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, 52(5), pp. 942-959. (doi: 10.1112/blms.12375)

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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
Glasgow Author(s) Enlighten ID:Mampusti, MIchael and Whittaker, Professor 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
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 52(5): 942-959
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