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14
Circle packing: a mathematical tale
 Notices Amer. Math. Soc
, 2003
"... The circle is arguably the most studied object in mathematics, yet I am here to tell the tale of circle packing, a topic which is likely to be new to most readers. These packings are configurations of circles satisfying preassigned patterns of tangency, and we will be concerned here with their creat ..."
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Cited by 27 (2 self)
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The circle is arguably the most studied object in mathematics, yet I am here to tell the tale of circle packing, a topic which is likely to be new to most readers. These packings are configurations of circles satisfying preassigned patterns of tangency, and we will be concerned here with their creation, manipulation, and interpretation. Lest we get off on the wrong foot, I should caution that this is NOT twodimensional “sphere ” packing: rather than being fixed in size, our circles must adjust their radii in tightly choreographed ways if they hope to fit together in a specified pattern. In posing this as a mathematical tale, I am asking the reader for some latitude. From a tale you expect truth without all the details; you know that the storyteller will be playing with the plot and timing; you let pictures carry part of the story. We all hope for deep insights, but perhaps sometimes a simple story with a few new twists is enough—may you enjoy this tale in that spirit. Readers who wish to dig into the details can consult the “Reader’s Guide ” at the end. Once Upon a Time … From wagon wheel to mythical symbol, predating history, perfect form to ancient geometers, companion to π, the circle is perhaps the most celebrated object in mathematics.
The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity
 ICM Proceedings
, 2006
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Expansion complexes for finite subdivision rules
 I, Conform. Geom. Dyn
"... Abstract. This paper gives applications of earlier work of the authors on the use of expansion complexes for studying conformality of finite subdivision rules. The first application is that a onetile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching 0) has an ..."
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Cited by 12 (5 self)
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Abstract. This paper gives applications of earlier work of the authors on the use of expansion complexes for studying conformality of finite subdivision rules. The first application is that a onetile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching 0) has an invariant partial conformal structure, and hence is conformal. The paper next considers onetile single valence finite subdivision rules. It is shown that an expansion map for such a finite subdivision rule can be conjugated to a linear map, and that the finite subdivision rule is conformal exactly when this linear map is either a dilation or has eigenvalues that are not real. Finally, an example is given of an irreducible finite subdivision rule that has a parabolic expansion complex and a hyperbolic expansion complex. 1.
CONSTRUCTING SUBDIVISION RULES FROM RATIONAL MAPS
"... Abstract. This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f is a critically finite rational map with no periodic critical points, then for any sufficiently large integer n the iterate f ◦n is the subdivision map of ..."
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Abstract. This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f is a critically finite rational map with no periodic critical points, then for any sufficiently large integer n the iterate f ◦n is the subdivision map of a finite subdivision rule. We are interested here in connections between finite subdivision rules and rational maps. Finite subdivision rules arose out of our attempt to resolve Cannon’s Conjecture: If G is a Gromovhyperbolic group whose space at infinity is a 2sphere, then G has a cocompact, properly discontinuous, isometric action on hyperbolic 3space. Cannon’s Conjecture can be reduced (see, for example, the CannonSwenson paper [5]) to a conjecture about (combinatorial) conformality for the action of such a group G on its space at infinity, and finite subdivision rules were developed to give models for the action of a Gromovhyperbolic group on the 2sphere at infinity. There is also a connection between finite subdivision rules and rational
SUBDIVISION RULES AND VIRTUAL ENDOMORPHISMS
"... Abstract. Suppose f: S 2 → S 2 is a postcritically finite branched covering without periodic branch points. If f is the subdivision map of a finite subdivision rule with mesh going to zero combinatorially, then the virtual endomorphism on the orbifold fundamental group associated to f is contracting ..."
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Abstract. Suppose f: S 2 → S 2 is a postcritically finite branched covering without periodic branch points. If f is the subdivision map of a finite subdivision rule with mesh going to zero combinatorially, then the virtual endomorphism on the orbifold fundamental group associated to f is contracting. This is a first step in a program to clarify the relationships among various notions of expansion for noninvertible dynamical systems with branching behavior. Let T 2 = R 2 /Z 2 denote the real twodimensional torus, equipped with the Euclidean Riemannian metric ds 2 inherited from the usual metric on R 2, and suppose f: T 2 → T 2 is a continuous orientationpreserving covering map. It is wellknown that a necessary and sufficient condition for f to be homotopic to a covering map
DIMENSION OF ELLIPTIC HARMONIC MEASURE OF SNOWSPHERES
, 2009
"... A metric space S is called a quasisphere if there is a quasisymmetric homeomorphism f: S 2 → S. We consider the elliptic harmonic measure, i.e., the push forward of 2dimensional Lebesgue measure by f. It is shown that for certain self similar quasispheres S (snowspheres) the dimension of the ellipt ..."
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A metric space S is called a quasisphere if there is a quasisymmetric homeomorphism f: S 2 → S. We consider the elliptic harmonic measure, i.e., the push forward of 2dimensional Lebesgue measure by f. It is shown that for certain self similar quasispheres S (snowspheres) the dimension of the elliptic harmonic measure is strictly less than the Hausdorff dimension of S. This result is obtained by representing the self similarity of a snowsphere by a postcritically finite rational map, and showing a corresponding result for such maps. As a corollary a metric characterization of Lattès maps is obtained. Furthermore, a method to compute the dimension of elliptic harmonic measure numerically is presented, along with the (numerically computed) values for certain examples.
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, 2012
"... This Dissertation is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact scholarsarchive@byu.edu. ..."
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This Dissertation is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact scholarsarchive@byu.edu.