Conformal Geometry and Dynamics

Author(s): Philip L. Bowers; Kenneth Stephenson
Journal: Conform. Geom. Dyn. 1 (1997), 58-86.
MSC (1991): Primary 05B45, 30C30; Secondary 30F20
Posted: November 14, 1997
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Abstract: 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.

1.: L. V. Ahlfors and L. Sario, Riemann surfaces, Princeton University Press, Princeton, New Jersey, 1960. MR 22:5729
2.: A. F. Beardon, A primer on Riemann surfaces, London Math. Soc. Lecture Note Series, vol. 78, Cambridge University Press, Cambridge, 1984. MR 87h:30090
3.: Alan F. Beardon and Kenneth Stephenson, The uniformization theorem for circle packings, Indiana Univ. Math. J. 39 (1990), 1383-1425. MR 92b:52038
4.: -, Circle packings in different geometries, Tohoku Math. J. 43 (1991), 27-36. MR 92a:52023
5.: Philip L. Bowers and Kenneth Stephenson, A branched Andreev-Thurston theorem for circle packings of the sphere, Proc. London Math. Soc. (3) 73 (1996), 185-215. MR 97d:52027
6.: James Cannon, The combinatorial Riemann mapping theorem, Acta Mathematica 173 (1994), 155-234. MR 95k:30046
7.: James W. Cannon, The theory of negatively curved spaces and groups, Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces (Tim Bedford, Michael Keane, and Caroline Series, eds.), Oxford Science Publications, Oxford-New York-Tokyo, 1991 (comments on circle packing, pp. 349-351). CMP 92:02
8.: Tomasz Dubejko and Kenneth Stephenson, Circle packing: experiments in discrete analytic function theory, Experimental Mathematics 4 (1995), no. 4, 307-348. MR 97f:52027
9.: Zheng-Xu He and Oded Schramm, Hyperbolic and parabolic packings, Discrete Computational Geom. 14 (1995), 123-149. MR 96h:52017
10.: -, On the convergence of circle packings to the Riemann map, Invent. Math. 125 (1996), 285-305. MR 97i:30009
11.: J. W. Cannon and E. L. Swenson, Recognizing constant curvature discrete groups in dimension 3, preprint, 1996.
12.: J. W. Cannon, W. J. Floyd, and Walter Parry, On the conformal invariance of tiling-systems, preprint, 1996.
13.: -, Sufficiently rich families of planar rings, preprint, 1996.
14.: -, Squaring rectangles: the finite Riemann Mapping Theorem, Contemporary Math. 169 (1994), 133-212. MR 95g:20045
15.: Richard Kenyon, Tilings and discrete Dirichlet problems, preprint, 1996.
16.: O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Springer-Verlag, New York, 1973. MR 49:9202
17.: Olli Lehto, Univalent functions and Teichmüller spaces, Springer-Verlag, New York, 1987. MR 88f:30073
18.: Gareth McCaughan, A recurrence/transience result for circle packings, Proc. Amer. Math. Soc., to appear. CMP 95:11
19.: Burt Rodin and Dennis Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geometry 26 (1987), 349-360. MR 90c:30007
20.: William P. Thurston, Groups, tilings and finite state automata, The Geometry Center Research Report GCG-1 (1989), Summer 1989 AMS Colloquium Lectures.

Retrieve articles in Conformal Geometry and Dynamics with MSC (1991): 05B45, 30C30, 30F20

Philip L. Bowers
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306--3027
Email: bowers@gauss.math.fsu.edu

Kenneth Stephenson
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996--1300
Email: kens@math.utk.edu

DOI: 10.1090/S1088-4173-97-00014-3
PII: S 1088-4173(97)00014-3
Keywords: Tiling, subdivision rules, circle packing, conformal maps
Received by editor(s): April 28, 1997
Received by editor(s) in revised form: August 21, 1997
Posted: November 14, 1997
Additional Notes: The second author gratefully acknowledges support of the National Science Foundation and the Tennessee Science Alliance
Copyright of article: Copyright 1997, American Mathematical Society

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