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

TL;DR

This paper introduces a recomposition of the Penrose aperiodic protoset, demonstrating that the new protoset is inherently aperiodic and exploring its dynamic properties through an iterative process parallel to Penrose deflation.

Contribution

It presents a novel recomposition of the Penrose protoset that forms an independent aperiodic set and analyzes its dynamic behavior and limiting geometry.

Findings

The recomposed protoset is inherently aperiodic without adjacency rules.

An iterative process on Ammann tilings parallels Penrose deflation.

The process converges to a limit in local geometry.

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.