Circle Packing: A Mathematical Tale Kenneth Stephenson he circle is arguably the most studied ob- There is indeed a long tradition behind our ject in mathematics, yet I am here to tell story. Who can date the most familiar of circle the tale of circle packing, a topic which packings, the “penny-packing” seen in the back- is likely to be new to most readers. These ground of Figure 1? Even the “apollonian gasket” Tpackings are configurations of circles (a) has a history stretching across more than two satisfying preassigned patterns of tangency, and we millennia, from the time of Apollonius of Perga to will be concerned here with their creation, manip- the latest research on limit sets. And circles were ulation, and interpretation. Lest we get off on the never far from the classical solids, as suggested by wrong foot, I should caution that this is NOT two- the sphere caged by a dodecahedron in (b). Equally dimensional “sphere” packing: rather than being ancient is the αρβηλoς´ or “shoemaker’s knife” in fixed in size, our circles must adjust their radii in (c), and it is amazing that the Greeks had already tightly choreographed ways if they hope to fit proved that the nth circle c has its center n together in a specified pattern. n In posing this as a mathematical tale, I am ask- diameters from the base. This same result can be ing the reader for some latitude. From a tale you found, beautifully illustrated, in sangaku, wooden expect truth without all the details; you know that temple carvings from seventeenth-century Japan. the storyteller will be playing with the plot and tim- In comparatively recent times, Descartes estab- ing; you let pictures carry part of the story. We all lished his Circle Theorem for “quads” like that in hope for deep insights, but perhaps sometimes a (d), showing that the bends bj (reciprocal radii) of simple story with a few new twists is enough—may four mutually tangent circles are related by + + + 2 = 2 + 2 + 2 + 2 you enjoy this tale in that spirit. Readers who wish (b1 b2 b3 b4) 2(b1 b2 b3 b4) . Nobel to dig into the details can consult the “Reader’s laureate F. Soddy was so taken by this result that Guide” at the end. he rendered it in verse: The Kiss Precise (1936). With such a long and illustrious history, is it Once Upon a Time … surprising or is it inevitable that a new idea about From wagon wheel to mythical symbol, predating circles should come along? history, perfect form to ancient geometers, com- panion to π, the circle is perhaps the most celebrated Birth of an Idea object in mathematics. One can debate whether we see many truly new Kenneth Stephenson is professor of mathematics at the Uni- ideas in mathematics these days. With such a rich versity of Tennessee, Knoxville. His email address is history, everything has antecedents—who is to say, [email protected]. for example, what was in the lost books of Apollo- The author gratefully acknowledges support of the National nius and others? Nonetheless, some topics have Science Foundation, DMS-0101324. fairly well-defined starting points. 1376 NOTICES OF THE AMS VOLUME 50, NUMBER 11 Our story traces its origin to William Thurston’s famous Notes. In constructing 3-manifolds, Thurston proves that associated with any triangulation of a sphere is a “circle packing”, that is, a configuration of circles which are tangent with one another in the pattern of the triangulation. Moreover, this packing is unique up to Möbius transformations and inver- sions of the sphere. This is a remarkable fact, for the pattern of tangencies—which can be arbitrarily in- (a) (b) tricate—is purely abstract, yet the circle packing su- perimposes on that pattern a rigid geometry. This is a main theme running through our story, that circle packing provides a bridge between the combinatoric on the one hand and the geometric on the other. Although known in the topological community C2 C3 through the Notes, circle packings reached a sur- C1 prising new audience when Thurston spoke at the 1985 Purdue conference celebrating de Branges’s C proof of the Bieberbach Conjecture. Thurston had 0 recognized in the rigidity of circle packings some- thing like the rigidity shown by analytic functions, and in a talk entitled “A finite Riemann mapping (c) (d) theorem” he illustrated with a scheme for con- structing conformal maps based on circle packings. He made an explicit conjecture, in fact, that his “fi- Figure 1. A long tradition. nite” maps would converge, under refinement, to a classical conformal map, the type his Purdue au- bear with me while I introduce the essentials needed to follow the story. dience knew well. As if that weren’t enough, Thurston even threw in an iterative numerical • Complex: The tangency patterns for scheme for computing these finite Riemann map- circle packings are encoded as abstract pings in practice, with pictures to back it all up. simplicial 2-complexes K; we assume K So this was the situation for your storyteller as he is (i.e., triangulates) an oriented topo- listened to Thurston’s Purdue talk: a most surpris- logical surface. ing theorem and beautiful pictures about patterns of circles, an algorithm for actually computing them, • Packing: A circle packing P for K is a and a conjectured connection to a favorite topic, an- configuration of circles such that for alytic function theory. This storyteller was hooked! each vertex v ∈ K there is a corre- As for antecedents, Thurston found that his the- sponding circle cv , for each edge orem on packings of the sphere followed from prior v,u∈K the circles cv and cu are (ex- work by E. Andreev on reflection groups, and some ternally) tangent, and for each posi- years later Reiner Kuhnau pointed out a 1936 proof tively oriented face v,u,w∈K the by P. Koebe, so I refer to it here as the K-A-T (Koebe- mutually tangent triple of circles Andreev-Thurston) Theorem. Nonetheless, for our cv ,cu,cw is positively oriented. purposes the new idea was born at Purdue in 1985, and our tale can begin. • Label: A label R for K is a collection of putative radii, with R(v) denoting the Internal Development label for vertex v. Once a topic is launched and begins to attract a com- Look to Figure 2 for a very simple first example. munity, it also begins to develop an internal ecology: Here K is a closed topological disc and P is a special language, key examples and theorems, central euclidean circle packing for K. I show the carrier themes, and—with luck—a few gems to amaze the of the packing in dashed lines to aid in matching uninitiated. circles to their vertices in K; there are 9 interior and The main players in our story, circles, are well 8 boundary circles. Of course the question is how known to us all, and we work in familiar geomet- to find such packings, and the key is the label R ric spaces: the sphere P, the euclidean plane C, and of radii—knowing K, the tangencies, and R, the the hyperbolic plane as represented by the unit disc sizes, it is a fairly simple matter to lay out the D. Working with configurations of circles, how- circles themselves. In particular, circle centers play ever, will require a modest bit of bookkeeping, so a secondary role. The computational effort in DECEMBER 2003 NOTICES OF THE AMS 1377 K P Figure 2. Compare packing P to its complex K. circle packing lies mainly in computing labels. It is • Miscellany: A packing is univalent if in these computations that circle packing directly its circles have mutually disjoint inte- confronts geometry and the local-to-global theme riors. A branch circle cv in a packing P plays out. Here, very briefly, is what is involved. is an interior circle whose angle sum is 2πn for integer n ≥ 2; that is, its petals • Flower: A circle cv and the circles tan- wrap n times around it. A packing P is gent to it are called a flower. The or- branched if it has one or more branch dered chain c , ··· ,c of tangent cir- v1 vk circles; otherwise it is locally univalent. cles, the petals, is closed when v is an (Caution: Global univalence is assumed interior vertex of K. for all circle packings in some parts of the literature, but not here.) Möbius Angle Sum: The angle sum θ (v) for • R transformations map packings to pack- vertex v, given label R, is the sum of the ings; a packing is said to be essentially angles at c in the triangles formed by v unique with some property if it is the triples c ,c ,c in its flower. v vj vj+1 unique up to such transformations. In Angle sums are computed via the the disc, a horocycle is a circle inter- appropriate law of cosines; in the nally tangent to the unit circle and may euclidean case, for example, be treated as a circle of infinite hyper- bolic radius. (R(v) + R(u))2 + (R(v) + R(w))2 − (R(u) + R(w))2 θ (v) = arccos , R 2(R(v) + R(u))(R(v) + R(w)) v,u,w You are now ready for the internal art of circle packing. Someone hands you a complex K. Do there where the sum is over all faces con- exist any circle packings for K? How many? In which taining v.