This paper examines a recomposition of the rhombic Penrose aperiodic protoset due to Robert Ammann. We show that the three prototiles that result from the recomposition form an aperiodic protoset in their own right without adjacency rules and that every tiling admitted by this protoset (here called an Ammann tiling) is mutually locally derivable with a Penrose tiling. Although these Ammann tilings are not self-similar, an iteration process inspired by Penrose composition is defined on the set of Ammann tilings that produces a new Ammann tiling from an existing one, and the exact relationship to Penrose composition is examined. Furthermore, by characterizing each Ammann tiling based on a corresponding Penrose tiling and the location of the added vertex that defines the recomposition process, we show that repeated Ammann iteration proceeds to a limit for the local geometry.
Dr. Arlo Caine, Department of Mathematics and Statistics, Cal Poly Pomona
Healey, Vivian Olsiewski
"A Family of Recompositions of the Penrose Aperiodic Protoset and its Dynamic Properties,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 12
, Article 7.
Available at: https://scholar.rose-hulman.edu/rhumj/vol12/iss2/7