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Aperiodicity, rotational tiling spaces and topological space groups

Hunton, John; Walton, James J

Aperiodicity, rotational tiling spaces and topological space groups Thumbnail


Authors

John Hunton

JAMES WALTON JAMES.WALTON@NOTTINGHAM.AC.UK
Assisstant Professor



Abstract

We study the rotational structures of aperiodic tilings in Euclidean space of arbitrary dimension using topological methods. Classical topological approaches to the study of aperiodic patterns have largely concentrated just on translational structures, studying an associated space, the continuous hull, here denoted Ωt. In this article we consider two further spaces Ωr and ΩG (the rotational hulls) which capture the full rigid motion properties of the underlying patterns. The rotational hull Ωr is shown to be a matchbox manifold which contains Ωt as a sub-matchbox manifold. We develop new S-MLD invariants derived from the homotopical and cohomological properties of these spaces demonstrating their computational as well as theoretical utility. We compute these invariants for a variety of examples, including a class of 3-dimensional aperiodic patterns, as well as for the space of periodic tessellations of R 3 by unit cubes. We show that the classical space group of symmetries of a periodic pattern may be recovered as the fundamental group of our space ΩG. Similarly, for those patterns associated to quasicrystals, the crystallographers' aperiodic space group may be recovered as a quotient of our fundamental invariant.

Citation

Hunton, J., & Walton, J. J. (2021). Aperiodicity, rotational tiling spaces and topological space groups. Advances in Mathematics, 388, Article 107855. https://doi.org/10.1016/j.aim.2021.107855

Journal Article Type Article
Acceptance Date May 10, 2021
Online Publication Date Jul 2, 2021
Publication Date Sep 17, 2021
Deposit Date Jul 1, 2021
Publicly Available Date Jul 3, 2022
Journal Advances in Mathematics
Print ISSN 0001-8708
Electronic ISSN 1090-2082
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 388
Article Number 107855
DOI https://doi.org/10.1016/j.aim.2021.107855
Public URL https://nottingham-repository.worktribe.com/output/5749789
Publisher URL https://www.sciencedirect.com/science/article/abs/pii/S0001870821002942?via%3Dihub

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