A Probabilistic Proof of Thurston's Conjecture on Circle Packings
نویسندگان
چکیده
منابع مشابه
Circle Packings in the Approximation of Conformal Mappings
Connections between circle packings and analytic functions were first suggested by William Thurston [T2], who conjectured that the conformai mapping of a simply connected plane domain Q to the unit disc A could be approximated by manipulating hexagonal circle configurations lying in Q. The conjecture was confirmed by Rodin and Sullivan [RS]. Their proof relies heavily on the hexagonal combinato...
متن کاملSolving Beltrami Equations by Circle Packing
We use Andreev-Thurston's theorem on the existence of circle packings to construct approximating solutions to the Beltrami equations on Riemann surfaces. The convergence of the approximating solutions on compact subsets will be shown. This gives a constructive proof of the existence theorem for Beltrami equations.
متن کاملOn the Convergence of Circle Packings to the Riemann Map
Rodin and Sullivan (1987) proved Thurston’s conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby providing a refreshing geometric view of Riemann’s Mapping Theorem. We now present a new proof of the Rodin–Sullivan theorem. This proof is based on the argument principle, and has the following virtues. 1. It applies to more general packings. The Ro...
متن کاملRigidity of Infinite (circle) Packings
The nerve of a packing is a graph that encodes its combinatorics. The vertices of the nerve correspond to the packed sets, and an edge occurs between two vertices in the nerve precisely when the corresponding sets of the packing intersect. The nerve of a circle packing and other well-behaved packings, on the sphere or in the plane, is a planar graph. It was an observation of Thurston [Thl, Chap...
متن کاملOn the Local-global Principle for Integral Apollonian 3-circle Packings
In this paper we study the integral properties of Apollonian-3 circle packings, which are variants of the standard Apollonian circle packings. Specifically, we study the reduction theory, formulate a local-global conjecture, and prove a density one version of this conjecture. Along the way, we prove a uniform spectral gap for the congruence towers of the symmetry group.
متن کامل