An aperiodic monotile that forces nonperiodicity through a local dendritic growth rule

Mampusti, M. and Whittaker, M. F. (2019) An aperiodic monotile that forces nonperiodicity through a local dendritic growth rule. arXiv, (Unpublished)

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Publisher's URL: https://arxiv.org/abs/1903.01158

Abstract

We introduce a new type of aperiodic hexagonal monotile; a prototile that admits infinitely many tilings of the plane, but any such tiling lacks any translational symmetry. Adding a copy of our monotile to a patch of tiles must satisfy two rules that apply only to adjacent tiles. The first is inspired by the Socolar--Taylor monotile, but can be realised by shape alone. The second is a local growth rule; a direct isometry of our monotile can be added to any patch of tiles provided that a tree on the monotile connects continuously with a tree on one of its neighbouring tiles. This condition forces tilings to grow along dendrites, which ultimately results in nonperiodic tilings. Our local growth rule initiates a new method to produce tilings of the plane.

Item Type:Articles
Status:Unpublished
Refereed:No
Glasgow Author(s) Enlighten ID:Mampusti, MIchael and Whittaker, Dr Michael
Authors: Mampusti, M., and Whittaker, M. F.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:arXiv
Copyright Holders:Copyright © 2019 The Authors
Publisher Policy:Reproduced with the permission of the Author

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
3006550Applications of space filling curves to substitution tilingsMichael WhittakerEngineering and Physical Sciences Research Council (EPSRC)EP/R013691/1M&S - Mathematics