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Self-Similar Polygonal Tiling

by Michael Barnsley and Andrew Vince

Year of Award: 2018

Publication Information: The American Mathematical Monthly, Volume 124, Number 10, December 2017, Pages 905-921.

Summary: Many mathematicians are familiar with the magical beauty of Penrose tilings. These famous aperiodic tilings of the plane involve two primitive shapes: the “kite” and the “dart.” They are non-local in the sense that one cannot distinguish between the uncountably many distinct Penrose tilings based upon examining any finite region of the plane. What about similar tilings that involve only one primitive shape? This fantastic article investigates the fascinating possibilities. It begins with a careful study of the tilings that arise from the “Golden Bee,” an unusual six-sided polygon closely related to the Golden Ratio. The authors then proceed to a general construction of self-similar polygonal tilings. Remarkably, many of their polygons are irregular in appearance and some are not even convex. Nevertheless, they still manage to tile the plane in startling and unusual ways. Many examples are studied, each of which is accompanied by dazzling full-color artwork.

Response from the Authors:

We are surprised, delighted, and honored to receive this award. Since our graduate days, we have admired the aim of the Monthly: like an orchestra performance, not all the music has to be new, but the performance should be excellent. The first engaging mathematical endeavor, for one of us, was working on the problems in the Problem Section. Like other authors of articles in the Monthly, our intent was to convey our passion for the subject. Underlying the patches of tilings seen every day in art, design, architecture and nature, lie mysteries concerning the long range order of the extended tilings of the Euclidean plane. As an easily stated example, there are uncountably many distinct golden b tilings of the plane, yet it appears that there is essentially a unique tiling of a quadrant by copies of the golden b tiles. Writing this paper was a pleasure. Moreover, it has been almost a decade since the two authors began collaborating, and this has certainly been rewarding. Along with the MAA, we believe that mathematicians do well to use all possible tools to advocate for their work: pictures and words, as well as carefully crafted definitions, statements, and proofs.

About the Authors:

Michael F. Barnsley received his BA in mathematics from Oxford University in 1968 and his PhD in Theoretical Chemistry from the University of Wisconsin in 1972. He is now an emeritus professor in the Mathematical Sciences Institute at the Australian National University where he researches and teaches fractal geometry. In previous lives, he was an itinerant post-doc in England, France, and Italy (1973-1979); a professor in the School of Mathematics at Georgia Institute of Technology (1979-1991); and a cofounder and chief scientist of Iterated Systems, Inc. (1987-1998), where he led the development of fractal image compression technology. He is fascinated by the interplay of nature and mathematics.

Andrew Vince received his PhD in mathematics from the University of Michigan. A professor in the Mathematics Department at the University of Florida since 1981, he has held visiting positions at Chancellor College in Malawi, Makerer University in Uganda, Universitӓt Kaiserlautern in Germany, Dokuz Eylül University in Turkey, Massey University in New Zealand, and the Australian National University. His mathematical interests lie in combinatorics and in discrete and fractal geometry. He enjoys outdoor sports such as hiking, bicycling, and kayaking.