Fractal dual substitution tilings

Frank, N. P., Webster, S. B.G. and Whittaker, M. (2016) Fractal dual substitution tilings. Journal of Fractal Geometry, 3(3), pp. 265-317. (doi:10.4171/JFG/37)

Frank, N. P., Webster, S. B.G. and Whittaker, M. (2016) Fractal dual substitution tilings. Journal of Fractal Geometry, 3(3), pp. 265-317. (doi:10.4171/JFG/37)

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Abstract

Starting with a substitution tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles have fractal boundary. We show that each of the new tilings is mutually locally derivable to the original tiling. Thus, at the tiling space level, the new substitution rules are expressing geometric and combinatorial, rather than topological, features of the original. Our method is easy to apply to particular substitution tilings, permits experimentation, and can be used to construct border-forcing substitution rules. For a large class of examples we show that the combinatorial dual tiling has a realization as a substitution tiling. Since the boundaries of our new tilings are fractal we are led to compute their fractal dimension. As an application of our techniques we show how to compute the \v{C}ech cohomology of a (not necessarily border-forcing) tiling using a graph iterated function system of a fractal tiling.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Whittaker, Dr Michael
Authors: Frank, N. P., Webster, S. B.G., and Whittaker, M.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Research Group:Analysis
Journal Name:Journal of Fractal Geometry
Publisher:European Mathematical Society Publishing House
ISSN:2308-1309
ISSN (Online):2308-1317
Copyright Holders:Copyright © 2016 European Mathematical Society Publishing House
First Published:First published in Journal of Fractal Geometry 3(3):265-317
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher
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