A “regular” pentagonal tiling of the plane
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- by Philip L. Bowers and Kenneth Stephenson PDF
- Conform. Geom. Dyn. 1 (1997), 58-86 Request permission
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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- Philip L. Bowers
- Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306–3027
- MR Author ID: 40455
- Email: firstname.lastname@example.org
- Kenneth Stephenson
- Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996–1300
- MR Author ID: 216579
- Email: email@example.com
- 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
- © Copyright 1997 American Mathematical Society
- Journal: Conform. Geom. Dyn. 1 (1997), 58-86
- MSC (1991): Primary 05B45, 30C30; Secondary 30F20
- DOI: https://doi.org/10.1090/S1088-4173-97-00014-3
- MathSciNet review: 1479069