Fibonacci direct product variation tilings

TL;DR

This paper explores the classification and properties of 48 variations of Fibonacci tilings created through direct product modifications, focusing on their measure-theoretic and topological characteristics.

Contribution

It provides a detailed analysis and classification of the 48 direct product variation tilings, including their topological conjugacy and cut-and-project set descriptions.

Findings

All variations are measure-theoretically isomorphic.

They can be described as cut and project sets with specific windows.

The paper classifies these tilings with respect to topological conjugacy.

Abstract

The direct product of two Fibonacci tilings can be described as a genuine stone inflation rule with four prototiles. This rule admits various modifications, which lead to 48 different inflation rules, known as the direct product variations. They all result in tilings that are measure-theoretically isomorphic by the Halmos--von Neumann theorem. They can be described as cut and project sets with characteristic windows in a two-dimensional Euclidean internal space. Here, we analyse and classify them further, in particular with respect to topological conjugacy.