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.

Author Bio

Vivian Olsiewski Healey started this research at the Canisius College REU and subsequently continued it at the University of Notre Dame with the guidance of her advisor, Professor Arlo Caine. The results presented here formed a portion of her senior honors thesis. She is now a math graduate student at Brown University studying discrete geometry and sphere packing.