An aperiodic monotile that forces nonperiodicity through dendrites

Mampusti, M. and Whittaker, M. F. (2020) An aperiodic monotile that forces nonperiodicity through dendrites. Bulletin of the London Mathematical Society, (doi: 10.1112/blms.12375) (Early Online Publication)

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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:Early Online Publication
Refereed:Yes
Glasgow Author(s) Enlighten ID:Mampusti, MIchael and Whittaker, Dr Mike
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:Bulletin of the London Mathematical Society
Publisher:Wiley
ISSN:0024-6093
ISSN (Online):1469-2120
Published Online:28 June 2020
Copyright Holders:Copyright © 2020 The Authors
First Published:First published in Bulletin of the London Mathematical Society 2020
Publisher Policy:Reproduced in accordance with the publisher copyright policy
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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