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Conformal Geometry and Dynamics

ISSN 1088-4173



A “regular” pentagonal tiling of the plane

Authors: Philip L. Bowers and Kenneth Stephenson
Journal: Conform. Geom. Dyn. 1 (1997), 58-86
MSC (1991): Primary 05B45, 30C30; Secondary 30F20
Published electronically: November 14, 1997
MathSciNet review: 1479069
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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.

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Additional Information

Philip L. Bowers
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306–3027
MR Author ID: 40455

Kenneth Stephenson
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996–1300
MR Author ID: 216579

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
Published electronically: November 14, 1997
Additional Notes: The second author gratefully acknowledges support of the National Science Foundation and the Tennessee Science Alliance
Article copyright: © Copyright 1997 American Mathematical Society