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Conformal Geometry and Dynamics

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Conformal tilings I: foundations, theory, and practice


Authors: Philip L. Bowers and Kenneth Stephenson
Journal: Conform. Geom. Dyn. 21 (2017), 1-63
MSC (2010): Primary 52C23, 52C26; Secondary 52C45, 68U05
DOI: https://doi.org/10.1090/ecgd/304
Published electronically: January 10, 2017
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Abstract: This paper opens a new chapter in the study of planar tilings by introducing conformal tilings. These are similar to traditional tilings in that they realize abstract patterns of combinatorial polygons as concrete patterns of geometric shapes, the tiles. In the conformal case, however, these geometric tiles carry prescribed conformal rather than prescribed euclidean structure. The authors develop the topic from the ground up: definitions and terminology, basic theory on existence, uniqueness and properties, numerous experiments and examples, comparisons to traditional tilings, patterns unique to conformal tiling, and details on computability through circle packing. Special attention is placed on aperiodic hierarchical tilings and on connections between abstract combinatorics on one hand and their geometric realizations on the other. Many of the motivations for studying tilings remain unchanged, not least being the pure beauty and intricacy of the patterns.


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Additional Information

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

Kenneth Stephenson
Affiliation: Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996
Email: kstephe2@utk.edu

DOI: https://doi.org/10.1090/ecgd/304
Received by editor(s): November 20, 2014
Received by editor(s) in revised form: August 26, 2015
Published electronically: January 10, 2017
Additional Notes: The second author gratefully acknowledges support of a Simons Foundation Collaboration Grant
Article copyright: © Copyright 2017 American Mathematical Society