Conformal Geometry and Dynamics of the American Mathematical Society
The paper introduces conformal tilings, wherein tiles have specified conformal shapes. The princi... more The paper introduces conformal tilings, wherein tiles have specified 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 illustrations of the paper. Moreover, it is shown that under refinement the discrete tiles converge to their true conformal shapes, shapes for which no other approximation 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.
Conformal Geometry and Dynamics of the American Mathematical Society
This is the second in a series of papers on conformal tilings. The overriding themes here are loc... more This is the second in a series of papers on conformal tilings. The overriding themes here are local isomorphisms, hierarchical structures, and the conformal "type" problem. Conformal tilings were introduced by the authors in 1997 with a conformally regular pentagonal tiling of the plane. This and even more intricate hierarchical patterns arise when tilings are repeatedly subdivided. Deploying a notion of expansion complexes, we build two-way infinite combinatorial hierarchies and then study the associated conformal tilings. For certain subdivision rules the combinatorial hierarchical properties are faithfully mirrored in their concrete conformal realizations. Examples illustrate the theory throughout the paper. In particular, we study parabolic conformal hierarchies that display periodicities realized by Möbius transformations, motivating higher level hierarchies that will emerge in the next paper of this series.
... This conjecture was subsequently proven by Burt Rodin and Dennis Sullivan [6]. Thurston terme... more ... This conjecture was subsequently proven by Burt Rodin and Dennis Sullivan [6]. Thurston termed the conjectured result a "Finite Riemann Mapping Theorem", theintuition being, at least in part, that since the conformal map carries infinitesimal circles to infinitesimal circles, one ...
... JW Cannon Department of Mathematics Brigham Young University Provo, Utah 84602 USA e-mail: Ca... more ... JW Cannon Department of Mathematics Brigham Young University Provo, Utah 84602 USA e-mail: [email protected] WJ Floyd Department of Mathematics Virginia Tech Blacksburg, VA 24061 USA e-mail: [email protected] ... by Wade Trappe and Lawrence Washington ...
This paper presents a geometric algorithm for approximating radii and centers for a variety of un... more 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 R 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.
LetF 0 , F i ,F 2 ,... be relatively closed disjoint subsets of the unit disk % = {\z\ < 1} with ... more LetF 0 , F i ,F 2 ,... be relatively closed disjoint subsets of the unit disk % = {\z\ < 1} with the closure in Ql of F o u F t u F 2 u ... having logarithmic capacity zero. An inner function 4> is constructed with the property that for each n, a lies in F n if and only if a is assumed as a value by (f) precisely n times, counting multiplicities. A bounded analytic function on the unit disk % is called an inner function provided its radial limits have modulus 1 almost everywhere on the unit circle. Every inner function <p factors uniquely as a product of inner functions <Kz) = B(z)S(z), ze<%, where B is the Blaschke factor of $. Specifically, if <fr has a zero of order k at z = 0 and if a u a 2 ,... are its other zeros, counting multiplicities, then a.-a.-z r7ĵ \aj\ l
In 1985 William Thurston conjectured that one could use circle packings to approximate conformal ... more 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 individual 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. Contents *The author gratefully aknowledges the support of the National Science Foundation and the Tennessee Science Alliance and the hospitality of the Seminario Matematico e Fisico di Milano.
Transactions of the American Mathematical Society, 1988
An explicit construction using Riemann surfaces and Brownian motion is given for an inner functio... more An explicit construction using Riemann surfaces and Brownian motion is given for an inner function in the unit disc which is not a finite Blaschke product yet belongs to the little Bloch space 3 §^. In addition to showing how an inner function can meet the geometric conditions for 3Sq, this example settles an open question concerning the finite ranges of inner functions: the values which it takes only finitely often are dense in the disc.
Conformal Geometry and Dynamics of the American Mathematical Society
The paper introduces conformal tilings, wherein tiles have specified conformal shapes. The princi... more The paper introduces conformal tilings, wherein tiles have specified 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 illustrations of the paper. Moreover, it is shown that under refinement the discrete tiles converge to their true conformal shapes, shapes for which no other approximation 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.
Conformal Geometry and Dynamics of the American Mathematical Society
This is the second in a series of papers on conformal tilings. The overriding themes here are loc... more This is the second in a series of papers on conformal tilings. The overriding themes here are local isomorphisms, hierarchical structures, and the conformal "type" problem. Conformal tilings were introduced by the authors in 1997 with a conformally regular pentagonal tiling of the plane. This and even more intricate hierarchical patterns arise when tilings are repeatedly subdivided. Deploying a notion of expansion complexes, we build two-way infinite combinatorial hierarchies and then study the associated conformal tilings. For certain subdivision rules the combinatorial hierarchical properties are faithfully mirrored in their concrete conformal realizations. Examples illustrate the theory throughout the paper. In particular, we study parabolic conformal hierarchies that display periodicities realized by Möbius transformations, motivating higher level hierarchies that will emerge in the next paper of this series.
... This conjecture was subsequently proven by Burt Rodin and Dennis Sullivan [6]. Thurston terme... more ... This conjecture was subsequently proven by Burt Rodin and Dennis Sullivan [6]. Thurston termed the conjectured result a "Finite Riemann Mapping Theorem", theintuition being, at least in part, that since the conformal map carries infinitesimal circles to infinitesimal circles, one ...
... JW Cannon Department of Mathematics Brigham Young University Provo, Utah 84602 USA e-mail: Ca... more ... JW Cannon Department of Mathematics Brigham Young University Provo, Utah 84602 USA e-mail: [email protected] WJ Floyd Department of Mathematics Virginia Tech Blacksburg, VA 24061 USA e-mail: [email protected] ... by Wade Trappe and Lawrence Washington ...
This paper presents a geometric algorithm for approximating radii and centers for a variety of un... more 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 R 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.
LetF 0 , F i ,F 2 ,... be relatively closed disjoint subsets of the unit disk % = {\z\ < 1} with ... more LetF 0 , F i ,F 2 ,... be relatively closed disjoint subsets of the unit disk % = {\z\ < 1} with the closure in Ql of F o u F t u F 2 u ... having logarithmic capacity zero. An inner function 4> is constructed with the property that for each n, a lies in F n if and only if a is assumed as a value by (f) precisely n times, counting multiplicities. A bounded analytic function on the unit disk % is called an inner function provided its radial limits have modulus 1 almost everywhere on the unit circle. Every inner function <p factors uniquely as a product of inner functions <Kz) = B(z)S(z), ze<%, where B is the Blaschke factor of $. Specifically, if <fr has a zero of order k at z = 0 and if a u a 2 ,... are its other zeros, counting multiplicities, then a.-a.-z r7ĵ \aj\ l
In 1985 William Thurston conjectured that one could use circle packings to approximate conformal ... more 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 individual 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. Contents *The author gratefully aknowledges the support of the National Science Foundation and the Tennessee Science Alliance and the hospitality of the Seminario Matematico e Fisico di Milano.
Transactions of the American Mathematical Society, 1988
An explicit construction using Riemann surfaces and Brownian motion is given for an inner functio... more An explicit construction using Riemann surfaces and Brownian motion is given for an inner function in the unit disc which is not a finite Blaschke product yet belongs to the little Bloch space 3 §^. In addition to showing how an inner function can meet the geometric conditions for 3Sq, this example settles an open question concerning the finite ranges of inner functions: the values which it takes only finitely often are dense in the disc.
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