English

Forcing nonperiodic tilings with one tile using a seed

Metric Geometry 2025-06-24 v2 Combinatorics

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.

Keywords

Cite

@article{arxiv.2109.09384,
  title  = {Forcing nonperiodic tilings with one tile using a seed},
  author = {Bernhard Klaassen},
  journal= {arXiv preprint arXiv:2109.09384},
  year   = {2025}
}

Comments

18 pages, 19 figures, to appear in: European Journal of Combinatorics 2021

R2 v1 2026-06-24T06:07:49.446Z