UNIFbowers/Papers/dessin.pdf · UNIF ORMIZING DESSINS 3 shaded faces to the lo w er half-plane, and appropriate iden ti cations are made along the in terv als [1; 0] [0 1] [1], then

UNIFORMIZING DESSINS AND BELY�I MAPS

VIA CIRCLE PACKING

PHILIP L. BOWERS AND KENNETH STEPHENSON

Abstract. Grothendieck's theory of Dessins d'Enfants involves combinatorially de-

termined a�ne, re ective, and conformal structures on compact surfaces. In this

paper the authors establish the �rst general method for uniformizing these dessin

surfaces and for approximating their associated Bely�� meromorphic functions.

The paper begins by developing a discrete theory of dessins based on circle packing.

This theory is surprisingly faithful, even at its coarsest stages, to the geometry of the

classical theory, and it displays some new sources of richness; in particular, algrebraic

number �elds enter the theory in a new way.

The paper goes on to show that the discrete dessin structures converge to their

classical counterparts under a hexagonal re�nement scheme. In addition, since the

discrete objects are computable, circle packing provides opportunities both for routine

experimentation and for large scale explicit computation. A range of examples up to

genus 4 is given in the paper, and an appendix addresses implementation issues.

1. Introduction

We are concerned in this paper with structures on triangulated surfaces. Our moti-vation is the theory of Dessins d'Enfants, traced to Grothendieck [13], an intriguingblend of algebra, combinatorics, conformal geometry, and complex function theory.The creation mythology of the topic posits a child innocently \drawing" on a topolog-

ical surface. Unbeknownst (presumably) to the child, that simple drawing determinesan algebraic number �eld and its Galois group, a conformal stucture on the surface, aclass of meromorphic functions, and a group of companion drawings. This becomes astory, then, of various rigid algebraic, analytic, and geometric structures, inextricablyintertwined, but all owing from simple combinatorics.The central aim of the study for algrebraists has been a deeper understanding of the

absolute Galois group Gal(Q =Q) and the famous \inverse Galois problem". Of course,any theory bringing together so many topics will invariably inspire other goals, and anextensive literature has developed. We particularly recommend the proceedings [29]for the 1993 Luminy conference for a broad view. Interests are both theoretical andpractical: along with the discussion of \braid towers" and Shimura varieties are the

Date: December 12, 1997.The second author gratefully acknowledges support of the National Science Foundation and the

Tennessee Science Alliance.

1

2 PHILIP L. BOWERS AND KENNETH STEPHENSON

e�orts of physicists to connect dessins with matrix models in computational studies ofRiemann surfaces for string theory.

The authors of the present paper, having worked in circle packing, were drawn tothe topic principally through triangulations: combinatorics also lead to rigid geometricstructures via circle packing. A rich theory has developed around circle packingssince their introduction by Thurston in [33] and [34], with particular connections toconformal geometry. One familiar with these developments cannot help but recognizethe many parallels with at least the combinatoric and geometric aspects of the theoryof dessins. The links promise to enrich the theory and practice of both topics. Inaddition, there is a visual and numerical side to circle packing which might contributeto the theory of dessins | perhaps introducing an experimental aspect.

To introduce the viewpoint of this paper, it will help to review brie y the elementsof the theory of dessins. A dessin d'enfant D is basically a �nite connected graph ona (compact orientable) surface S. Associated with D is a canonical triangulation Tof S; this is our fundamental combinatorial data. A simple genus 0 dessin and itstriangulation are given in Figure 1.

Figure 1. A simple dessin and associated triangulation

The triangulation T in turn imposes a conformal structure on S making it into aRiemann surface SD; this is accomplished by constructing a model for S by pastingtogether euclidean unit-sided equilateral triangles in the pattern encoded in T | aso-called equilateral surface. The triangulation is such that alternate faces may beshaded; if the unshaded faces are mapped conformally to the upper half-plane, the

UNIFORMIZING DESSINS 3

shaded faces to the lower half-plane, and appropriate identi�cations are made along theintervals [1; 0]; [0; 1]; [1;1], then one obtains a meromorphic function BD : SD �! S2

branching only over f0; 1;1g. The structures are conveniently packaged in the so-called Bely�� pair (SD; BD); indeed, the dessin D itself is in there, being (isomorphic to)the set B�1

D [0; 1] � SD. Moreover, the Riemann surface SD, as an algebraic surface,has a de�ning equation whose coe�cients lie in an algebraic number �eld F (a �nitealgebraic extension of the rationals).For surfaces of positive genus, the results may be summarized in a very striking form.

The most di�cult of the implications is due to Bely�� , so this is commonly referred toas

Bely��'s Theorem. For a Riemann surface R of genus g � 1, the following statementsare equivalent:(a) There exists a de�ning equation for R whose coe�cients lie in an algebraic

number �eld F over the rationals.(b) There exists a nonconstant meromorphic function f : R �! S2 which branches

only over the points f0; 1;1g.(c) R is conformally equivalent to an equilateral surface.

In other words, we have a distinguished class of Riemann surfaces which simultane-ously enjoys algebraic, function-theoretic, and combinatoric characterizations. Equiv-alent to all these, R = SD for some dessin D. Thus, all these consequences ow froma simple drawing.

Let us now turn to circle packing and what we will refer to as the \discrete" setting.A circle packing is a con�guration of circles realizing a speci�ed pattern of tangen-cies. It enjoys dual combinatoric and geometric natures: the \pattern" of tangenciesis encoded as an abstract triangulation of a surface, while circle radii provide the ge-ometry. We paraphrase the central theoretical pivot, with terminology to be explainedlater.

Theorem. Let T be a simplicial triangulation of a compact oriented topological surfaceS. Then there exists a unique Riemann surface homeomorphic to S that supports aunivalent circle packing P with the combinatorics of T .

Thus circle packings provide an alternate way in which abstract combinatorics deter-mine a rigid conformal structure.Let's now begin again with a dessin D on a surface S. Generate the associated trian-

gulation T as before (though for technical reasons we work instead with the barycentricsubdivision of T ). The Circle Packing Theorem provides a conformal structure for S,giving a Riemann surface, denoted sD, which supports a circle packing P for T . Again,we may shade alternate faces of T as de�ned by the circles. Circles corresponding tothe unshaded faces can be identi�ed with circles in the sphere packing the upper half-plane, those corresponding to the shaded faces with circles packing the lower half-plane,


and the edge circles can be appropriately identi�ed along the intervals [0; 1]; [1;1]; and[1; 0] to form a circle packing Q of S2. The identi�cation of P with Q yields a map-ping bD : sD �! S2 which acts as a discrete meromorphic function branching only over0; 1;1. Thus we arrive at a discrete Bely�� pair (sD; bD) associated with our dessin.

In summary, then, a dessin can be associated with rigid conformal data in TWOparallel ways, via classical Bely�� pairs or via their discrete analogues.

First Objective: to develop the discrete theory, emphasizing these par-allels in combinatorics and geometry.

However, our discrete objects not only mimic their classical counterparts, but alsoapproximate them. We prove that certain re�nements Tn of a triangulation T , whileinducing the identical classical Bely�� pair (SD; BD), will lead to new discrete Bely�� pairs

(s(n)D ; b

(n)D ) based on \�ner" circle packings. We prove that under successive re�nement,

(s(n)D ; b

(n)D ) �! (SD; BD); as n!1:

In other words, the Riemann surfaces s(n)D converge to SD in Teichm�uller space, while

the discrete Bely�� maps b(n)D converge uniformly on compacta to BD. The discrete

objects have the advantage that they are e�ectively computable.

Second Objective: to prove that the objects of the discrete theory gener-

ated by circle packings uniformize classical dessin surfaces and approximateclassical Bely�� maps.

It should be noted that there are varying uses of the term \uniformize" in the dessinliterature: For a given dessin surface SD, this often refers to the structure it inheritsfrom a classical triangle group by modding out a covering group; this is actually a con-formal structure on the punctured surface SDnV , where V is the set of vertices of T .On the other hand, we uniformize the full surface SD. Despite very concrete descrip-tions of the surfaces, it has been almost impossible to compute such uniformizationsin the past, aside from certain highly symmetric situations.

We might exploit the visual nature of circle packings here with an early example |details will come later. Let's consider the dessin of Figure 1. Figure 2(a) illustratesthe circle packing of S2 associated with three stages of re�nement of T ; the edges of Tare drawn in, with the heavy edges being those of the dessin D, and appropriate facesare shaded. Figure 2(b) illustrates the packings of the upper/lower half planes at the

corresponding re�nement stage. The discrete meromorphic function b(3)

D identi�es eachcircle on the left with a corresponding circle on the right. (This de�nes a 4-sheetedcovering of S2, so each circle on the right is the image of four circles on the left.)The discrete dessin is nearly conformally correct, and in positive genus examples, thesurface will be nearly equal in modulus to the classical surface. This illustrates our


S2

(a) (b)

bD(3)

Figure 2. A discrete Bely�� map

Third Objective: to provide a variety of examples and discuss the com-putational issues associated with the theory.

A notable loss in the discrete setting is the direct connection to number �elds. Recallthat Bely�� 's Theorem provides an elegant algebraic characterization of the countabledense set of \equilateral" points of Teichm�uller space. There is as yet no characteri-zation of the \circle packing" points, also countable and dense; this characterizationappears to be a challenging issue. The circle packing points are, of course, indirectlyassociated with number �elds via the equilateral surfaces. However, it also turns outthat algebraic numbers enter directly, but in a new way | as entries in the coveringgroup in PSL(2; C ). There are a number of open questions here, and we will return tothis issue in the last section of the paper.

Now for a brief outline of our paper. Section 1 is concerned with dessins in theclassical setting (we use the term \classical" only to distinguish it from the discretesetting), and we review the basic terminology, notation, and theory used in the sequel.In Section 2 we develop the parallels in the discrete setting, beginning with the basictheory of circle packing and ending with the discrete Bely�� pairs.Section 3 contains the paper's main theoretical results. We introduce the key no-

tions of re ective structures and their conformal subdivisions for triangulated surfaces.This permits re�nement of a triangulation without changing its associated conformalstructure. We then prove (Theorems 4.5 and 4.10) that the discrete Bely�� pairs for


successive re�nements of a dessin triangulation converge to the classical Bely�� pair forthe dessin. In other words, using circle packings we simultaneously uniformize theassociated Riemann surface and approximate its Bely�� map.Section 4 is a Menagerie of examples, organized by genus. We encourage the reader to

make an early visit. We illustrate several examples from the dessin literature, classicalsurfaces such as Klein's and Picard's, and new examples up to genus 4. These showthe type and quality of information available in this discrete approach. We discusscomputational and experimental issues in Section 5, including a \dessin modi�cation"procedure which, coupled with speedy computation and display, may provide newinsights into the behavior of dessins. (For readers wanting more computational details,we have included an Appendix discussing alogrithms, run times, visualization, and soforth. The illustrations in the paper were generated using packages CirclePack andDesPack available from the second author.)The paper concludes with a summary of the discrete setting, its faithfulness to its

classical counterpart, and directions for further development. This work also raisesseveral questions about circle packing: the characterization of packable Riemann sur-faces, the surprising accuracy of even coarse packing structures, and new connectionsto algebraic numbers.The authors express their thanks to Jack Quine, Robert Varley, Chuck Collins, and

David Singerman for helpful conversations and to Alan Beardon, Keith Carne, and theUniversity of Cambridge for their hospitality during the authors' sabbaticals when thiswork began.


2. Dessins d'Enfants

Our work involves various procedures for constructing Riemann surfaces (with extrastructure) from purely combinatorial data. In this section we review the fundamen-tals of Grothendieck's theory of dessins d'enfants which motivate the developments ofthe paper. Dessin theory is of recent origin and we refer to it as \classical" only todistinguish it from the coming \discrete" theory.Authors di�er slightly in the terminology they employ for dessins d'enfants. We

adopt that of [29, p.4] and refer the reader to [29] for more extensive treatments of thetopic and, in particular, for a discussion of the deep and intimate connections betweendessins d'enfants, Bely�� maps, surfaces de�ned over number �elds, and the absoluteGalois group Gal(Q =Q). Here we recall terminology, �x notation, and state resultswhich set the pattern for the remainder of the paper.

2.1. Dessins d'enfants.

De�nition. A dessin d'enfant, or dessin for short, is an oriented, closed (com-pact and connected with empty boundary) topological surface S equipped with a �niteembedded graph D for which

1. the graph D is connected,

2. the complement SnD is a (�nite) collection of open 2-cells,3. the vertices of D are 2-colorable; i.e., the vertex set V decomposes as the disjoint

union of two nonempty collections, V0 and V1, such that every edge of D has onevertex in V0 and the other in V1.

Requirement (3) implies that D has no loops, though there may be multiple edges.The vertices in Vi are called the i-vertices, for i = 0; 1. In diagrams of dessins, we usea solid dot � to mark 0-vertices and a small � to mark 1-vertices. We consider dessins(S1; D1) and (S2; D2) to be equivalent if there exists an orientation preserving home-omorphism of S1 to S2 whose restriction to D1 is a color-preserving graph isomorphismof D1 to D2. We make no distinction between equivalent dessins.A dessin (S;D), or just D if S is understood from the context, is pre-clean if each

1-vertex has valence at most 2 and is clean if each 1-vertex has valence equal to 2.There is a standard way to get a clean dessin from a graph G | what Grothendieckdescribed as a \drawing". For simplicity, nearly all our examples will be of this type,although the methods apply to all dessins, and even more general situations. If G isembedded in a surface S and (1) and (2) hold, one obtains a clean dessin D by addinga � to each edge, so V0 is the set of original vertices of G and V1 the set of addedvertices. There is no loop-free restriction on the graph G.By triangulation of a surface S we mean a regular cellular decomposition T of S

with each 2-cell a topological triangle so that two 2-cells, when they meet, meet in asubcomplex of the 1-skeleton T (1). If it happens that two faces can meet only at asingle vertex or along a single edge, we refer to T as a simplicial triangulation ofS. In general, however, a triangulation might have faces meeting along the union of


two sides, along one common side and a common opposite vertex, along three commonvertices but no sides, or even along all three sides. Notice that we view a triangulationas a structure in situ, as a collection of actual subsets of S | vertices consistingof points of S, edges consisting of arcs in S, and faces consisting of 2-cells in S.Abstract triangulations | purely combinatorial objects | will always have surfacesas geometric realizations. The degree of a triangulation is the maximum of the degreesof its vertices; that is, the largest number of edges emenating from any one vertex.Associated with a dessin (S;D) is a canonical triangulation of S, denoted T =

T (D), described as follows. The vertex set of T is the disjoint union V0 [ V1 [ V1,where V0 and V1 are as before and V1 consists of a collection of points from SnD, onepoint from each (2-cell) component of SnD. The edges of T consist of the edges of Dalong with edges formed by connecting vertices in V0 [ V1 with those of V1 as follows.For v 2 V0 [ V1, let � be a disc neighborhood of v that meets D only at points ofst(v;D); i.e., at points in the edges of D incident with v. The edges of D divide �into d open sectors, where d = deg(v;D). Add, for each sector �, an edge emanatingfrom v, meeting D only at v, and traveling along an arc through � into the 2-cellcomponent of SnD containing � until it meets the vertex in V1 in that component.We proceed, of course, so that new edges, if they meet, meet only at the vertices inV1. For the general dessin on the left in Figure 1, the procedure produces the cellulardecomposition of S = S2 on the right. As we shall do subsequently, we have markedthe 1-vertices, the elements of V1, with open dots �. The open faces of T consistof the components of XnE, where E denotes the union of the edges of T . Notice thateach face of T is a topological triangle with three vertices, one of each type, and acircuit of three edges, one of type �|||�, one of type �|||�, and another of type�|||�. The faces are naturally partitioned into two collections by the orientation onS, depending on whether or not the circuit �|||�|||�|||� around the boundary ofa face is compatible with the orientation the face inherits from the orientation of S. Ifthe circuit �|||�|||�|||� is compatible, we call the face a (+)triangle; otherwise a(�)triangle. In diagrams, we shall shade in the (�)triangles, as in Figure 1. Notice thateach vertex of T has even degree, and no two (+)triangles (resp. (�)triangles) share acommon edge. Any two canonical triangulations of (S;D) are not only combinatorallyequivalent, but isotopic, so we may speak of the canonical triangulation of (S;D).

2.2. The Equilateral Structure. Up to this point, a dessin represents purely com-binatorial data. However, it can be realized geometrically as a piecewise equilateralsurface.

De�nition. Let T be a triangulation of the topological surface S (possibly with bound-

ary). An equilateral metric structured on T is a piecewise euclidean distancefunction on S in which each edge of T is isometric with the unit interval [0; 1] andeach face of T is isometric with a euclidean unit equilateral triangle. The surface Sequipped with such a metric is termed an equilateral surface structured on T , and

is denoted as jT jeq.


One can always construct such a metric, de�ning it �rst on the 1-skeleton, extendinglocally so that each face becomes a euclidean unit equilateral triangle, and �nallyde�ning the distance between points as the length of a shortest path between them,which always exists. We could equally well use edges of length " > 0, in which case wedenote the surface by jT j"eq; this simply respresents a scaling of the metric by ".One next de�nes a compatible conformal structure. Each edge e 2 T is shared by two

faces f�; let Ue � S denote the open set f �+[e�[f ��. De�ne � C to be the interior of

the union4[4, where4 is a unit equilateral triangle with vertices h0; 1; (1+p3i)

2i. From

the de�nition of our equilateral metric, it is evident that there exists an orientation-preserving, isometric map 'e of Ue onto . The charts f(Ue; 'e) : e 2 T g provideSnV , where V is the vertex set of T , with a covering by compatible analytic chartsdetermining a complex atlas. This atlas can be augmented by introducing chartsinvolving appropriate power functions in neighborhoods of the isolated singularities atthe vertices V (see, e.g., [3, x3.3]). The resulting atlas de�nes a conformal structurefor all of S.

De�nition. In general, if T is a triangulation of a topological surface S, the Riemannsurface associated with the equilateral surface structured on T is called the equilateral

surface for T and denoted ST . In case T = T (D), the canonical triangulation fordessin D, we refer to this Riemann surface as the dessin surface and denote it bySD.

Note that T will be treated as an in situ triangulation of ST . As such, it is uniquelydetermined up to conformal automorphisms and will be treated as �xed; when nec-essary, a convenient normalization may be imposed. We will be investigating the\re ective" nature of these structures in the sequel.

2.3. The Bely�� Map. A Bely�� map of a Riemann surface R is a meromorphic func-tion B : R �! S2 that is rami�ed only over 0, 1, and 1. (In other words, if B fails tobe locally one-to-one at a point z 2 R, then B(z) equals 0; 1; or 1.)Using Schwarz re ection, one constructs a canonical Bely�� map for a given dessin

surface SD as follows. A dessin face f 2 T (D) is a topological triangle with a conformalstructure inherited from SD; consequently there exists a unique conformal mapping ofthe interior of f onto an open hemisphere of S2 that extends continuously to @f andcarries the j-vertex to j 2 S2, j = 0; 1;1. Orientation guarantees that (+)trianglesare mapped to the upper half-plane, (�)triangles to the lower. De�ne B on the interiorof each face f to be the associated conformal map. If faces f� share an edge e of T (D),recall the map 'e : Ue �! de�ned above. If g = B � '�1e , then the restriction of

g to 4 � (resp. 4 � ) is a conformal map onto the upper (resp. lower) half-plane and g extends to the interval [0; 1] and is real there. By Schwarz re ection, gextends analytically to a conformal map on all of . In particular, then, B extendsanalytically across the open edge e and is locally one-to-one. That is, B is a smoothanalytic covering map of SDnV onto the thrice punctured sphere S2nf0; 1;1g. The


isolated singularities are removable, with values in f0; 1;1g so B is a meromorphicfunction on SD which can branch only over f0; 1;1g.

De�nition. Given a dessin (S;D), the meromorphic function B : SD �! S2 de�ned

as above is termed the Bely�� map for D and will be denoted BD. Note that BD

is uniquely determined up to conformal automorphisms of SD. The pair (SD; BD) is

called a (classical) Bely�� pair. The set B�1D [0; 1] � SD is D and in the presence of

the conformal structure will be called a conformally correct dessin.

Note that any pair (R;B), where R is a compact Riemann surface and B : R �! S2

is a Bely�� map, will be a Bely�� pair for the dessin D = B�1[0; 1] � R.

In summary, then, the classical theory takes one from elementary and purely com-binatorial data to a rigid geometric surface and an explicit associated meromorphicfunction. Recall that Bely�� 's Theorem further associates with each Bely�� pair analgebraic number �eld; we will not make direct use of this.

3. Discrete Dessins via Circle Packing

In this section we develop the discrete parallels of the classical objects in the previoussection. First we review the background on circle packings pertinent to our task ofuniformization; the basic references for this material are [4, 7, 10, 14, 24, 28, 31, 33].For background on Riemann surfaces, see [3, 11].

3.1. Circle Packings. Circle packings are con�gurations of circles realizing spec-i�ed patterns of tangency. The key to understanding them lies in recognizing theirdual natures, geometric in the radii of the circles, but combinatoric in their requiredpattern of contact. We �rst discuss geometry on Riemann surfaces, then the encodingof the combinatorics.

Recall that every Riemann surface admits an essentially unique complete metricof constant curvature 0 or �1 that is compatible with its conformal structure in thevery strong sense that isothermal local coordinates determined by the metric cover thesurface with analytic charts from the complete complex atlas of the surface [35]. Thismetric will be termed an intrinsic metric for the surface and is unique up to isometry.One of the most useful properties of an intrinsic metric is that it is conformal, meaningthat the angle between two smooth curves on the surface that is determined by theRiemannian metric agrees with the angle determined by any appropriate chart in thecomplex atlas of the surface.Let � be an intrinsic metric for the Riemann surface R. A metric disk in R of radius

r > 0 and center z 2 R is a topological 2-cell D in R for which

D = fw 2 R : �(z; w) � rg:


The boundary of D, @D = fw 2 R : �(z; w) = rg, is then a topological circle and issaid to be a metric circle in R with center z. The term circle will always refer to ametric circle in the intrinsic metric of the appropriate Riemann surface.All our circle packings will be associated with triangulations. More formally, suppose

K is an abstract simplicial 2-complex that triangulates an orientable topological sur-face; we use the abbreviated term complex. A con�guration P of circles in a Riemannsurface R is a circle packing for K provided (i) P contains a circle cv associated witheach vertex v 2 K(0), (ii) cv is (externally) tangent to cw at a single point whenever vwis an edge of K, and (iii) the geodesic segments connecting the centers of cu, cv, and cwbound a metric triangle in R whenever uvw is a face of K. The geometric 2-complexin R determined by connecting centers of tangent circles in P by geodesic segments iscalled the carrier of P , written carr(P ).In this paper, P will always be univalent, meaning that the circles have mutually

disjoint interiors, and carr(P ) �lls R. In this case, carr(P ) provides a concrete real-ization of the abstract complex K as a geodesic triangulation of the surface R, eachface a metric triangle with geodesic sides. Considerably more striking is the fact thatthe abstract complex K actually determines uniquely the Riemann surface R; this isthe content of the following basic existence-uniqueness result, a proof of which may befound in [4, 33]. We state the result for compact orientable surfaces.

Circle Packing Theorem. Let K be an oriented abstract simplicial complex trian-gulating a compact, connected, oriented topological surface S. Then S admits uniquelythe structure of a Riemann surface R that supports a univalent circle packing P for Krespecting the orientations; moreover, the circle packing P is unique up to conformalautomorphisms of R.

If K and S are as in the Circle Packing Theorem, the Riemann surface R will bedenoted by jKjcp and the packing P , with circles measured in the intrinsic metric ofjKjcp (constant curvature 0, �1), will be denoted PK. We call jKjcp the circle packingsurface determined by K and refer to its conformal structure as a discrete conformalstructure; it is a perfectly legitimate conformal structure, the adjective \discrete"simply acknowledges its source. When S has genus 0, then jKjcp is the Riemannsphere; an additional normalization of PK will typically be necessary, since circle radiiand centers (and hence carr(PK)) are not invariant under conformal automorphisms.

Circle packings exist in great variety, and mappings among them have been shownto display quantitative and qualitative properties associated with analytic functions.Indeed, the proof of the Circle Packing Theorem follows closely the lines of classicalcovering theory and the Riemann Mapping Theorem. We require only parts of thisbroader theory.Let's begin with covering theory. A complex K triangulating a compact orientable

surface S can always be \lifted" to a simply connected complex eK triangulating theuniversal covering surface of S. The Discrete Uniformization Theorem of [4] gives


an essentially unique maximal univalent circle packing eP for eK. If eK is �nite, theneK = K, eP = P packs the Riemann sphere, and we are �nished. Otherwise, eP isan in�nite packing whose carrier is either the euclidean or the hyperbolic plane. If

we denote this plane by D, then eP is unique up to elements of Aut(D), the conformalautomorphisms (M�obius transformations) of D. The simplicial deck transformations of

the abstract covering eK �! K are identi�ed with a discrete group � � Aut(D) whichleaves eP invariant. Therefore, eP=� may be identi�ed with a circle packing P for K inthe Riemann surface R = D=�. Even the radii are una�ected by this identi�cation,since the intrinsic metric of R is precisely that inherited from the metric of D. Whenwe come to examples in our Menagerie, the only practical way to display packings ofsurfaces is to identify them with fundamental domains within their universal coveringpackings.

Next, let us consider maps between circle packings, the identi�cation map eP �!eP=� = P being a prime example. In general, let P and Q be circle packings lying in(possibly distinct) Riemann surfaces. A map f : P �! Q which preserves tangenciesand orientation is termed a discrete analytic function; if Q lies on the Riemannsphere, f is called a discrete meromorphic function, and we will see examples of

these shortly. The projection � : eP �! eP=� is a discrete covering map. The mapf clearly induces a simplicial map of carr(P ) to carr(Q). We will abuse notation anduse f to denote both the set map f : P �! Q and the associated point mappingf : carr(P ) �! carr(Q) de�ned to carry the center of each circle c 2 P to the center off(c) 2 Q and then extended via barycentric coordinates. The function f is basicallya simplicial map which acquires the geometry imposed by the circle packings of itsdomain and range.

Remark. Natural barycentric coordinates for geodesic triangles exist in each of ourclassical geometries; in the hyperbolic setting, they are most easily described in thehyperboloid model, where geodesics are the intersections of planes through the originwith the hyperboloid sheet (see [35, 27]). In particular, as a point mapping, f is acontinuous, orientation preserving, light interior mapping of carr(P ). It maps eachedge of carr(P ) \convexly" onto an edge of carr(Q) in the metric sense. We will seelater that lower bounds on the angles of the faces in carr(P ) and carr(Q) also givebounds on the quasiconformal dilitation of f on the interiors of the faces.

Discrete analytic functions can fail to be locally one-to-one, even between univalentpackings. Consider f : P �! Q with P and Q univalent. Let c0 2 P ; the ower of c0consists of c0 as the center and the sequence fc1; c2; � � � ; ckg of petal circles. The petalsare tangent to c0, successively tangent to one another, and wrap around c0 once in thepositive direction. The image circles ff(c1); f(c2); � � � ; f(ck)g in Q will necessarily bepetals of the ower of f(c0). By orientation they must wrap in the positive directionabout f(c0); if they wrap n � 2 times around f(c0), we say that f has a (discrete)branch point at c0 of order n�1. For instance, if 12 \petals" around c0 are mapped


to four about f(c0), the result is a branch point of order 2, with each petal of f(c0)the image of three petals of c0.Those curious about the practical side of circle packing should see the Appendix,

which brie y discusses numerical algorithms, implementation, and software. For fur-ther connections between circle packing and analytic functions, see [4, 10, 28] andreferences therein.

3.2. Discrete Dessins. Suppose that a �xed dessin (S;D) is given. We are now inposition to create its associated discrete Bely�� pair.Recall that the canonical triangulation T = T (D) may not be simplicial. For this

and several other practical reasons we perform a barycentric subdivision of T anddenote the resulting complex by K = K(D).

De�nition. Given a dessin (S;D), the Circle Packing Theorem implies existence of

the Riemann surface jK(D)jcp, homeomorphic to S, supporting a circle packing PKfor K. We denote this Riemann surface by sD and refer to it as the coarse circle

packing surface for D and refer to the packing PK(D) as the coarse circle packing

PD.

Observe that T is represented in K: its 0-, 1-, and1-vertices are among the verticesof K; each edge of T is a chain of edges (at this stage, two) of K; and each \dessin"face is now a union of faces (six) of K. Thus sD constitutes the \surface" part of adiscrete Bely�� pair and provides a geometry for T . When sD is the Riemann sphere,a normalization of PD is needed. For this purpose, we will designate 0-, 1-, and 1-vertices v0; v1; v1 of T and apply an automorphism to ensure that these points ofcarr(PD) are located at 0; 1, and 1, respectively, on S2.For the meromorphic function part we must �rst describe the standard coarse circle

packing Q for the Riemann sphere: Choose three points on a topological circle on atopological sphere; this breaks the circle into three segments and de�nes a triangulationT of the sphere, the two faces sharing a common boundary. Its barycentric subdivision�T will be denoted by H = H0 and de�nes an abstract simplicial triangulation of thetopological sphere. The Circle Packing Theorem guarantees a packing Q for H on theRiemann sphere S2. This Q is shown on the right in Figure 3, with a normalizationplacing the vertices of T at 0; 1;1 and with shading of the triangle of T (six faces ofH) forming the lower half-plane. (Our conventional orientation of S2 in R3 means thatthe \lower" half-plane is the left hemisphere.)We are now in position to de�ne a discrete meromorphic function bD : PD �! Q.

The vertices of T have been decomposed into sets V0, V1, and V1. Identify each circleassociated with a 0-vertex to the circle of Q centered at 0, each 1-vertex to the circleof Q centered at 1, and each 1-vertex to the circle of Q centered at 1. Tangenciesand orientation clearly dictate that the seven circles of PD de�ning any (+)triangle ofT are carried to the seven circles of Q forming the unshaded upper half-plane, whilethe seven forming any (�)triangle are carried to the seven forming the shaded lowerhalf-plane.


De�nition. Given a dessin (S;D), the associated coarse discrete Bely�� pair isthe pair (sD; bD) consisting of the coarse circle packing surface for D and the discrete

meromorphic function bD : SD �! S2.

The coarse pair for the simple dessin of Figure 1 is illustrated in Figure 3. The surfacesD is just S2 in this genus 0 case, but the locations of its vertices are determined bythe packing PD, and these de�ne the dessin itself, shown with the thick line.

S2

(a) (b)

bD

Figure 3. A coarse discrete Bely�� pair

All nontrivial discrete Bely�� maps will have branch points. For instance, Figure 3 isassociated with the dessin of Figure 1; it has an 1-vertex which belongs to 6 faces ofT = T (D), alternately (+) and (�)triangles. These six are mapped by bD to the twotriangles forming the upper and lower half-planes, respectively, in Q; consequently, this1-vertex is a branch point of order two for bD. This behavior, mimicking the classicalsituation, is typical: a dessin vertex v which belongs to 2n faces of T engenders abranch point of order n� 1 in the Bely�� map. Moreover, one can easily check that bDdoes not branch at vertices of K which are not dessin vertices (i.e., vertices of T ). Ina clean dessin, all the 1-vertices are simple branch points (order 1).It is important to recognize that the discrete Bely�� pair (sD; bD) and the classical

Bely�� pair (SD; BD) are qualitatively indistinguishable. The surfaces sD and SD arehomeomorphic to one another, since each is homeomorphic to the topological surfaceS. In fact there exist homeomorphisms � : S �! sD and � : S �! SD which respectthe embedded T 's. The maps sD � � : S �! S2 and SD � � : S �! S2 are open,continuous, orientation preserving maps sharing valence and branch structures. In


fact, sD � � and SD � � are isotopic maps. Topologically and combinatorially, thesepairs can't be distinguished.

3.3. Hexagonal Re�nement. The discrete objects created so far are termed \coarse"only because they involve so few circles for carrying the geometric information. Thereis a rule-of-thumb in circle packing: the �ner the circle packing, the closer its geometricbehavior to classical conformal behavior. So we need packings with more numerousand smaller circles if we hope to approach continuous behavior. We deploy a processcalled hexagonal-re�nement.We use � to indicate the hex subdivision operator which, when applied to a

complex K triangulating a surface S, yields the complex �K also triangulating S. Itse�ect is illustrated in Figure 4: one �rst adds a vertex to the middle of each edge ofK and then adds three edges in each face of K to connect pairs of the new vertices.Thus, each triangle (face) of K is broken into four triangles in �K. The vertices ofK remain, with their degrees unchanged; each new vertex, being in the edge of twofaces, has degree six, hence the adjective \hexagonal". Repeated applications will beindicated with powers, �nK; three stages of hex re�nement of a triangle are shown inthe �gure.

original stage-1 stage-2 stage-3

Figure 4. The hexagonal subdivision operation

In the case of a dessin (S;D), we have the complex K = �T triangulating S, thecomplex H triangulating the sphere, and the coarse discrete Bely�� map bD : PD �! Qbetween their packings. We simply hex re�ne K and H, obtain circle packings for there�nements, and arrive at a \re�ned" map between them.In particular, by the Circle Packing Theorem, there is a Riemann surface j�Kjcp

homeomorphic to S which supports a circle packing for �K; denote the Riemann

surface by s(1)

D and the packing by P(1)

D . Likewise, there is a circle packing on S2 for the

re�nement �H, to be denoted Q(1) and normalized to keep the original three verticesat 0; 1;1, respectively. It is elementary to see that bD, as a simplicial map between K


and H, induces a simplicial map between �K and �H; in fact, on the vertex set, the

new map extends the original. This de�nes the circle packing map from P(1)

D to Q(1)

which we denote by b(1)

D . We therefore arrive at a new discrete Bely�� pair (s(1)

D ; b(1)

D ).We may repeat this re�nement process any �nite number of times.

De�nition. Given a dessin (S;D) and n � 0, the stage-n discrete Bely�� pair

associated with (S;D) is the pair (s(n)D ; b

(n)D ) obtained from the coarse pair (sD; bD) by n

successive stages of hexagonal re�nement and repacking. The associated stage-n circle

packing of s(n)D is P

(n)D , the stage-n packing of the sphere is Q(n), so b

(n)D : P

(n)D �! Q(n).

It will be a standing assumption in the genus 0 case that P(n)D has been normalized

as follows: for each i 2 f0; 1;1g an i-vertex vi has been designated; a M�obius trans-

formation has been applied to the sphere so that vertex vi of carr(P(n)D ) is located at

i, for i = 0; 1;1.A stage-n discrete Bely�� pair will again be qualitatively indistinguishable from the

other stages and from the classical Bely�� pair. In particular, the branch points of b(n)D

occur at dessin vertices (the original vertices inherited from T ) and branch orders are

preserved under re�nement (because degrees of vertices are preserved). The b(n)D have

identical branch structures, and, as maps from S to S2, will be isotopic to one another.One would expect the pairs to be distinct, however, in their quantitative structure. A

packing P(n+1)

D is not a direct \re�nement" of P(n)D (no one has discovered a good way

to re�ne circle packings as one could, say, squares), only the combinatorics are beingre�ned. In fact, there is, in the absense of special symmetries, no reason to expect

the Riemann surfaces s(n+1)

D and s(n)D to be conformally equivalent; in other words, the

geometries are di�erent.Figure 2 in the Introduction is the stage-3 discrete Bely�� pair associated with the

coarse pair of Figure 1. It is intuitively clear that �ner packings will carry moregeometric information than their coarser predecessors. We make this very precise inSection 4.

3.4. Geometric Lemmas. The bounded geometry associated with circle packings iscritical to our uniformization work. It is a standard feature of the theory dating fromthe seminal work of Rodin and Sullivan. In the following, (S;D) is our dessin, as usual;however, the results apply equally well to the more general triangulations of the nextsection. Here, d = maxf6; deg(K)g = deg(Kn); n � 1.The bounded geometry enters primarily through the Ring Lemma [28].

Lemma 3.1. There exists a constant � = �(d) > 0, depending only on d, so that the

following holds: If S has positive genus, then the angles of the faces of carr(P(n)D ) are

bounded below by � for all n � 0. If S has genus 0, then these angles are boundedbelow by � for n su�ciently large.

Proof. In the euclidean case, the Ring Lemma tells us that in a univalent packing,the ratio of the radii of two neighboring (interior) circles is bounded by a constant


depending only on degree. This easily converts to a lower bound on angles of facesin the carrier, since they are formed by triples of neighboring circles. The argumentsextend easily to cover the hyperbolic setting.On the sphere, however, the Ring Lemma fails; the ratio of radii for neighbors may

be made arbitarily large by applying a suitable M�obius transformation to the packing.Our normalization becomes crucial: we show in Lemma 3.4 below that for su�cientlylarge n, the spherical radii of circles of PKn

will be bounded by �=6. In particular, any ower of circles will lie in a hemisphere. Stereographic projection of that hemisphereto the plane, application of the euclidean Ring Lemma, and some elementary geometry

shows that the angles of faces of carr(P(n)D ) will be bounded below by a positive constant

depending only on d.

To convert the bounded geometry of packings to the distortion of associated maps,we need this geometric lemma, whose proof is standard. References to the literatureon quasiconformal maps will be given in the next section.

Lemma 3.2. Let T and T 0 be geodesic triangles, each lying in the sphere, the eu-clidean, or the hyperbolic plane. Assume � > 0 is a lower bound for the angles of Tand for the angles of T 0, measured in their respective geometries. Then the barycentric

map g : T �! T 0 is �-quasiconformal for a � � 1 depending only on �.

Recall that as point mappings, the discrete Bely�� map b(n)D carries faces of carr(P

(n)D )

barycentrically onto faces of carr(Q(n)). By Lemmas 3.1 and 3.2, the restriction of

b(n)D is �-quasiconformal on open faces. By removability of analytic arcs and isolated

singularities, b(n)D is �-quasiregular. In conclusion, we have the following:

Proposition 3.3. There exists a constant � = �(d) > 1 depending only on d so that

the following holds: If S has positive genus, then b(n)D : s

(n)D �! S2 is �-quasiregular for

all n � 0. If S has genus zero, then b(n)D is �-quasiregular for all su�ciently large n.

Our last result of the section relies on the Rodin and Sullivan Length-Area Lemmaalong with the Ring Lemma. It con�rms one's intuition that the radii of circles inre�ned packings should go to zero. We actually require only the genus 0 case in thispaper (for use in the proof of Lemma 3.1 above). However, the idea of the proof easilygeneralizes and may be useful in other circumstances.When S has genus 0, we require the standard normalization of packings noted earlier.

When S has genus 1, the euclidean metric of s(n)D is determined only up to a scalar,

so we may assume that diam(s(n)D ) = 1. No normalization is needed when S has genus

greater than one.

Lemma 3.4. Let rn be the maximum radius among the circles of the normalized pack-

ing P(n)D . Then rn �! 0 as n!1.

Proof. We begin on the sphere. Everything follows from very local considerations, so�x a vertex v of T . Let denote the simple closed edge-path in K running through the


neighbors of v. In K, this path separates v from at least two of the three designatedvertices v0; v1; v1 used in the normalization of the packings. Without loss of generality,assume separates v from v1 and v1.As K is hex re�ned, the new complexes Kn have additional chains of vertices layered

between v and (which is also re�ned); in a sense, this region combinatorially \fattens"

as n grows. For each n let pn denote the collection of circles from P(n)D corresponding to

vertices of Kn separated from 1 and 1 by and let p0n be the same collection with thecircle cv;n for v removed. Observe that the combinatorics of p0n depend only on d andn; it is a combinatorial annulus of particularly simple type which we describe below.If An � S2 denotes the annulus bounded by @cv;n and , then p0n � An. Below wewill use p0n to establish a lower bound Mn on the modulus of An which goes to in�ntiywith n. In other words, cv;n is seen to be separated from 1 and 1 on S2 by an annuluswhose modulus goes to 1 with n. We may conclude by standard arguments aboutextremal annuli that the spherical radius of cv;n must go to zero, with rate governedby d. This will conclude the genus 0 case.Since we don't have the Ring Lemma on the sphere, we project p0n to the plane.

Observe that the circles of p0n break naturally into n disjoint chains of circles. Letthe corresponding edge-paths be denoted 1; 2; � � � ; n, ordered so that j separatesv from j+1. Let l1; l2; � � � ; ln+1 denote the combinatorial lengths of these paths. 1is just the edge-path through the neighbors of v in Kn, so l1 = deg(v;K) � d. Weobtain easy bounds on the successive combinatorial lengths because these vertices, allresulting from hex re�nement, are of degree six. In particular, one can show thatlj � j; j = 1; � � � ; n. That allows us to write down the quantity pertinent to theLength-Area Lemma, ([28, p. 353]); namely de�ne kn = kn(d) by

kn(d) = [1

d(1 +

1

2+1

3+ � � �+ 1

n)]�1=2 � [

1

l1+

1

l2+ � � �+ 1

ln]�1=2:

Since the harmonic series diverges, kn goes to zero as n grows.The argument of the Length-Area Lemma is only indirectly about moduli of ring

domains; we need to do some adjustment. First, stereographically project pn (whichavoids1) to a packing qn of the plane. Repack qn as a \maximal" or Andreev packingbqn in D with the circle c corresponding to v at the origin (see [4]). The chains of circlesassociated with the j in pn now correspond to chains of circles separating c from theboundary of the unit disc. By the argument in the proof of the Length-Area Lemma,radius(c) � kn. As n grows, c gets smaller and with a little help from the (euclidean)Ring Lemma one can show carr(bq 0n) nearly �lls the annulus D nc. Thus for su�cientlylarge n we have

log(1=kn) � 2Mod(carr(bq 0n)):The Ring Lemma and Lemma 3.2 above imply

Mod(carr(bq 0n)) � �Mod(carr(q0n)):


Under stereographic projection to S2, this latter modulus does not change; the imageof carr(q0n) is a subset of An, so �nally,

Mod(carr(q0n)) � Mod(An):

Set Mn = 2�log(1=kn). This depends only on d, Mn ! 1 as n ! 1, and Mn �

Mod(An), as desired. With this, we are done with the spherical case.

The positive genus cases are minor variations. In genus 1, the packing pn may beassumed, with an appropriate lift under the covering map, to lie in C with cv;n atthe origin and lying (by our diameter normalization) within D . The argument of theLength-Area Lemma directly implies that radius(cv;n) goes to zero. When genus isgreater than one, pn lifts to D and we can put cv;n at the origin. Since p0n separatescv;n from the unit circle, the euclidean radius of cv;n goes to zero as before; since itis centered at the origin, the same applies to its hyperbolic radius. The bounds onlydepend on d, so we are done.

Ever �ner control of the geometry emerges under re�nement, as we will see in thenext section when we apply Rodin and Sullivan's Hexagonal Packing Lemma.

4. Uniformizing Dessins

In this, the central theoretical section of the paper, we connect the classical and discreteBely�� pairs. Our work will apply to general triangulations. In the classical settingwe introduce the notion of \re ective surfaces" and their re�nement by \conformalsubdivision". We then show that it is a small step from a re�ned classical structure toa re�ned circle packing structure.

4.1. Re ective Structures and Conformal Subdivisions. The \equilateral" dessinsurfaces are examples of what we will refer to as re ective triangulations. We workonly with triangulations in this paper, but since similiar ideas have been used recentlyin other settings, we o�er a more general de�nition.

De�nition. A regular cell decomposition C of a Riemann surface R is re ective if,for every edge e of C in the boundary of two closed faces (2-cells) f�, there existsa conformal re ection of f+ [ f� across e; explicitly, there exists an idempotent

homeomorphism re that is anticonformal on int(f+ [ f�), re(f�) = f�, the restrictionof re to e is the identity on e, and re exchanges the vertices of f� that do not lie onthe edge e. In particular, all cells have the same number of vertices. We say that Cendows R with a re ective structure.

In general, the decomposition of a topological surface S into topological n-gons im-plies existence of a unique conformal structure in which the decomposition is re ectiveand the cells are conformally regular n-gons, as in x 2.2. In [6], the plane was endowedwith a re ective pentagulation, a refective structure based on pentagonal cells, andwork on cubulated manifolds suggests uses for re ective squarings.


Remark. The conformal re ection re across the edge e is not globally de�ned and,indeed, generically will not extend anticonformally beyond its domain of de�nitionf+ [ f�. Even in a covering surface, the re ection does not necessarily extend, asis required, e.g., in Wolfart's \mirror-invariant" triangulations in [36]. Though re isnecessarily angle-preserving at the vertices of f� forming the endpoints of e, it generallywill not preserve angles at the remaining vertices of f+ and f�. Notice that if v is avertex of C of degree d, then the angle between consecutive edges of C incident at v asdetermined by the complex structure on R is precisely 2�

d.

Example. Let (S;D) be a dessin with classical Bely�� map BD : SD �! S2. Recall ournotation that SD = ST , where T = T (D) is the canonical triangulation for D. ThenT is a re ective triangulation of SD: indeed, for any edge e, let Be be the restriction ofBD to int(f+[f�), where f� are the faces of T containing e. Then re is the continuousextension to f+ [ f� of B�1

e � C �Be, where C denotes complex conjugation.

Proposition 4.1. Let C and C 0 be combinatorially isomorphic, re ective cell decompo-sitions of the respective Riemann surfaces R and R0. Then any (orientation-preserving)combinatorial isomorphism of C with C 0 may be realized as a conformal isomorphismof R with R0 that takes C to C 0.The elementary proof, based on the Schwarz Re ection Principle and removability

of isolated singularities, is left to the reader. We state a useful corollary:

Corollary 4.2. Let T be a re ective triangulation of the Riemann surface R. Then

RT , the Riemann surface determined by an equilateral metric structured on T , is con-formally equivalent to R.

A triangulation T of an oriented topological surface not only imposes a conformalstructure on the surface, but simultaneously realizes itself as a re ective triangulationin situ on the resulting Riemann surface. In our method of uniformizing equilateralsurfaces we rely crucially on the fact that T admits arbitrarily �ne subdivisions, whatwe call re�nements, which themselves form re ective triangulations of this very sameRiemann surface.A subdivision of the triangulation T of the topological surface S is a triangulation

T 0 of S such that each cell (vertex, edge, or triangle) of T 0 is contained in some cell ofT . We are interested in subdivisions of a very special type.

De�nition. If T 0 subdivides the triangulation T of the oriented topological surface S,

then T 0 is a conformal subdivision of T if there is a subdivision of the re ectivetriangulation T of the Riemann surface ST that is itself re ective in ST and combina-torially isomorphic to T 0.

Lemma 4.3. If T 0 is a conformal subdivision of T , then the associated equilateralsurfaces ST and ST 0 are conformally equivalent. Further, if T 0 and T are realizedsimultaneously as re ective triangulations of this Riemann surface with T 0 subdividingT , and if e is an edge of T 0 contained in the edge E of T , then the conformal re ection


re is the restriction of the conformal re ection rE to the two faces of T 0 contiguousalong e.

The �rst assertion of the lemma is important in uniformization, since it will allowus to replace the Riemann surface ST by ST 0 , realized as the equilateral surface jT 0jeq.However, \conformality" involves an equally important but separate issue: T 0 and Tmust be realizable simultaneously as re ective triangulations in that common Riemannsurface, with T 0 subdividing T in situ. As a cautionary example to keep in mind, everytriangulation T of a topological 2-sphere induces the same conformal structure, namely,that of the Riemann sphere S2. However, for a generic subdivision T 0 of T , the in situtriangulations T 0 and T will be incompatible.

Proof of 4.3. Since T 0 is a conformal subdivision of T , there is a subdivision of there ective triangulation T of the Riemann surface ST that is itself re ective and com-binatorially isomorphic with T 0. An application of Proposition 4.1 guarentees that theRiemann surfaces ST and ST 0 are conformally equivalent, and the �rst assertion of thelemma follows. The second assertion follows from the fact that conformal re ectionsare uniquely determined in a neighborhood of their �xed point sets by those very �xedpoint sets.

We are interested primarily in barycentric subdivision and the hexagonal re�ne-ments described earlier. These easily are seen to be conformal subdivisions, but forcompleteness we prove the next result, which provides many additional examples.

Proposition 4.4. Let � be an abstract triangulation of a triangle t that has combina-torial dihedral symmetry. Let T 0 be a subdivision of T such that the restriction of T 0 toeach face of T is combinatorially equivalent to � . Then T 0 is a conformal subdivisionof T .

Proof. Denote the vertices of t as u0, u1, u2 and recall that j� jeq denotes t equipped withan equilateral metric structured on � . Identifying t with a euclidean unit equilateraltriangle 4, there is a unique homeomorphism � : j� jeq ! 4 which �xes u0; u1; u2 andis conformal on the interior of j� jeq (recall that the interior of j� jeq inherits a conformalstructure compatible with its equilateral metric via charts associated with pairs ofcontiguous equilateral faces, as in x 2.2). By replacing the abstract triangulation � oft with the image triangulation �(�), we may assume, without loss of generality, that� is a re ective triangulation of the equilateral triangle 4 that is invariant under theaction of the dihedral isometry group D2�3 of 4. This means, of course, that each edgee of � that does not lie in the boundary @4 determines a conformal re ection re acrosse of its two contiguous faces.In the Riemann surface ST , realized concretely as the equilateral surface jT jeq, let T�

denote the triangulation formed by isometrically mapping the re ective triangulation� of 4 onto each face of T (each a euclidean unit equilateral triangle). The fact that� is D2�3-invariant guarantees that this gives a well-de�ned triangulation. We claimthat T� is a re ective triangulation of ST . Indeed, every edge e of T� whose interior


lies in the interior of a face of T admits a conformal re ection re of its contiguousfaces by construction. Further, if e is an edge of T� contained in an edge of T , thenthe facts that � is D2�3-invariant and T itself is re ective in ST imply the existenceof a conformal re ection re. In fact, if the edge e of T� is contained in the edge Eof T , then, because of the dihedral symmetry across the altitudes of the equilateraltriangles, we have re = rEj(f+ [ f�), where f� are the faces of T� contiguous along e.Thus T� is a re ective triangulation of ST that subdivides T and, hence, T 0, which iscombinatorially equivalent to T� , is a conformal subdivision of T .De�nition. If � and T are as in the proposition, �T denotes a subdivision (or re-�nement) obtained by subdividing each face of T according to the combinatorics of� .

α β γ δ

Figure 5. Conformal subdivision rules

Figure 5 illustrates four D2�3-invariant triangulations of 4, denoted �; �; , and �.The �rst three are conformally correct as shown, since the re ective edges are straighteuclidean line segments. The case � is more generic and its conformally correct formis shown in Figure 6; its re ective edges are some unknown curves, neither euclideanline segments nor euclidean circular arcs. (This image of Figure 6 was obtained byapplying essentially the same methods developed below.)The uniformization algorithm we develop only requires a sequence of re�nements

whose meshes go to zero, so it will su�ce to work with just the hexagonal and barycen-tric rules, � and �, respectively. We remind the reader of a trait of hex re�nementthat is of crucial importance; namely, hex re�nement does not raise degree at theoriginal vertices, whereas the others in Figure 5 either double or triple those degrees.We therefore rely on re�nement sequences of the form f�n�T g, signifying n successivehex re�nements of an initial barycentric subdivsion of T . The reader will recognizethese re�nements from Section 3. The notation and terminology of re�nements applyequally well, of course, to simplicial complexes which triangulate surfaces.


Figure 6. The re ective subdivision �(�)

4.2. Uniformizing Equilateral Surfaces. This section establishes the theoreticaltools for uniformizing equilateral surfaces, in particular, dessin surfaces. We use manystandard results regarding Riemann surfaces and quasiconformal mappings; good ref-erences are [19, 11, 20].Let T be a triangulation of the compact, connected, oriented topological surface

S. Our task is to uniformize the re ective surface ST determined by the equilateralsurface jT jeq. For practical reasons we work with the barycentric subdivision �T .Since barycentric subdivision is conformal, the equilateral surface j�T jeq is in thesame conformal class as ST . By identifying ST as the equilateral surface j�T jeq andsubdividing each unit equilateral face of j�T jeq into equilateral triangles of side length"(n) = 1

2nvia n successive hex re�nements, we realize ST as jTnj"(n)eq , where T0 = �T

and Tn+1 = �Tn for n � 0. Thus, fTng is a sequence of re ective triangulations of STinto triangles, equilateral as seen in the piecewise euclidean metric of jT0jeq, which arere�nements of one another in situ, and whose meshes approach zero in the metric ofjT0jeq as n!1.Note that, unlike the intrinsic metric, the piecewise euclidean metric on the Riemann

surface ST determined by the triangulation T0 generally is not conformal. Though itis compatible with the complex structure of ST in the sense that it provides analyticcharts for ST that cover all but the �nitely many vertices of T0, and it is conformal onthe complement of these vertices, nonetheless, it fails to be conformal at the non-sixdegree vertices of T0. Indeed, though the measure of every angle in every face of T0is �

3when measured in the euclidean geometry of the equilateral metric, the measure

that the Riemann surface ST ascribes to a particular angle is 2�d, where d is the degree

in T0 of the associated vertex.


To aid in the readability of what follows, we continue to use abstract simplicialcomplexes for the combinatorics on the discrete side of things, as in x3. Thus K = K0

is the simplicial 2-complex which is combinatorially isomorphic to T0 = �T , and foreach n � 1, Kn = �nK is obtained from K by n successive hex re�nements and iscombinatorially isomorphic to Tn.

We are now in position to link the classical and discrete objects. Fix n � 0. Onthe classical side we have the Riemann surface ST with its re ective triangulation Tn;on the discrete side, the Riemann surface jKnjcp determined by the circle packing PKn

.Since Kn and Tn have been canonically identi�ed, we may consider a vertex v of Tnsimultaneously as a vertex of Kn. We now de�ne the key homeomorphisms hn fromST onto jKnjcp.De�nition. Let hn to be a simplicial homeomorphism

hn : ST �! jKnjcp; n = 0; 1; � � �from the re ective triangulation Tn of ST to the carrier carr(PKn

) in jKnjcp that takes avertex v in Tn to the center zv of the corresponding circle cv in jKnjcp and then extendsvia barycentric coordinates to map each equilateral triangle uvw in Tn homeomorphi-cally onto the geodesic triangle zuzvzw in jKnjcp.(When we face the situation in which jKnjcp is the sphere, we will have to impose anormalization on PKn

to ensure that hn is unambiguous.)Our approximation results are nicely formulated in the language of Teichm�uller space

when S has positive genus. Recall that the Teichm�uller space Teich(S) of the surface Sconsists of equivalence classes [g] of homeomorphisms g from S onto Riemann surfacesSg, where the homeomorphisms g : S �! Sg and h : S �! Sh are equivalent wheneverthe homeomorphism h � g�1 is homotopic to a conformal isomorphism of Sg onto Sh.The Teichm�uller distance between points [g] and [h] of Teich(S) is

�([g]; [h]) = 12log inf �(q);

where �(q) is the (global quasiconformal) dilatation of q, and q ranges over all qua-siconformal mappings in the homotopy class of h � g�1. This Teichm�uller metric �is complete, convex, externally convex, and Teich(S) with the �-metric topology ishomeomorphic to a �nite-dimensional euclidean space | to R2 if S is a genus 1 sur-face, and to R6m�6 if S is a genus m � 2 surface. These and other properties of theTeichm�uller metric may be found in [5, 18, 21, 20, 23, 25].Now the Riemann surface ST is exactly the topological surface S with a maximal

complex atlas determined by pairs of contiguous faces from the triangulation T realizedas euclidean equilateral triangles. Thus the identity mapping �S : S ! ST determinesthe point [�S] of Teich(S). Further, the homeomorphisms hn, by forgetting the complexstructure on ST , are mappings de�ned on S and so provide points [hn] of Teich(S).We will show that the homeomorphisms hn are quasiconformal maps with uniformlybounded dilatations; even though those dilatations are bounded away from unity, we


have the following result, which shows convergence in moduli of the circle packingsurfaces jKnjcp to the surface ST . In many arguments to follow there is a �nite setV � of points of ST which must be treated di�erently; namely, V � denotes the set ofvertices of T0 that have degree not equal to six.

Theorem 4.5. If S has nonzero genus, then in the Teichm�uller space Teich(S),

limn!1

[hn] = [�S]:

The pointwise dilatations of the maps hn are bounded above and converge to unityuniformly on compact subsets of ST nV �.

Proof. We have seen in the results of x3.4 that circle packings induce geometry in away that controls distortion in a quasiconformal sense. We use this in conjunctionwith the fact that the Teichm�uller metric � is proper, meaning that closed �-boundedsubsets of Teich(S) are compact. This follows, for instance, from the facts that �is complete and convex and Teich(S), being homeomorphic to a euclidean space, islocally compact.By Lemmas 3.1 and 3.2, the restrictions of the homeomorphism hn to the interiors

of the equilateral faces of ST are �-quasiconformal, for some � depending only onthe degree of T0. By removability of analytic arcs and isolated singularities [19], eachhomeomorphism hn : ST �! jKnjcp is therefore �-quasiconformal. It follows that thesequence [hn] in Teich(S) is contained in the closed �-ball of radius 1

2log� about the

point [�S ].(The reader might hope that the global quasiconformal dilatations of the hn con-

verge to unity as n ! 1; however, this generally is not the case. Examine the anglechange under hn at a vertex an 2 Tn adjacent a vertex v 2 V �. Let fn be a face ofTn with v and an as vertices. Since Tn is re ective and fn is one of exactly six facesthat share the vertex an, the angle of fn at an has measure precisely �

3in ST , for all

n � 1. On the other hand, the corresponding face f 0n of carr(PKn) is asymptotically

a euclidean isosceles triangle having angle 2�dat v, meaning that its angle at an ap-

proaches (d�2)�=2d, where d = deg(v; T0) = deg(v;Kn). When d 6= 6, this value is not�3. Fortunately, such unwanted distortion is restricted to ever-shrinking neighborhoods

of the vertices of V �.)We shall argue that every convergent subsequence of the sequence [hn] converges to

[�S]. This and the fact that � is proper then imply that every subsequence of [hn] has[�S] as a limit point, which in turn implies that the original sequence [hn] converges to[�S]. Let n(1); n(2); : : : be a strictly increasing sequence of positive integers for whichthe sequence [hn(i)] converges to the point [h : S ! Sh] of Teich(S). Then there is asequence of quasiconformal homeomorphisms

qi : jKn(i)jcp �! Sh


with qi homotopic to h � h�1n(i)

and whose global dilatations �(qi) converge to unity.

We have the following diagram of homeomorphisms, with the upper triangle triviallycommuting and the lower triangle commuting up to homotopy.

ST

��hn(i)

E DB C 'ioo

S

77�S

oooooooooooooooo //hn(i)

''h OOO

OOOOOO

OOOOOO

O jKn(i)jcp

��qi

Sh

Our aim is to show that the point [h] of Teich(S) is equal to [�S], which will beaccomplished by showing that a subsequence of the sequence of quasiconformal home-omorphisms 'i = qi�hn(i) : ST �! Sh converges uniformly to a conformal isomorphism

of ST onto Sh that is homotopic to h � ��1S . Since the maps hn(i) are �-quasiconformaland the maps qi have dilatations �(qi) converging to unity, all but possibly �nitely manyof the maps 'i are 2�-quasiconformal. Standard arguments in [19] about covergence ofsequences of quasiconformal mappings of plane domains applied in the universal cover-ing surfaces of the surfaces ST and Sh imply that there is a subsequence of the sequence'i that converges uniformly to a 2�-quasiconformal homeomorphism ' : ST �! Sh.We frame this fact as a separate lemma, which appears, along with its proof, at the endof this proof. Since each map 'i is homotopic to the �xed map h � ��1S , it follows thatthe limit mapping ' is homotopic to h� ��1S . Modulo the veri�cation of the lemma, ourproof that [h] is equal to [�S ] is complete once we show that this limit quasiconformalhomeomorphism ' is in fact conformal.Let E be any compact subset of ST missing the vertices V �. Our aim is to show that

the dilatation of hn on E goes uniformly to unity as n grows. Let N be an arbitrarypositive integer and choose n so large that each point z of E is centered in a simplyconnected neighborhood Uz formed by 2N generations of the hexagonal grid withinthe re ective triangulation Tn and missing V �.The set Uz is conformally equivalent to the carrier of 2N generations of a regular

hexagonal circle packing, as illustrated in Figure 7(a) for N = 4. (In fact, in theequilateral structure j�T jeq of ST , Uz and this carrier are isometric if the circles aregiven radius 1

2n+1 .) Write pz;n for the circle packing within PKncorresponding to Uz.

Let us suppose �rst that S has genus one. Lift pz;n under the covering map ofjKnjcp to a circle packing qz;n in C . The restriction of hn to Uz, with image liftedto C , is precisely the circle packing map from 2N generations of a regular hexagonalcircle packing to qz;n, as suggested in Figure 7. By Rodin and Sullivan's HexagonalPacking Lemma, [28], the maximum pointwise dilatation of hn in a neighborhood of


z is bounded by a quantity depending only on N which goes to unity as N ! 1.Therefore, the maximum of the pointwise dilatations of the restriction of hn to E

converges to unity, as desired.

(a) (b)

hnUz

qz,n

Figure 7. hn restricted to 2N hexagonal generations

If S has genus greater than one, our lifted packings qz;n lie in D . However, euclideangeometry is the small scale limit of hyperbolic geometry. By Lemma 3.4, the hyperbolicradii of the circles of qz;n are going uniformly to zero as n grows, so the arguments of theprevious paragraph still apply and, again, the pointwise dilatations of the restrictionof hn to E converges to unity.We have now veri�ed the second part of the theorem. With this it is easy to see

that any limit mapping ' of a subsequence of the maps 'i is conformal. Indeed, sincethe maximum dilatations �(qi) converge to unity, the pointwise dilatations of the maps'i = qi�hn(i) converge to unity uniformly on compact subsets of ST nV �. By restrictingto simply connected compact subsets of ST nV � and working in the universal coveringsurface of Sh, we may invoke [19, Theorem II.5.3] to conclude that the restriction ofthe limit mapping ' to any such compact subset is (1 + �)-quasiconformal for every� > 0. It follows that ' is conformal on ST nV � and, by removability of isolatedsingularities, we conclude that ' is a conformal homeomorphism. This completes theproof of Theorem 4.5, modulo veri�cation of the next lemma.

Lemma 4.6. Suppose R and R0 are compact, positive genus Riemann surfaces and'i is a sequence of �-quasiconformal homeomorphisms of R onto R0. Then there is asubsequence of the 'i's that converges uniformly on R to a �-quasiconformal homeo-morphism of R onto R0.


Proof. Let

� : D �! R and �0 : D �! R0

be the universal covering projections, where D denotes appropriately either C with itsusual metric or D with the Poincar�e metric. The groups � and �0 of covering trans-formations of the respective coverings � and �0 are groups of M�obius transformationsacting properly discontinuously and by isometries on D. Let F 0 be the Dirichlet re-gion for �0 centered at the origin and, for each i, let zi be an element of F 0 for which�0(zi) = 'i(�(0)). Let �i : D �! D be the lift of the map 'i � � that sends 0 to zi,which satis�es �0 � �i = 'i � �.

D //�i

��

D

��0

R //'i R0

Since the 'i's are �-quasiconformal, so too are their lifts �i, to which we may thenapply the convergence theorems of [19, xII.5]. Our claim is that these convergenceresults imply the existence of a �-quasiconformal mapping � : D �! D that is thelimit of a subsequence of the �i's whose convergence is uniform on compact subsets ofD. It is a straightforward exercise using the discontinuity of the actions of � and �0 onD to show that any such limit mapping � takes �bers of � to �bers of �0 and thereforeinduces a �-quasiconformal homeomorphism ' = �0 ��1 that is the limit mappingof a subsequence of the 'i's. Moreover, the convergence of the subsequence of the 'i'sto ' is uniform since the convergence of the subsequence of the �i's to � is uniform oncompact subsets of D, in particular, uniform on a compact fundamental domain for �.The veri�cation of the claimed existence of � and its nondegeneracy involves lookingseparately at two cases, genus one and genus greater than one.In the genus one, or parabolic, case, D = C . Letting D = C � f0g, the restriction

of �i to D omits the two values zi and 1 of the extended plane bC . Their sphericalseparation is bounded below because zi 2 F 0 and F 0 is bounded in C , and so the

family f�ijDg of �-quasiconformal mappings of D into bC is normal. It follows thata subsequence of these mappings converges uniformly (in the spherical metric) on

compact subsets of D to a limit mapping � : D �! bC . Since the points zi = �i(0)are all in the compact set F 0, by passing to a further subsequence if necessary, we mayassume that the points zi converge to a point w 2 F 0, allowing us to de�ne �(0) = w.

Thus a subsequence of the maps �i converges to � : C ! bC . There are exactly threepossibilities: the limit function � is either a constant mapping, a mapping of C onto

two points of bC , or a �-quasiconformal mapping of C . We leave it to the reader toeliminate the �rst two possibilities by using the fact that the maps �i cover the maps'i.


For higher genus, the hyperbolic case, D = D , we get normality of the family f�ig forfree, and therefore some subsequence converges uniformly on compact subsets of D toa mapping �. The limit function � is either a constant mapping of D onto a boundarypoint of D or a �-quasiconformal mapping of D onto itself. Since the points �i(0) alllie in the compact set F 0, the �rst possibility is precluded and we are done.

The convergence in Theorem 4.5 can be stated in a more concrete form if we applythe reasoning just used with this lemma. As we did there, let D denote the universalcovering surface of ST (D or C ), � : D �! ST the (analytic) universal coveringprojection, and � � Aut(D) the covering group. The triangulation T in ST lifts under� to an in�nite triangulation T 1 of D invariant under �. Within T 1 one can identifya �nite union F of triangles which forms a simply connected fundamental domain ofthe covering. This concrete setting requires normalization: identify vertices �0; �1 2 Fso that u0 = �(�0) and u1 = �(�1) are neighboring vertices of T ; we will alwaysapply a conformal automorphism to D so that �0 is located at the origin and �1 ison the positive real axis in D. The conformal data pertaining to ST is encoded in F ,and in particular, in the locations of the boundary vertices of F . From these (andknowledge of the side-pairings of F) one can generate the covering group, which is (upto conjugation) uniquely associated with the point [�S] of Teich(S).We can carry out the analogous process for each Riemann surface jKnjcp. For n � 0,

its universal covering surface is again D; we write �n : D �! jKnjcp for the coveringprojection, �n for the covering group. We may identify a fundamental domain in D,call it Fn, which corresponds combinatorially (under our usual identi�cation of Kn with

T ) to F ; designated vertices �(n)0 ; �

(n)1 2 Fn lie over u0; u1, respectively, and we impose

the same normalization as before. Note that Fn is combinatorially just the re�nement�n�F . The ordered list of vertices about @F occur (in order) among the vertices of@Fn, each � of @F occurs as a vertex �n 2 @Fn so that v = �(�) 2 T corresponds tov = �n(�n) 2 Kn.Lifting the maps de�ned earlier from the surface level up to D, one obtains maps

Hn de�ned by

Hn = ��1n � hn � �; Hn(0) = 0; Hn(�1) > 0:

Each Hn is a homeomorphism of D with Hn(F) = Fn. Standard arguments yield thefollowing corollary to Theorem 4.5.

Corollary 4.7. The homeomorphisms Hn : D �! D converge uniformly on compactsubsets of D to the identify function. In particular, for each boundary vertex � of F ,the sequence f�ng of corresponding boundary vertices of Fn satis�es

limn!1

�n = �:


Likewise, for the covering groups,

limn!1

�n = �;

where convergence is in the usual topology of Aut(D).In the case that our equilateral surface is associated with a dessin (S;D), this corol-

lary tells us that within the universal covering surface one can essentially see the

convergence of the discrete dessin surfaces s(n)D to the classical dessin surface SD. The

reader will �nd several examples in the Menagerie.

When S is a genus zero surface, Teich(S) is a singleton and the limit result ofTheorem 4.5 is automatic, and uninteresting. Here ST = S2, and the maps hn :S2 �! S2 are not de�ned in a conformally invariant fashion. However, our next resultshows that appropriate normalizations of the hn will again be quasiconformal, withdilatations converging to unity o� V �; in other words, we still have rigidity waiting tobe exploited.For purposes of normalization, �x distinct vertices v0; v1; v1 of the triangulation T .

We assume that each packing PKnhas been adjusted by a conformal automorphism of

S2 to put vi at i; thus hn(vi) = i, for i = 0; 1;1. (When T = T (D), we have alreadydesignated such vertices.)

Theorem 4.8. If S has genus zero, then the (normalized) maps hn : S2 �! S2 con-verge uniformly to a conformal automorphism, with the pointwise dilatations of the hnconverging to unity uniformly on compact subsets of S2nV �.

Proof of theorem. By Lemmas 3.4 and 3.1, for N su�ciently large, each map hn is�-quasiconformal. [19, Theorem II.5.1] immediately implies that fhng forms a normalfamily of maps, and [19, Theorem II.5.3] implies that every limit function of this familyis �-quasiconformal. Lemma 3.4 implies that the local geometry of the packings PKn

is in�nitesimally euclidean. Therefore, in arguments similar to those in the proof ofTheorem 4.5, the Hexagonal Packing Lemma implies the uniform convergence of thedilatations to unity on compact subsets missing V �, and this in turn implies that everylimit function of the family fhng is conformal. If f and g are two such limit functions,then the composition g � f�1 is a conformal mapping of the Riemann sphere that �xesthe three points 0; 1, and1, hence, f = g. This and the normality of the family implythat the maps hn converge uniformly to a conformal automorphism.

4.3. Convergence of the Bely�� Maps. Given a dessin (S;D) and applying The-orem 4.5 to the triangulation T (D), we conclude that the associated discrete dessin

surfaces s(n)D converge to the classical dessin surface SD in Teichm�uller space. We now

establish the convergence, in an appropriate sense, of the discrete Bely�� maps b(n)D to

the classical Bely�� map BD.First, one should note the discrete re ective structure within jKnjcp that parallels

the classical re ective structure of ST . This analogue is visually apparent in packing


illustrations, such as Figure 2 and Figure 3 or later illustrations in the Menagerie. Forcontiguous shaded and unshaded faces, the re ection across the shared edge shouldsimply interchange the carriers of the packings of the two faces; under re�nement wewould expect this map to become increasingly more anticonformal. We already have inthe homeomorphisms hn the appropriate machinery. The image hn(T ) of the re ectivetriangulation T of ST provides a triangulation of jKnjcp. Given an edge e 2 T andcontiguous faces f�, recall the edge re ection re in ST ; the discrete edge re ection isde�ne by

r(n)e = hn � re � h�1n :

This is an idempotent quasiconformal homeomorphism interchanging hn(f�) preciselyas suggested visually.

Theorem 4.5 (and Theorem 4.8 in the genus 0 case) implies that r(n)e is nearly

anticonformal away from V �. To be more precise, recall that our de�nition of the mapshn begins with the surface ST realized concretely as the equilateral surface j�T jeq, sothe face f+ is realized in j�T jeq as the union of six equilateral triangles forming a ateuclidean hexagon. The homeomorphism hn is then obtained only after n iterations

of hex re�nement. Let e(n) and f(n)� denote the images hn(�

n�e) and hn(�n�f�) in

the circle packing surface jKnjcp. Let U be a (small) neighborhood of the vertices ofV �\�n�(f+[f�). There are at most nine such vertices, up to four forming the originalvertices of the two faces f� and �ve (4-degree vertices) forming the barycenters of theoriginal edges of these two faces. Let Un denote the image hn(U) and let �n denote

the maximum dilatation of r(n)e taken over the set (f

(n)+ [ f

(n)� )nUn. Theorems 4.5 and

4.8 imply the following result.

Proposition 4.9. For any edge e of T and neighborhood U as above, the correspond-ing sequence �n of dilatations converges to unity.

It is important to observe from this result that when numerically uniformizing re- ective surfaces, one obtains not only approximate fundamental regions, as in Corol-lary 4.7, but also a triangulation of that fundamental region which has the combina-torics of T and approximates the conformally correct re ective triangulation in thesurface ST . In the zero genus case, the numerical uniformization gives a triangulationof the Riemann sphere into nearly re ective triangles whose vertices and edges are closeto their conformally correct positions. In the case that T is the canonical triangulationT (D) of a dessin D, the numerical uniformization provides the approximate locationsof points that branch over 0; 1; and 1 under a Bely�� map, as well as the approximatepre-image of the extended real line under that map. The authors know of no othergeneral method for obtaining such approximations.

Recall from Proposition 3.3 that the discrete Bely�� maps are �-quasiregular. Thenext result guarantees that the Bely�� map BD may be approximated as closely asdesired by these discrete ones.


Theorem 4.10. Given a dessin (S;D) and the associated discrete Bely�� maps b(n)D ,

de�ne the maps

�n = b(n)D � hn : ST �! S2; n � 0:

Then the sequence f�ng converges uniformly on ST to the Bely�� map BD. The pointwisedilatations of the maps �n converge to unity uniformly on compact subsets of ST nV �.

Proof. The proof is similar to those of Theorems 4.5 and 4.8. Fix a (+)triangle f ofthe dessin triangulation T = T (D). As in the proof of Threorem 4.8, the Ring Lemmaand [19, Theorem II.5.1] imply that the restrictions of the maps �n to f form a normalfamily of �-quasiconformal maps, for some constant � � 1. By [19, Theorem II.5.5],every limit function of this family must be either a �-quasiconformal homeomorphismor a mapping onto a single point in the extended real line. We verify in the nextparagraph that the former possibility holds for every limit function. Assuming thisfor the moment, an argument as in the proof of Theorem 4.8, using the fact that anylimit function must take the i-vertex of f to the point i, for i = 0; 1;1, implies thatthe restrictions of the �n's to f converge to a conformal map taking the boundary

@f to the extended real line bR . Immediately, this limit function must be the uniqueconformal mapping of f onto the upper half plane that takes the i-vertex of f to i.This is precisely the restriction of the Bely�� map BD to f . The same argument worksif f is any (�)triangle. It follows that the maps �n converge uniformly to the Bely��map BD. The dilatation convergence (already used when the argument of Theorem4.8 was applied) follows as in the proof of Theorem 4.5.We now verify that any limit function of the normal family of maps described in

the previous paragraph takes at least one value in the (open) upper half plane. Recallthe abstract simplicial complex H triangulating the sphere, de�ned in x 3; let v denotethe vertex corresponding to the barycenter of the upper half plane, one of the twofaces of H. Recalling that v is then contained in all the hex re�nements �nH of H,let z(n) denote the center of the circle cv in the circle packing Q(n) for Hn = �nH.

The de�nitions of hn and b(n)D guarantee that the restriction of each map �n to the

(+)triangle f satis�es �n(bz ) = z(n), where bz is the (unique) conformal barycenter off . Since the restriction of each map �n to f is a �-quasiconformal homeomorphismonto the upper half plane, it su�ces to show that the points z(n) are bounded awayfrom the extended real line. In fact, however, the z(n) form a constant sequence:each z(n) is precisely the point e�i=3, which is the (unique) conformal barycenter of theupper half plane thought of as a triangle with vertices 0,1, and1 and sides forming theextended real line. This follows from the fact that the circle packing Q(n) inherits ordersix dihedral symmetry from that of Hn; this symmetry permutes the circles centeredat 0,1, and 1 and hence �xes the circle corresponding to the barycenter bz.


5. A Menagerie of Dessins d'Enfants

For convenience, the Menagerie is organized by genus. As opportunities arise, wepoint out various pertinent features, many of which will apply in several settings. Wewill refer back to these examples when we discuss computational issues in Section 6,but at this point we should remark on the two senses in which these examples are\approximations". First, all are subject to the usual round-o� and truncation errors ofnumerical computation and display; second, even were we to have perfect informationon a given discrete dessin, that dessin might only approximate its classical companion.

5.1. Genus 0. In the case of the sphere, the conformal structure engendered by adessin is not at issue | the sphere has only one. We have latitude only in the normal-ization, and we have agreed to the convention that designated i-vertices are placed ati for i = 0; 1;1. In such a normalized situation, the conformally correct dessin, thelocation of zeros, ones, and poles, and various other metric information are examplesof data of interest.

�� XX

S2

Figure 8. Dessin 1 and its stage-4 packing in S2

5.1.1. Example 1. We begin with the very simple dessin D of Figure 8(a). This is theonly example we give which fails to be \clean"; it has a \free" edge ending in a �,so its Bely�� map will not branch at this 1-vertex. The stage-4 packing P

(4)

D is shownin Figure 8(b). As will be standard, we shade appropriate faces and mark the dessinitself with a heavy line.


5.1.2. Example 2. We have used a clean dessin only slightly more complicated thanthe previous example for purposes of illustration earlier in the paper; we denote itExample 2. The dessin and its triangulation are displayed in Figure 1, its stage-3discrete Bely�� map in Figure 2, and its coarse discrete Bely�� map in Figure 3. InFigure 9 we project its stage 3 packing to the plane for comparison to Figure 1.

Figure 9. Stereographic projection of Dessin 2

5.1.3. Example 3. We add further edges to the dessins of Examples 1 and 2 to getDessin 3 of Figure 10(a). (See x6.1 concerning \dessin moves".) Stereographic pro-jections of the discrete dessin as embedded by the coarse and �rst four stages of hexre�nement are given in Figure 11, illustrating the evolving shape. Detail around thehead at stage-4 is shown in Figure 12.This dessin, due to Gunter Malle (see [16]) has a known and nontrivial orbit. Recall

that dessins are associated with number �elds. The Galois group of the number �eldprovides an action on dessins, and the Galois orbit of a dessin is the smallest collectionof dessins closed under the Galois action. In the case of Example 3, the Galois orbitconsists of Dessin 3 and the dessin of Figure 10(b) | in other words, the Galois actionswitches between the left-arm person and her right-arm companion!

5.1.4. Example 4. Trees form the only general class of dessins for which approximationmethods have been developed. Dessin 4 is a tree from [29]. The embedded image of

Figure 13 is from the stereographic image of P(4)

D and should be compared to theimage [29, p. 112]. Using Grobner basis methods, Couveignes and Granboulan canprovide extreme accuracy for embedded trees, enough to eventually recover coe�cients


(a) (b)

Figure 10. Dessin 3: A genus 0 Galois orbit

Figure 11. Successively �ner stages of the right-armed dessin

of de�ning equations. (The equation for Dessin 4 has integer coe�cients of over twentydigits!) This far surpasses the accuracy available via circle packing. However, theirmethods involve certain initial guesses for locations of vertices and the addition ofvertices to existing trees. Circle packing might prove very helpful in this process; seeespecially x6.1.

5.2. Genus 1. A dessin has genus 1 if it is drawn on the topological 1-torus T. Thetorus has the euclidean plane as its universal cover, and T endowed with a conformalstructure is typically identi�ed with a fundamental domain in C . The lift to the planeof any point on the conformal torus forms a doubly periodic lattice, and the ratios of


Figure 12. Stereographic projection of the head of Dessin 3

Figure 13. A dessin tree

pairs of complex numbers generating that lattice parameterize the conformally distinctmarked tori. We will see this played out in its discrete form here.We concentrate on a dessin introduced by Shabat and Voevodsky [30] and thoroughly

analyzed by them and others. In this case the associated number �eld, the de�ning


equation, the Galois orbit, and the parameters of the associated tori are known, af-fording one an opportunity to compare the discrete and classical information.

5.2.1. Example 5. Dessin 5 is illustrated in Figure 14(a), where opposite sides of therectangle are identi�ed in the usual way so that the dessin is seen to lie on T.

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

(a) (b)

Figure 14. Dessins 5 and 6: A genus 1 Galois orbit

From Dessin 5 one obtains the canonical triangulation of S = T and its barycentricsubdivision K. There is a unique conformal torus sD, the coarse dessin surface, whichsupports the circle packing PK for K; in this genus 1 case, the metric is euclidean,hence only de�ned up to a multiplicative positive constant.As in the classical setting, one may represent sD by lifting to the universal covering

surface, C . In fact, everything lifts: the triangulation, the complex, the circle packing,the metric, and the dessin. The results for the coarse packing of Dessin 5 are illustratedin Figure 15(a), with a fundamental domain highlighted.We are in position to estimate the conformal modulus of the discrete torus. The

four corners of the fundamental domain in Figure 15(a) correspond to the same pointof sD and de�ne the covering lattice in C . From the packing centers one can reado� approximations to a pair of complex numbers which generate the lattice. Forconvenience we have placed the lower edge of the packing with ends at 0 and 1 inFigure 15(a); the complex number associated with the left side is approximately � �0:249612 + 0:968346i; we will refer to this information in x6.Of course we have the option of re�ning our circle packing for a more accurate

approximation of the classical dessin. Using 2 stages of hex re�nement and repacking

leads to the fundamental domain of Figure 15(b) for s(2)

D . Again, the lower corners


(a)

(b)

Figure 15. Fundamental domains for genus 1 packings


have been placed at 0 and 1, and the complex number for the other side can be reado� as approximately � � 0:248308 + 0:968683i.

5.2.2. Example 6. The Galois orbit for Dessin 5 contains one other dessin, shown in

Figure 14(b). Dessin 6 is laid out using the stage-2 packing s(2)

D in Figure 16. Thisillustrates the general fact (see [15]) that conjugate dessins share the same numbers ofdessin faces, edges, and vertices. (Their circle packings consequently share the samenumber of circles at each re�nement stage.) Dessins 5 and 6 should be compared, bothin their schematic and embedded forms, to [2, Fig. 8, p. 209] and [30, p. 215].

Figure 16. The Galois conjugate of Dessin 5

5.3. Genus 2. With higher genus we move into the hyperbolic realm. The dessinsurfaces are covered now by the hyperbolic plane, which we will represent as the unitdisc D endowed with the Poincar�e metric ds = 2=(1� jzj2)jdzj of constant curvature -1. In a manner completely analogous to the previous genus 1 setting, one may lift allstructures from the surface to a fundamental domain F of the covering. The coveringgroup � is now a discrete nonabelian group of automorphisms of D . We will illustratewith three examples.


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��A B

C

D

CD

A

B

A

B

C

DA

B

C

D

(a) (b)

Figure 17. Dessins 7 and 8: Two dessins of genus 2

5.3.1. Example 7. Dessin 7 is shown in Figure 17(a); it is important to note the iden-ti�cations of the sides which makes this into a surface of genus 2. When triangulated,Dessin 7 contains 28 faces. The associated coarse circle packing is shown in Figure 18(a)and the stage-3 re�nement in Figure 18(b). In each case, a 0-point has been placedat the origin and only fundamental domains F0 and F3, respectively, are shown; thenumbering of boundary edges for the side-pairings is indicated on the coarse packing.In each case, the action of the covering group �n generates a tiling of the hyperbolicplane with isometric images of Fn.A few observations are in order. The eight circles at the cusps on the boundary of

F0 are in fact eight lifts of one circle; in particular, although their radii are euclideanlydi�erent because of their distances from the boundary, they share a common hyperbolicradius. The angles at the cusps sum to 2�, and in the fundamental domain for theclassical surface would all be precisely �=4. A cautionary note: the edges betweencusps in these �gures are approximating analytic arcs; though they may appear to beapproximate geodesics, this need not be the case.This dessin provides an opportunity to illustrate Corollary 4.7. In particular, the

circle packing for a discrete dessin provides numerical estimates of the covering mapsof the associated classical dessin surface. Estimation occurs in two stages which wenow discuss. (The same general considerations applied in the euclidean setting ofExample 5.)Step 1: First is the estimation of the covering maps for a discrete dessin surface.

In Figure 18(a), let fc1; � � � ; c8g denote the eight cusp points of F0. Each side-pairingof F0 is associated with an element 2 �0 which carries one ordered pair of thesepoints to another: : (ci; cj) 7! ( (ci); (cj)): This information alone determines the


(a)

(b)

DD

A

B

A B

C

D

C

Figure 18. Coarse and stage-3 fundamental domains for Dessin 7


automorphism . In turn, such side-pairing automorphisms generate �0. There-fore, the locations and pairings of the eight points cj determine �0. In theory, thisinformation is exact, but in practice, of course, the data is subject to roundo� errorsin the computation of packing radii, the location of the centers, and the subsequentcomputation of the automorphisms.Step 2: The second stage involves the use of successively �ner circle packings so

that their covering maps converge to the classical covering maps. Figure 18(b) is thestage-3 hex re�nement, involving 5374 circles. The computed locations of the eightcusp points give estimates for the associated side-pairing maps. Matrices in GL(2; C )representing the A, B, C, and D side-pairings (see Figure 18) are given by, respectively,�

0:610254� 0:705705i �0:725003 + 0:416873i�0:725003� 0:416873i 0:610254 + 0:705705i

��0:389727 + 0:416402i 0:511033 + 0:158976i0:511033� 0:158976i �0:389727� 0:416402i

��0:426400 + 0:422296i �:0443194 + 0:336691i�:0443194� 0:336691i �0:426400� 0:422296i

��0:532299� 0:413640i 0:605959� 0:0348912i0:605959 + 0:0348912i 0:532299 + 0:413640i

�It seems clear that estimates of the type obtained here do not provide perfect infor-

mation | a precise number �eld, for example | about the dessin surface. However,they do provide what would normally be considered as fundamental information forone to \know" its conformal structure. The data should be su�cient, for instance,to invoke some of the available programs for working numerically with Riemann sur-faces, such as the CARS program of the HCM network on Computational ConformalGeometry and the Symbolic Computation Group at Florida State University.Example 7 is a favorite of the authors because it was the �rst hyperbolic example

attempted: we were surprised with the speed, beauty, and accuracy of the process.The accuracy still seems remarkable, and we comment on this in the next section.

5.3.2. Example 8. Dessin 8, shown in Figure 17(b), di�ers from the previous example inits side-pairings. The coarse packing is shown in Figure 19(a), and a visual comparisonwith Figure 18 suggests that these surfaces are conformally very close. Are they thesame surface? Is this, in fact, the same dessin?

5.3.3. Example 9. Here we show a somewhat more generic genus 2 dessin; the dessinand its faces, as embedded by a stage-2 packing, are laid out in Figure 19(b). Oneside-pairing has been highlighted to demonstrate that this is clearly not a standardfundamental domain bounded by geodesics.This dessin was constructed to provide one simple handle and one with richer com-

binatorics. The resulting asymmetry in their conformal structures is evident. (One\handle" consists of the two cells at the top, the other of the nine lower cells.) This


(a)

(b)

A

B

C

D A

B

C

D

Figure 19. Dessins 8 and 9


highlights a central issue: To what extent can one anticipate conformal implicationsdirectly from the combinatorics?

5.4. Higher Genera. We begin with two very classical surfaces: the Klein \Haupt-�gur" of [12] and the Picard curve. Historically, these resulted from other considera-tions, and only in hindsight are associated with \dessins". We also display a genericgenus 4 dessin to show the level of complexity that our methods can handle. The mainhurdle in preparing such examples lies in specifying the dessin combinatorics.

5.4.1. Example 10. Klein's \Haupt�gur" is a genus 3 surface which has played a sem-inal role in classical geometry and function theory. It is the modular curve X(7) ofdegree seven and has an automorphism group of order 168 = 84(g � 1), the maximalpossible in genus g = 3. This curve also has an honored place in the visual history,thanks to the beautiful illustrations, now over a century old, produced for Klein.Klein's surface R is the compact surface H =�(7), where H is the upper half-plane

and �(7) is the subgroup of PSL(2;Z) consisting of elements congruent to �id modulo7. See, for example, [17, 22]. A natural triangulation G on R (which one can obtaingeometrically, combinatorially, or via the group structure) is a 7-regular graph. Aclassical fundamental domain for R can be indenti�ed within a tiling of the hyperbolicplane by equilateral hyperbolic triangles having angles 2�=7.We take (R;G) as our dessin; the circle packing (not shown) provides the image

in Figure 20(a), an image familiar to analysts for more than a century. Figure 20(b)shows the associated dessin overlaying the carrier of the same circle packing. For thereader's bene�t, this latter illustration indicates the side-pairings associated with thesurface.Note that the dessin faces of the dessin surface are geodesic (2; 3; 7) triangles due

to the ubiquitous symmetries of G. In particular, even with the coarse circle packing,these illustrations have perfect accuracy (up to the usual computational roundo�). Itis because the dessin faces are geodesic triangles that this picture could be constructeda century ago | this will be in contrast to our next examples.

5.4.2. Example 11. Next is the genus 3 \Picard" curve y3 = x4 � 1. We �rst employa dessin taken from Shabat and Voevodsky [30, p. 217]; this is pictured with thefundamental domain from the coarse packing in Figure 21(a).The Picard curve can also be constructed by methods similar to Klein's surface, using

the fact that it has a very rich automorphism group, one of order 48. In [26], J. R. Quinebuilds the fundamental domain of Figure 21(b) using (2; 3; 12) triangles. The side-pairings are determined by labeling the sides counterclockwise, from 1 (as indicated)to 24; each even-numbered side k is then paired with side k + 7. It turns out thatthis triangulation arises from a dessin distinct from the one of Shabat and Voevodsky;the triangles forming the fundamental domain are rearranged in Figure 21(c) to moreeasily picture this alternate dessin. For the side-pairings, number the sides as beforefrom 1 and identify each odd-numbered side k with side k + 5.


(a)

(b)7

9

2

4

1

3

5

5

7

9

2

4

1

3

Figure 20. Klein's Haupt�gur


(a)

(b) (c)

1

9

72

6

5

3

7

4

1

58

2

49

3

8

6

1 1

Figure 21. Two dessins for the Picard surface


It is convenient to have two genus 3 dessins determining the same conformal struc-ture, since it a�ords us opportunities to judge the quality of circle packing approx-imations. (Is there some reason to anticipate this circumstance based solely on thecombinatorics of the two dessins?) Comparing the triangulations, note that the trian-gles of Figure 21(b) (and (c)) are, as with the fundamental domain for Klein's surfacein Figure 20(a), geodesic triangles, so this fundamental domain is essentially exact. Onthe other hand, the triangles of Figure 21(a) may appear to be geodesic, but in factde�nitely are not. (One can see, for example, that in re ections of dessin faces acrosscertain edges, the opposite angle on one side will be �=12, while that on the other is�=4.) The image of Figure 21(a), obtained from a coarse circle packing, is, as far aswe know, only approximate.

5.4.3. Example 12. Our �nal example is a garden variety genus 4 dessin. The maindi�culty with higher genera is largely that of encoding the combinatorics. Here weuse the \pair of pants" paradigm.

Figure 22. A standard pair of pants

A pair of pants is basically a sphere with three round discs removed; any compacthyperbolic Riemann surface can be decomposed as a disjoint union of a �nite numberof pairs of pants (with appropriate structure parameters) identi�ed along boundarycomponents. We will use the basic pair of pants shown in Figure 22, with conformalstructure determined by two dessin-like 2-cells. A genus 4 surface constructed from 6identical pairs of pants, twelve 2-cells, is shown in Figure 23.Observe that at this point we have a highly symmetric surface, especially so since

the particular pair of pants we are using has the greatest possible symmetry (only


Figure 23. A symmetric genus 4 surface

countably many surfaces can be constructed from from copies of this pair). We havealso built in some global symmetry for easy visualization. (One consequence is, as withKlein's surface, that the discrete conformal structure is precisely equal to the classical.)We have modi�ed the dessin by adding edges and vertices (see x6.1); the new fun-

damental domain from the coarse packing is shown in Figure 24. To help with visu-alization, modi�cations were restricted to the \upper" half of the dessin and left-rightsymmetry was maintained. A region of �ne structure has also been blown up to showsuccessive levels of detail.

This concludes our small Menagerie. Many additional examples could be given,both from the dessin literature and from other areas. There are, for example, thegeneralizations H =�(k) of Klein's surface for k > 7 and a plethora of pair-of-pantsconstructions. We move on to a discussion of some computational issues; see theAppendix for details on implementation.


Figure 24. Details of a more generic genus 4 dessin


6. Computational Issues

The discrete theory of dessins based on circle packings | discrete conformal struc-tures, discrete coverings, discrete Bely�� maps, and so forth | is computationally ac-cessible because there exist both theoretical and practical algorithms for computingcircle packing radii in the various geometries. In other words, the discrete theory canbe implemented and investigated experimentally. We would like to emphasize thatexperiments might provide classical insight both through analogy and through explicitapproximation. Moreover, as we suggest in Section 7, the discrete theory has a certaincharm of its own.We must say to begin that we as yet have no �rm quantitative handle on the in-

ternal consistency of circle packing structures nor on the accuracy with which theyapproximate the classical structures. Having said that, however, our experience sug-gests that the discrete structures are internally very robust and remarkably close tothe classical structures. Our comments here will largely be directed to experimentalobservations (and potential observations) with Menagerie specimens. Comparisons toclassical objects are, of course, complicated by the fact that few of them are known inany practical sense.

First we observe that quite complicated dessins can be circle packed with relativeease and appear to give visually consistent discrete structures (e.g. fundamental do-mains and Bely�� maps). The main experimental hurdle typically lies in specifyingthe abstract combinatorics for the dessin (for which help is now available through the\dessin moves" described at the end of this section). From there, the experiments, car-ried out using CirclePack and DesPack, are fairly straightforward, modulo the usuallaboratory glitches. (For more details on software, algorithms, implementation, runtimes, references, and so forth, see the Appendix.)Recall that given a dessin (S;D) there is a countable collection of associated Bely��

pairs: namely, the discrete pairs (s(n)D ; b

(n)D ) for the various re�nement stages and the

classical pair (SD; BD) to which they converge (per Theorems 4.5, 4.8, and 4.10). We

address mainly the case of positive genus g and consider the surfaces s(n)D and SD; when

g = 0, roughly parallel considerations pertain to the Bely�� maps. We may place theaccuracy issues in three broad catagories: (I) How closely does a computed discreteconformal structure approximate the actual discrete conformal structure (given by theCircle Packing Theorem)? (II) How closely do the discrete conformal structures ofsuccessive re�nements approximate one another? (III) How closely does the discreteconformal structure at a given stage approximate the classical conformal structure?We can try to tease some insight from our examples.For issue I, Klein's surface, Example 10, with 336 dessin faces and a coarse packing

involving 1004 circles, provides a particularly good starting place. Due to its order 168automorphism group, the discrete conformal structures | at every level of re�nement| are in theory precisely equal to its classical conformal structure, with dessin faces


which are geodesic (2; 3; 7) triangles and a covering group which is (conjugate to) themodular group �(7). This gives several precise values to which one may comparecomputed ones.Consider the coarse packing displayed in Figure 20. As discussed in the Appen-

dix, the radii of the packing are computed so that the packing \angle sums" agree toapproximately 10 digits with the value 2�; the radii are then used to lay out (a fun-damental domain of) the packing. The ultimiate aim is accuracy in the placement ofthese circles. We carried out a few sample comparisons. (Incidently, the computationof this packing takes roughly 15 seconds on a Sparcstation 10.)

� Concerning the computed radii, note that symmetry implies that for eachi = 0; 1;1, all the circles associated with i-vertices should share the same radius.Using repackings starting from randomized initial labels suggests that computedradii will agree to 7 digits.

� To judge accuracy in the placement of centers, we laid down circles to buildthe closed chain of 324 boundary faces in the fundamental domain of Figure 20.This process is akin to analytic continuation. It suggests accuracy to 5 digits inplacing the boundary cusp points.

� Side-pairing maps represent key data for any covering and can be estimatedfrom locations of boundary vertices of the fundamental domain. A simple check,for example, compares the hyperbolic distances between endpoints of identi�edsides; in Figure 20, these distances typically agree to 6 decimal places.

Further experiments should suggest how these various measures of accuracy respondto the error tolerances set in the packing routines. There are other highly symmetriccases in which the discrete and classical dessins are precisely equal, such as one formof Example 11 and the symmetric predecessor to Example 12 (Figure 24). However,we will not pursue this further here.

To address issue II, the stability of discrete structures under re�nement, let us moveto more generic examples. Consistency has certainly been one of the most strikingfeatures of the computer experiments: in example after example the discrete structuresseem stable from the very �rst, coarsest packing. Figure 11 shows normalized discreteembeddings of the \one-armed" Dessin 4, from coarse through stage-4. The generalshape seems reliable from the beginning, though the proportions of head and legsemerge more slowly. This is typical of our genus 0 experiments. In positive genuscases, both the salient features and the details seem to stabilize even more quickly.For instance, let us return to the \peace" symbol of Example 7. We would expect

(although we have no proof) that the surfaces s(n)D from various stages are all distinct

from one another and from SD. In Figure 25, we have overlaid images of the boundariesof the fundamental regions for the coarse and for the stage-3 packings, respectively 82


and 5374 circles, and have enlarged the image of the cusp where they di�er most; theeuclidean distance between cusp points is approximately 0.003.

Figure 25. Comparing cusps for stages of Example 7

There is a point to be made about our discrete dessin drawings, particularly in genuszero. In a conformally correct dessin, the edges meeting at a vertex will meet with equalangles. In various of our illustrations of discrete dessins one can clearly see that thisfails (e.g. Figure 2 or Figure 12). This is due, in part to our drawing method: dessinsare drawn as piecewise geodesic arcs running through the centers of appropriate chainsof circles. Accuracy would presumably improve if they were drawn as splines, with therequirement that they run through appropriate sequences of tangency points and cometogether at vertices with equal angles. This makes little di�erence in the positive genuscases. However, in spherical geometry, circle centers are not conformally determined(they are not preserved under M�obius transformations); tangency points are, and theuse of splines can make noticable di�erences in dessin drawings. In any case, thepictures in the paper (and the package CirclePack) use the more computationallyconvenient circle centers.

Issue III concerns the accuracy of the descrete vis-a-vis the classical structures. Thestability of the discrete conformal structures under re�nement in conjunction with


Theorem 4.5 suggests some level of accuracy. Of course, further work needs to be doneon the numerical side prior to any claims. Nonetheless, the Menagerie provides someopportunities to address accuracy beyond the very symmetric cases such as Klein.In particular, the genus 1 dessin of Example 5 has been studied thoroughly and its

classical j-function value was computed in [30, p. 215], namely:

4(8 + 3p7)2(2�

p7)6(10� 3

p7)3 = 457208� 172564

p7 � 646:5707574:

In Section 5, we gave numerical estimates of the complex modulus � for the discretetorus at the coarse and at the stage-2 levels. We put these values throughMathematica

to obtain the estimated j-function values, j(�), with these results:

Coarse stage: 635:06238� 0:00035i Stage-2: 642:57533� 0:01011i;

representing errors of about 1:78 and 0:62 percent at the coarse and stage-2 levels,respectively.

Remark. The j-function itself may be constructed by a re ective process similar toBely�� maps, the only complication being that it has an in�nite order 1-vertex (apuncture) representing a logarithmic branch point; see [1]. Replacing the puncture bya vertex of high �nite degree, one can theoretically construct discrete approximationsto the j-function.

There are other opportunities to compare discrete and classical data, as with thetree of Example 4. We could also point to the interesting Picard surface, Example 11.In this instance we have two dessins which de�ne the same surface. For one, the dessinfaces are geodesic triangles, so as with the Klein surface, the discrete structures areall identical to the classical one. There seems no reason to expect this for the otherdessin, so again one can compare the approximate structures to the known classicalone.The reader may have observed in our examples that circles associated with dessin

vertices are often quite large, even in �ner stages; one might be led to question theprecision. This behavior, however, is wholly consistent with classical behavior. Recallthat the number n of faces incident to a dessin vertex v is 2(b + 1), where b is theorder of the branch point BD has at the point z 2 S2 associated with v. If b � 1, BD

compresses large neighborhoods about z into small neighborhoods about BD(z). Inother words, BD is approximately constant on a large neighborhood of z. This, ratherthan any loss of precision, is what the large circle at v re ects for the discrete mapping.The question of theoretical accuracy | how closely the discrete conformal structures

(resp. discrete Bely�� maps) for a dessin approximate the classical conformal structures(resp. classical Beyl�� maps) seems to be very deep. As we point out in the �nal section,approximation is not the only legitimate aim for studying the discrete structures.

6.1. Dessin Modi�cations. There is a small number of elementary dessin moves,reminiscent of Reidemeister moves in knot diagrams, which allow one to construct anarbitrary dessin on a compact surface from any beginning dessin. This is not di�cult to


establish, and the reader can furnish the details. We mention this because these dessinmoves are now not only practical in the experimental setting, but almost essential.Nearly all of our more complicated dessins have been built from simple seeds by dessinmoves which added complexity. In turn, the fact that moves can be implemented opensa new avenue for thinking about dessin structures and their relations to one another.The basic dessin moves are of three types: (I) adding/deleting vertices, (II) add-

ing/deleting free edges, and (III) adding/deleting separating edges. Recall that dessinsare connected and 2-colorable graphs, and the moves must preserve this. So, forexample, a type I move adding a vertex actually requires adding two vertices. Withthis caviat, the moves are largely self-explanatory. It is also useful, particularly workingin the category of clean dessins, to identify common \composite" moves, such as addinga double free edge or a loop at a 0-vertex, or adding a \bridge" across a 2-cell fromone 0-vertex to another. Only the add versions of these composite moves have beenimplemented in DesPack so far, and these were used in creating the Menagerie.Concrete examples of moves are illustrated in Figure 26. We started with the sim-

plest genus 1 dessin, two loops at a base point, and applied a sequence of \composite"moves leading to Dessin 6, the lower right hand �gure. The coarse dessin, circle pack-ing, and shaded dessin faces are shown at various intermediate stages, all normalizedto place the bottom corner circles at 0 and 1.

When D0 is obtained from the dessin D by a �nite sequence of dessin moves, eachof which is an \adding" move as opposed to a \deleting" move, we call D0 a dessinre�nement of D.

Proposition 6.1. Any two dessins D and D0 on an oriented closed surface S share acommon re�nement.

The proof is left to the reader. It helps to note that by applying a small isotopy toD0, one may assume that D and D0 are in general position. Their union may then beconverted into a 2-colorable graph of which they are both re�nements. A corollary tothis proposition is: On a topological surface S, any dessin D0 may be obtained from

any other dessin D by a �nite sequence of elementary dessin moves.

6.2. Graph Embedding. The carrier of a univalent circle packing in a surface pro-vides an embedding of its complex K (in fact, in the cases we have been considering,it even provides the geometry for the surface). In some cases it may not be the surfacewhich is of interest, but rather the embedding of the 1-skeleton K(1) as a graph. Indeed,circle packings are valuable for embedding fairly general locally planar graphs: the 1-skeletons of their carriers are geodesic, they can simultaneously provide geodesic dualorthogonal graphs, and graph edges meet at angles bounded below by constants de-pending only on degree. Graphs of considerable complexity can be generated automat-ically and randomly and the packings computed relatively quickly; packing algorithmsare known that work in polynomial time with the number of vertices.


Figure 26. Dessin moves generating Dessin 6


The circle packing techniques of the paper provide embeddings of general triangu-lated surfaces, polygonal surfaces, open and bordered surfaces, and are being applied inrecent work on conformal \tilings" (see [6]). Circle packings tend to be accompanied inall these settings by vestiges of classical analysis | random walks, discrete Laplacians,discrete extremal length, discrete analytic functions, and so forth.Some of the interest in dessins d'enfants stems from fact that they provide a count-

ably dense set of points of Teichm�uller space (see the next section), and that thesepoints follow an underlying organizational structure. Physicists, for instance, are in-terested in statistical and asymptotic analysis of surfaces.

7. The Discrete Option

Circle packing theories have displayed a remarkable knack for both mimicking andapproximating their continuous models in various areas of conformal geometry andanalytic function theory. It might be said that they provide a \quantum theory whichis classical in the limit". We wish to close by formulating some of the open issues inthe dessin setting. In particular, we would suggest a broad and somewhat independentview of the discrete theory, since it has attractions and raises questions of its own.

With regard to approximating the classical theory, it seems that the discrete the-ory has the potential to contribute in several ways; visualization, manipulation, dessinmodi�cation, and rough numerical estimation. Its main strength seems to lie in theexperimental setting which it provides for developing intuition and perhaps new the-oretical insights. It is not yet suitable for highly accurate numerical approximations(as are available, e.g., with trees [9]) and there is considerable work to be done ontheoretical accuracy, numerical precision, and rates of convergence. There is a rapidrise in computation times with successive re�nement, and even with numerical im-provements one might wonder about the long range practicality for actual estimation.This is somewhat counterbalanced, however, by the surprising accuracy of even coarsepackings; this accuracy is one of the more intriguing issues vis-a-vis the classical theory.Questions such as the Teichm�uller distance from SD to sD are of considerable interest,but probably very hard to pin down, even for individual dessins. Despite any currentnumerical limitations, however, note that circle packing is the only game in town forapproximating most conformal structures.

However, we want to leave these issues for a moment and move our thinking tothe discrete setting | after all, the qualitative features of the theory seem essentiallyunchanged. Suppose you were resigned to living entirely in the discrete world. Whatwould you see as the key intrinsic issues? What internal structures do the collectionsof surfaces and functions have? fail to have? How much algrebraic structure can onesee? What can one learn when your computer can actually build surfaces?To frame some of the discussion, we might consider dessins of a �xed positive genus g

as viewed from the standpoint of the Teichm�uller space Teich(g). On the discrete side,


we will restrict attention to the \course" dessin surfaces only | foregoing re�nements.Bely�� 's theorem and the density of algebraic numbers in the complex �eld imply thatthe set of (classical) dessin surfaces

T*(g) = fSD : genus(D) = gg � Teich(g)

form a countable dense subset. Indeed, these surfaces are characterized as those havingde�ning equations with coe�cients in a number �eld. This is one of their principalattractions; they provide a geometric playground for the study of algebraic numbersand Galois groups.There is now a parallel collection of discrete (coarse) dessin surfaces, which we will

denote by

t*(g) = fsD : genus(D) = gg � Teich(g):

This provides new playground equipment.There are many natural questions about t*(g). Note that it intersects T*(g), as with

the Klein surface (e.g., in the presence of su�cient symmetry, see Example 10). Aretheir any other intersections? Intuition suggests that these sets are essentially disjoint.Are there redundancies in t*(g), in the absence of symmetry? Within T*(g) there aresystemic redundancies: for instance, the 1-skeleton of T (D), treated as a graph, leadsto a new dessin D0, but Proposition 4.1 implies that SD = SD0. Other redundanciesare perhaps much deeper, as with the distinct dessins yielding the Picard surface ofExample 11. (Incidently, with modi�cations of the proof of Theorem 4.5, one can provethat t*(g) is a countable dense set in Teich(g); see also [7].)A quite fascinating open question concerns the characterization of the points of t*(g).

Here there is a tantalizing new connection to algebraic numbers proven by McCaughan[24]: The covering group for a point of t*(g) is conjugate to a subgroup of PSL(2;Q )

(resp. PSL(2;Q + iQ )) if g > 1 (resp. g = 1). That is, one may choose the coveringgroup so that its matrix representatives have algebraic entries. For example, in thegenus 1 cases, the modulus will be algebraic.The ability in the discrete setting to experiment with quite complicated dessins

suggests new issues. What, for example, are the algebraic implications of dessin mod-i�cation? Are there geometric descriptions of dessin Galois orbits, or is this truely analgebraic notion? Can one actually do arithmetic with surfaces? Just how suitableis this new playground equipment if you are interested in number �elds and Galoistheory?Some of the interest in dessins d'enfants, in string theory, for instance, stems largely

from the fact that dessins impose an \organizing principle" on the space of surfacesthrough the dense sets T*(g). The discrete theory may o�er de�nite advantages. Sur-faces of considerable complexity can be generated automatically and randomly and thepackings computed (at least at the coarse stage) relatively quickly, aiding in the sta-tistical and asymptotic analysis of surfaces. Circle packings seem to provide a natural


geometry of the type that nature might favor, and their re�nements are akin to thephysicist's renormalization.

In closing, the authors might suggest staying in this discrete world for a while longer| it has much of the geometry, combinatorics, and even possibly the number theoryof the classical setting, plus a certain charm all its own.

8. Appendix: Implementation

The examples of the paper were produced using two suites of computer programs:CirclePack, a graphically based program for creating, computing, manipulating, anddisplaying circle packings, and DesPack, a set of auxiliary programs for building dessinsand their associated data structures. These programs are available for use by othersfrom the second author.

8.1. A Quick Experiment. As an overview, let us follow a typical development cyclefor a dessin, say the genus 2 dessin of Example 7.

1. Start with the common fundamental domain for a genus 2 surface, schemati-cally an octagon with \normal form" side-pairings. This is the minimal dessin ofgenus 2 and its canonical triangulation involves 16 triangles. These are numberedand the adjacency relationships are entered in an ASCII �le (in a straightforwardformat) as input to DesPack. This �le contains what we will informally refer toas a \red" chain, a chain of faces enclosing a fundamental region.

2. DesPack is programmed with the composite dessin moves of x6.1; three ofthese convert the basic dessin into the dessin D of Figure 17.

3. DesPack now generates a simple ASCII �le representing a 2-complex K whichencodes the combinatorics of the coarse circle packing for D. It also produces alist of faces of K enclosing a fundamental domain and it speci�es the requisitegeometry, in this case, hyperbolic.

4. CirclePack reads the combinatorics K and computes the associated hyper-bolic packing label. This is a crucial step: theory gives the surface sD. Inpractice, this is represented as a unique \packing label" for K consisting of thehyperbolic radii for PK. CirclePack computes the packing label using a modi�-cation of an iterative algorithm suggested by Thurston, an algorithm which hasbeen proven to converge [4].

5. Using the hyperbolic radii, combinatorics, and red chain of the packing,CirclePack will lay out a con�guration P of circles in the hyperbolic planewhose carrier is the desired fundamental domain of the covering of sD.


6. DesPack and CirclePack can communicate to permit display of various infor-mation in the hyperbolic plane: e.g., drawing the dessin, shading faces, labelingthe 0- 1- and 1-points, etc.

7. CirclePack will generate the combinatorics Kn for one (or more) hexagonalre�nements of the packing for K. It can then compute the packing label for there�ned combinatorics, lay out the new fundamental domain, and in conjunctionwith DesPack, carry out various drawing operations.

The di�culty in building examples typically lies in specifying dessin combinatoricsand a red chain for visual layout. With this in hand, the remaining operations arequite fast. The sequence of commands to CirclePack is typically entered in a \script"�le, which can be shared with others and invoked automatically.

8.2. The algorithm. The most computationally intense stage in working with dis-crete dessins involves computation of packing radii for circle packing complexes K.The \packing" algorithm in CirclePack is iterative, similar in spirit to classical ralax-ation methods for solving the discrete Laplace equation: the routine repeatedly passesthrough the list of circles, adjusting the radius of each circle it visits to make it �twith its immediate neighbors. The algorithm is described brie y in [10, p. 316]; foradditional details and recent improvements in implementation due to Chuck Collins,see [8].The computation times increase rapidly with the number of circles and the desired

numerical accuracy. Tables 1 and 2 list the timings for several stages of most ofour examples. The \Iterations" refers to number of passes through the full set ofvertices, so its product with the previous column indicates the total number of radiiadjustments in the iterative repacking algorithm. The runs were carried out on DECAlphaStation 5/333 machine. The timings indicate CPU time for the repacking phaseonly, disregarding overhead, such as input/output. They were measured in seconds,so \0" in the last column indicates a repacking time of less than a second. Note thateven the most complex course packings required less than a second to pack.The examples in the table represent roughly 9 digit accuracy. To suggest the e�ect

of desired accuracy, Table 3 lists timings for the stage-3 Klein surface (64,508 circles)for tolerances set from 10�5 down to 10�10.The properties of packing labels and the behavior of the packing algorithm are quite

fascinating in their own right, and signi�cant improvements may be possible withdeeper understanding. For instance, radii adjustments may be modeled as discreteMarkov processes having to do with the ow of \curvature" among vertices in thepacking complex; see [32] and [31]. Packings of re�nements, important if one wishesto approximate the classical objects, may be improved through multi-grid methods inwhich the labels for coarser packings provide the initial guesses for labels of their re�ne-ments; see [10, x5]. Finally, the algorithms are amenable to signi�cant parallelization.


Example #Genus

Re�nement

Stage

Circle

count

Iterations CPU time

(min. m, sec. s)

Ex.#2, g = 0 coarse 25 31 0(lollypop) 1 97 66 0

2 385 319 03 1537 1325 10s4 6145 5017 2m 33s

Ex.#3, g = 0 coarse 74 27 0(one-arm) 1 290 68 0

2 1154 251 1s3 4610 1234 32s4 18434 5019 9m 15s

Ex.#4, g = 0 coarse 109 11 0(tree) 1 433 36 0

2 1729 104 1s3 6913 524 21s4 27650 2158 6m 15s

Ex.#5, g = 1 coarse 144 41 0

(p7) 1 576 94 0

2 2304 213 2s3 9216 580 26s4 36864 1656 5m 16s

Ex.#6, g = 1 coarse 144 55 0

(p7-conj.) 1 576 171 0

2 2304 389 4s3 9216 1083 50s4 36864 3374 11m 04s

Table 1. Computation times, genus 0 and 1

8.3. Accuracy. In Section 6 we gave some numerical results from our examples tosuggest the accuracy of the discrete methods. Recall that this involved both compu-tational and theoretical accuracy issues.As for computational accuracy, consider a given circle packing complex K trian-

gulating a compact surface. The associated packing label is e�ectively computable,meaning that in theory the packing algorithm can produce an approximation to thepacking label having any desired level of accuracy. In addition, geometric considera-tions and certain monotonicity properties associated with circle con�gurations provideupper estimates on the errors.In practice, the accuracy of computed packing labels is inferred from the accuracy

of \angle sums". The angle sum at a vertex v is the sum of the angles in the faces towhich v belongs; each of these angles is computed from the law of cosines using the


Example #Genus

Re�nement

Stage

Circle

count

Iterations CPU time

(min. m, sec. s)

Ex.#7, g = 2 coarse 82 36 0(peace) 1 334 98 0

2 1342 325 2s3 5374 1273 38s4 21502 4004 7m 54s

Ex.#8, g = 2 coarse 82 36 0(peace 2) 1 334 108 0

2 1342 346 2s3 5374 1100 33s4 21502 4568 9m 27s

Ex.#9, g = 2 coarse 250 96 0(generic) 1 1006 332 1s

2 4030 1146 25s3 16126 4234 6m 37s

Ex.#10, g = 3 coarse 1004 110 0(Klein) 1 4028 406 9s

2 16124 1544 2m 14s3 64508 4776 32m 18s

Ex.#11, g = 3 coarse 284 49 0(Picard : 1 1148 146 0Quine) 2 4604 494 12s

3 18428 1724 3m 8s4 73724 6049 46m 11s

Ex.#11, g = 3 coarse 140 34 0(Picard : 1 572 73 0Sabbat & 2 2300 261 3sVoevodsky) 3 9212 1048 55s

4 36860 3486 12m 23s

Ex.#12, g = 4 coarse 522 94 0(generic) 1 2106 248 3s

2 8442 952 46s3 33786 3238 11m 27s

Table 2. Computation times, higher genus

labels for v and the other two vertices. It measures how close the immediate neighborscome to wrapping round v.Inaccuracies also arises when the circle packing (or its fundamental domain) is ac-

tually laid out on the plane, the disc, or sphere (which is actually projected from thedisc). Due to inevitable errors in computed radii, there is no consistent way to laydown an associated con�guration of circles meeting the requisite tangency pattern. Inpractice, the circle centers are computed sequentially in successive generations from an


Tolerance 10�5 10�6 10�7 10�8 10�9 10�10

Iterations 1100 1978 2576 3422 4033 4776

CPU time 7m 13s 13m 19s 17m 20s 23m 14s 27m 6s 32m 18s

Table 3. E�ect of tolerance in stage-3 Klein surface times

initial circle; errors frequently tend to cancel one another, and an essentially consistentpacking generally emerges. When errors are too large, �ssures occur in the pattern asartifacts of the plotting sequence and one typically sends the radii through the packingroutines with a smaller error tolerance. (The tables above re ect 10 decimal placeaccuracy, though 5 places will generally yield a coherent packing.)

P Q

f

Figure 27. A discrete power map

8.4. Cone Points. We conclude by looking at discrete power maps. When d facesmeet at a vertex v of an equilateral surface, forming a \cone point", the local coordinatechart at v is the power map � : z 7! z6=d. Discrete structures mimic this behavior, asFigure 27 illustrates with d = 4. The packings P and Q have the same combinatorics.The radii of P prevent it from embedding in the plane, hence the arrow indicating amissing attachment. A packing operation leads to Q; the circles have been forced tochange sizes in order to atten out at v. The circle packing map f : P �! Q is thediscrete analogue of z 7! z6=d. In fact, with successive re�nements (and appropriate


normalizations), re�ned maps fn will converge to �. The \repacking" algorithm iswhere the combinatorics and the geometry debate with one another to achieve anembedding.

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