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Computational Aspects of M.C. Escher’s Ribbon Patterns

Abstract

At the heart of the ideas of the work of Dutch graphic artist M.C. Escher is the idea of automation; we consider a problem that was inspired by some of his earlier and lesser known work. Specifically, a motif fragment is a connected region contained in a closed unit square. Consider a union of motif fragments and call the result an Escher tile T. One can then construct a pattern in the Euclidean plane, as Escher did, with the set of horizontal and vertical unit length translations of T. The resulting pattern gives rise to infinitely many sets of motif fragments (each set may be finite or infinite) that are related visually by way of the interconnections across boundaries of the unit squares that underly the construction; a set of related motif fragments sometimes gives the appearance of a ribbon and thus the resulting pattern in the plane is called a ribbon pattern. Escher’s designs gave rise to beautiful artwork and inspired equally aesthetic combinatorial questions as well. In his sketchbooks, Escher coloured the ribbon patterns with pleasing results. Colouring the ribbon patterns led naturally to a question of periodicity: is there a prototile that generates a well-coloured pattern? The current work answers the question in the affirmative by way of tools from graph theory, algorithms, and number theory. We end with tools to help address questions of optimization and a list of open questions.

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Acknowledgements

The authors wish to thank Anne Condon, Will Evans, Joel Friedman, and Doris Schattschneider for their interest, helpful insights and support for this project. The first author is grateful to Steve Ogden for his implementation of our algorithm in \(\mathbb{R}^{2}\) and \(\mathbb{R}^{3}\), and to Rick Mabry and Stan Wagon for interesting discussions about their work in [12]. We are grateful to M.C. Escher for his thought provoking ideas and beautiful designs; all of the Escher tiles in this article are adaptations of tiles originally designed by Escher. Penultimately, a shorter version of this article appeared in the conference proceedings of FUN with Agorithms 2012 [9], and we thank the organizers for the opportunity to expand our paper. Finally, we thank the referees for their helpful comments in the current version.

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Correspondence to Ellen Gethner.

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A version of this article with colour graphics can be found at http://cse.ucdenver.edu/~egethner/Papers/.

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Gethner, E., Kirkpatrick, D.G. & Pippenger, N.J. Computational Aspects of M.C. Escher’s Ribbon Patterns. Theory Comput Syst 54, 640–658 (2014). https://doi.org/10.1007/s00224-013-9485-9

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Keywords

  • Escher
  • Ribbon patterns
  • Tiling
  • Geometric structure
  • Periodicity
  • Optimization