Kenneth Stephenson

Circle packing: experiments in discrete analytic function theory

Circle packings are configurations of circles with specified patterns of tangency, and lend themselves naturally to computer experimentation and visualization. Maps between them display, with surprising faithfulness, many of the geometric properties associated with classical analytic functions. This paper introduces the fundamentals of an emerging “discrete analytic function theory” and investigates connections with the classical theory. It then describes several experiments, ranging from investigation of a conjectured discrete Koebe ¼ theorem to a multigrid method for computing discrete approximations of classical analytic functions. These experiments were performed using CirciePack, a software package described in the paper and available free of charge.

A probabilistic proof of Thurston's conjecture on circle packings

In 1985 William Thurston conjectured that one could use circle packings to approximate conformal mappings. This was confirmed by Burt Rodin and Dennis Sullivan with a proof which relied on the hexagonal nature of the packings involved. This paper provides a probabilistic proof which accomodates more general combinatorics by analysing the dynamics of invididual circle packings. One can use reversible Markov processes to model the movement of curvature and hyperbolic area among the circles of a packing as it undergoes adjustement, much as one can use them to model the movement of current in an electrical circuit. Each circle packing has a Markov process intimately coupled to its geometry; the crucial local rigidity of the packing then appears as a a Harnack inequality for discrete harmonic functions of the process.

A \REGULAR" PENTAGONAL TILING OF THE PLANE

The paper introduces conformal tilings, wherein tiles have spec- ied conformal shapes. The principal example involves conformally regular pentagons which tile the plane in a pattern generated by a subdivision rule. Combinatorial symmetries imply rigid conformal symmetries, which in turn illustrate a new type of tiling self-similarity. In parallel with the conformal tilings, the paper develops discrete tilings based on circle packings. These faithfully reflect the key features of the theory and provide the tiling illustra- tions of the paper. Moreover, it is shown that under renement the discrete tiles converge to their true conformal shapes, shapes for which no other ap- proximation techniques are known. The paper concludes with some further examples which may contribute to the study of tilings and shinglings being carried forward by Cannon, Floyd, and Parry.

Chapter 11 – Circle Packing and Discrete Analytic Function Theory

Abstract Circle packings – configurations of circles with specified patterns of tangency – came to prominence with analysts in 1985 when Thurston conjectured that maps between such configurations would approximate conformal maps. The proof by Rodin and Sullivan launched a topic which has grown steadily as ever more connections with analytic functions and conformal structures have emerged. Indeed, the core ideas have matured to the point that one can fairly claim that circle packing provides a discrete analytic function theory .

The set of circle packing points in the Teichmüller space of a surface of finite conformal type is dense

W. Thurston initiated interest in circle packings with his provocative suggestion at the International Symposium in Celebration of the Proof of the Bieberbach Conjecture (Purdue University, 1985) that a result of Andreev[2] had an interpretation in terms of circle packings that could be applied systematically to construct geometric approximations of classical conformal maps. Rodin and Sullivan [11] verified Thurstons conjecture in the setting of hexagonal packings, and more recently Stephenson [12] has announced a proof for more general combinatorics. Inspired by Thurstons work and motivated by the desire to discover and exploit discrete versions of classical results in complex variable theory, Beardon and Stephenson [4, 5] initiated a study of the geometry of circle packings, particularly in the hyperbolic setting. This topic is a recent example among many of the beautiful and sometimes unexpected interplay between Geometry, Topology, and Cornbinatorics that is evident in much of the topological research of the past decade, and that has its roots in the seminal work of the great geometrically-minded mathematicians – Riemann, Klein, Poincare – of the last century. A somewhat surprising example of this interplay concerns us here; namely, the fact that the combinatorial information encoded in a simplicial triangulation of a topological surface can determine its geometry.

Spiral hexagonal circle packings in the plane

We discuss an intriguing geometric algorithm which generates infinite spiral patterns of packed circles in the plane. Using Kleinian group and covering theory, we construct a complex parametrization of all such patterns and characterize those whose circles have mutually disjoint interiors. We prove that these ‘coherent’ spirals, along with the regular hexagonal packing, give all possible hexagonal circle packings in the plane. Several examples are illustrated.

Circle packings in surfaces of finite type: an in situ approach with applications to moduli

ROBERT BRINKS [lo, 1 l] used circle packings to parametrize the deformation space of a Kleinian group and applied his results to prove that the “circle packing points” (closed Riemann surfaces that can be filled by circle packings) form a dense subset of moduli space. In [9], the authors combined techniques of Brooks, Thurston [28], and BeardonStephenson [4, 5) to extend Brooks’ result to surfaces of finite conformal type, closed surfaces with a finite number of punctures. The methods there involve the in situ manipulation of circle packings and rely heavily on certain canonical Brooks’ packings of quadrilaterals as developed in [lo], along with canonical infinite packings of cusp regions. The latter are rigid, but the Brooks’ packings act like shock absorbers, permitting the small adjustments to modulus that lead to circle packing points. Our purposes here are threefold: first, to develop more fully and systematically the in situ approach to circle packings in hyperbolic surfaces begun in [9]; second, to define canonical packings of annuli that have a flexibility reminiscent of Brooks’ packings of quadrilaterals; and thiid, to apply these in an extension of Brooks’ result to surfaces of finite topological type, closed surfaces having a finite number of punctures and a finite number of half-annular ends. Our main result is the following.

Inner divisors and composition operators

Abstract This characterization is stated and proved in much greater generality, beginning with subspaces of arbitrary L ∞ spaces and using a notion of inner divisors. Among the consequences are a variant of Wermers theorem on embedding disks in maximal ideal spaces and a result on linear isometrics of H ∞ . The ranges of composition operators C I when I is not necessarily inner are characterized. In particular, relatively closed sets E ⊂ c Δ of zero logarithmic and zero analytic capacity are characterized in terms of the algebras of bounded analytic functions on A invariant under corresponding Fuchsian groups. The paper concludes with an example of a uniformly closed subalgebra of H ∞ which contains the constants and the inner factors of its members, but is not of the form H ∞ I .

A linearized circle packing algorithm

Abstract This paper presents a geometric algorithm for approximating radii and centers for a variety of univalent circle packings, including maximal circle packings on the unit disc and the sphere and certain polygonal circle packings in the plane. This method involves an iterative process which alternates between estimates of circle radii and locations of circle centers. The algorithm employs sparse linear systems and in practice achieves a consistent linear convergence rate that is far superior to traditional packing methods. It is deployed in a MATLAB® package which is freely available. This paper gives background on circle packing, a description of the linearized algorithm, illustrations of its use, sample performance data, and remaining challenges.

Curvature Flow in Conformal Mapping

We use a simple example to introduce a notion of curvature flow in the conformal mapping of polyhedral surfaces. The inquiry was motivated by experiments with discrete conformal maps in the sense of circle packing. We describe the classical theory behind these flows and demonstrate how to modify the Schwarz-Christoffel method to obtain classical numerical confirmation. We close with some additional examples.