A Family of Recompositions of the Penrose Aperiodic Protoset and Its Dynamic Properties

Abstract

This paper describes 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. An interation process is defined on the space of Ammann tilings that produces a new Ammann tiling from an existing one, and it is shown that this process runs in parallel to Penrose deflation. Furthermore, by characterizing Ammann tilings based on their corresponding Penrose tilings and the location of the added vertex that defines the recomposition process, we show that this process proceeds to a limit for the local geometry.