Forcing nonperiodic tilings with one tile using a seed

TL;DR

This paper explores the possibility of using a single tile with seed placement to generate nonperiodic tilings, advancing the understanding of the einstein problem without relying solely on decorated tiles or strict matching rules.

Contribution

It introduces a class of spiral tilings and a non-spiral example linked to a seed-based approach, offering new insights into nonperiodic tilings with minimal tile types.

Findings

Linked spiral tilings to seed-based nonperiodicity

Identified weaker forms of the einstein problem

Discussed classical and new matching rules

Abstract

The so-called "einstein problem" (a pun playing with the famous scientist's name and the German term "ein Stein" for "one stone") asks for a simply connected prototile only allowing nonperiodic tilings without need of any matching rule. So far, researchers come only close to this demand by defining decorated prototiles forcing nonperiodicity of any generated tiling using matching rules. In this paper a class of spiral tilings (and one non-spiral example) is linked to a weaker form of the einstein problem where one or several seed tiles are used. Furthermore, the classical types of matching rules are listed and some new types are discussed.