growing challenge mathe-matics faces to incorporate new modes of researchinto its practices and literature.

Of course circle packing, like any mathematicaltopic, has many potential storylines. In that spirit,let me end our mathematical tale in a very tradi-tional way, namely, in the hope that it nourishesothers who can pass along their own stories intheir own words.

Reader’s GuideOnce: [12], [26], [28], [1], [30]. Birth: [25], [2], [35],[36], [13]. Internal: [27], [5], [4], [9], [21], [3], [22], [10].Dust: [19], [18], [34] (survey), [31]. Special: [32], [20](survey), [15], [7], [17], [29], [6], [23]. New Kid: [33](survey), [21], [19], [11], [8], [37], [14], [16]. ReachOut: [24]. (You can download a more complete bibliography and CirclePack from my website:http://www.math.utk.edu/~kens.)

References[1] DOV AHARONOV and KENNETH STEPHENSON, Geometric se-

quences of discs in the Apollonian packing, Algebrai Analiz, dedicated to Goluzin 9 (1997), no. 3, 104–140.

[2] E. M. ANDREEV, Convex polyhedra in Lobacevskiı space,Math. USSR Sbornik 10 (1970), 413–440 (English).

[3] ALAN F. BEARDON, TOMASZ DUBEJKO, and KENNETH STEPHEN-SON, Spiral hexagonal circle packings in the plane,Geom. Dedicata 49 (1994), 39–70.

[4] ALAN F. BEARDON and KENNETH STEPHENSON, The uni-formization theorem for circle packings, Indiana Univ.Math. J. 39 (1990), 1383–1425.

[5] ——— , The Schwarz-Pick lemma for circle packings,Illinois J. Math. 35 (1991), 577–606.

[6] A. I. BOBENKO, T. HOFFMANN, and B. SPRINGBORN, Minimalsurfaces from circle patterns: Geometry from combi-natorics, preprint, 2002; arXiv:math.DG/0305184.

[7] ALEXANDER I. BOBENKO and BORIS A. SPRINGBORN, Varia-tional principles for circle patterns and Koebe’s the-orem, preprint, 2002; arXiv:math.GT/0203250.

[8] PHILIP L. BOWERS and KENNETH STEPHENSON, Uniformizingdessins and Bely i maps via circle packing, Memoirs,Amer. Math. Soc., to appear.

[9] ——— , Circle packings in surfaces of finite type: Anin situ approach with applications to moduli, Topol-ogy 32 (1993), 157–183.

[10] ——— , A branched Andreev-Thurston theorem for cir-cle packings of the sphere, Proc. London Math. Soc. (3)73 (1996), 185–215.

[11] ——— , A “regular” pentagonal tiling of the plane, Con-form. Geom. Dyn. 1 (1997), 58–86.

[12] Robert Brooks, The spectral geometry of the Apol-lonian packing, Comm. Pure Appl. Math. 38 (1985),358–366.

[13] ——— , Circle packings and co-compact extensions ofKleinian groups, Invent. Math. 86 (1986), 461–469.

[14] JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, andWALTER R. PARRY, Constructing rational maps from sub-division rules, preprint, 2001.

[15] YVES COLIN DE VERDIERE, Une principe variationnel pourles empilements de cercles, Invent. Math. 104 (1991),655–669.

[16] CHARLES R. COLLINS, TOBIN A. DRISCOLL, and KENNETH

STEPHENSON, Curvature flow in conformal mapping,preprint, 2003.


[17] CHUCK COLLINS and KENNETH STEPHENSON, A circle pack-ing algorithm, Comput. Geom. 25 (2003), 233–256.

[18] TOMASZ DUBEJKO, Infinite branched packings and dis-crete complex polynomials, J. London Math. Soc. 56(1997), 347–368.

[19] TOMASZ DUBEJKO and KENNETH STEPHENSON, The branchedSchwarz lemma: A classical result via circle packing,Michigan Math. J. 42 (1995), 211–234.

[20] ——— , Circle packing: Experiments in discrete ana-lytic function theory, Experiment. Math. 4 (1995), no. 4,307–348.

[21] ZHENG-XU HE and ODED SCHRAMM, Fixed points, Koebeuniformization and circle packings, Ann. of Math. 137(1993), 369–406.

[22] ——— , The inverse Riemann mapping theorem for rel-ative circle domains, Pacific J. Math. 171 (1995),157–165.

[23] JIM HOSTE and MORWEN THISTLETHWAITE, Knotscape, Soft-ware package, http://www.math.utk.edu/~morwen.

[24] M. K. HURDAL, P. L. BOWERS, K. STEPHENSON, D. W. L. SUM-NERS, K. REHM, K. SCHAPER, and D. A. ROTTENBERG, Quasi-conformally flat mapping the human cerebellum, Med-ical Image Computing and Computer-AssistedIntervention—MICCAI’99 (C. Taylor and A. Colchester,eds.), vol. 1679, Springer, Berlin, 1999, pp. 279–286.

[25] PAUL KOEBE, Kontaktprobleme der Konformen Abbil-dung, Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl.88 (1936), 141–164.

[26] BENOIT B. MANDELBROT, The Fractal Geometry of Nature,W. H. Freeman, San Francisco, CA, 1982.

[27] BURT RODIN and DENNIS SULLIVAN, The convergence ofcircle packings to the Riemann mapping, J. Differen-tial Geom. 26 (1987), 349–360.

[28] TONY ROTHMAN, Japanese temple geometry, Sci. Amer.278, no. 5, (1998), 84–91.

[29] ODED SCHRAMM, Circle patterns with the combina-torics of the square grid, Duke Math. J. 86 (1997),347–389.

[30] F. SODDY, The Kiss Precise, Nature (June 20, 1936),1021.

[31] KENNETH STEPHENSON, Introduction to Circle Packingand the Theory of Discrete Analytic Functions, Cam-bridge University Press, forthcoming 2004.

[32] ——— , CirclePack software (1992–2002), http://www.math.utk.edu/~kens.

[33] ——— , Approximation of conformal structures via cir-cle packing, Computational Methods and Function The-ory 1997, Proceedings of the Third CMFT Conference(N. Papamichael, St. Ruscheweyh, and E. B. Saff, eds.),vol. 11, World Scientific, 1999, pp. 551–582.

[34] ——— , Circle packing and discrete analytic functiontheory, Handbook of Complex Analysis, Vol. 1: Geo-metric Function Theory (R. Kühnau, ed.), Elsevier, 2002.

[35] WILLIAM THURSTON, The geometry and topology of 3-manifolds, Princeton University notes, preprint.

[36] ——— , The finite Riemann mapping theorem, invitedtalk, an International Symposium at Purdue Universityon the occasion of the proof of the Bieberbach Con-jecture, March 1985.

[37] GEORGE BROCK WILLIAMS, Approximation of quasisym-metries using circle packings, Discrete Comput. Geom.25 (2001), 103–124.

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