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)

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, Professor Mike
• 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

Related URLs

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