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Aperiodicity, rotational tiling spaces and topological space groups

Hunton, J R and Walton, J J (2021) 'Aperiodicity, rotational tiling spaces and topological space groups.', Advances in mathematics., 388 . p. 107855.

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 . In this article we consider two further spaces and (the rotational hulls) which capture the full rigid motion properties of the underlying patterns. The rotational hull is shown to be a matchbox manifold which contains 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 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 . Similarly, for those patterns associated to quasicrystals, the crystallographers' aperiodic space group may be recovered as a quotient of our fundamental invariant.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives 4.0.
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1016/j.aim.2021.107855
Publisher statement:© 2021 This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Date accepted:10 May 2021
Date deposited:18 October 2021
Date of first online publication:02 July 2021
Date first made open access:02 July 2022

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