Advertisement

Theory of Computing Systems

, Volume 54, Issue 4, pp 640–658 | Cite as

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

  • Ellen Gethner
  • David G. Kirkpatrick
  • Nicholas J. Pippenger
Article
  • 272 Downloads

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.

Keywords

Escher Ribbon patterns Tiling Geometric structure Periodicity Optimization 

Notes

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.

References

  1. 1.
    Berger, R.: The undecidability of the domino problem. Mem. Am. Math. Soc. 66, 72 (1966) Google Scholar
  2. 2.
    Cohen, E., Megiddo, N.: Recognizing properties of periodic graphs. In: Applied Geometry and Discrete Mathematics. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, pp. 135–146. Am. Math. Soc., Providence (1991) Google Scholar
  3. 3.
    Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, vol. 138. Springer, Berlin (1993) CrossRefzbMATHGoogle Scholar
  4. 4.
    Dan, D.: On a tiling scheme from M.C. Escher. Electron. J. Comb. 4(2), R23 (1997). Research Paper. Approx. 11 pp. The Wilf Festschrift (Philadelphia, PA, 1996) Google Scholar
  5. 5.
    Escher, G.: Potato printing: a game for winter evenings. In: Coxeter, H.S.M., Emmer, M., Penrose, R., Teuber, M. (eds.) M.C. Escher: Art and Science, pp. 9–11. North-Holland, Amsterdam (1986) Google Scholar
  6. 6.
    Fowler, J.J., Gethner, E.: Counting Escher’s m×m ribbon patterns. J. Geom. Graph. 10(1), 1–13 (2006) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Gardner, M.: Penrose Tiles to Trapdoor Ciphers and the Return of Dr. Matrix. MAA Spectrum. Math. Assoc. of America, Washington (1997) zbMATHGoogle Scholar
  8. 8.
    Gethner, E.: On a generalization of a combinatorial problem posed by M.C. Escher. In: Proceedings of the Thirty-Second Southeastern International Conference on Combinatorics, Graph Theory and Computing, Baton Rouge, LA, 2001, vol. 153, pp. 77–96 (2001) Google Scholar
  9. 9.
    Gethner, E., Kirkpatrick, D.G., Pippenger, N.: M.C. Escher wrap artist: aesthetic coloring of ribbon patterns. In: FUN, pp. 198–209 (2012) Google Scholar
  10. 10.
    Gethner, E., Schattschneider, D., Passiouras, S., Fowler, J.J.: Combinatorial enumeration of 2×2 ribbon patterns. Eur. J. Comb. 28(4), 1276–1311 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Itten, J.: The Art of Color: the Subjective Experience and Objective Rationale of Color. Wiley, New York (1974) Google Scholar
  12. 12.
    Mabry, R., Wagon, S., Schattschneider, D.: Automating Escher’s combinatorial patterns. Math. Educ. Res. J. 5, 38–52 (1996) Google Scholar
  13. 13.
    Osborne, H.: The Oxford Companion to Art. Clarendon, Oxford (1970) Google Scholar
  14. 14.
    Pisanski, T., Schattschneider, D., Servatius, B.: Applying Burnside’s lemma to a one-dimensional Escher problem. Math. Mag. 79(3), 167–180 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Robinson, R.M.: Undecidability and nonperiodicity for tilings of the plane. Invent. Math. 12, 177–209 (1971) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Schattschneider, D.: Escher’s combinatorial patterns. Electron. J. Comb., 4(2), R17 (1997). Research Paper. Approx. 31 pp. The Wilf Festschrift (Philadelphia, PA, 1996) MathSciNetGoogle Scholar
  17. 17.
    Schattschneider, D.: M.C. Escher: Visions of Symmetry. Harry N. Abrams, New York (2004) Google Scholar
  18. 18.
    Sysło, M.M.: On cycle bases of a graph. Networks 9(2), 123–132 (1979) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    van Emde Boas, P.: The convenience of tilings. In: Complexity, Logic, and Recursion Theory. Lecture Notes in Pure and Appl. Math., vol. 187, pp. 331–363. Dekker, New York (1997) Google Scholar
  20. 20.
    Wang, H.: Notes on a class of tiling problems. Fundam. Math. 82, 295–305 (1974/75). Collection of articles dedicated to Andrzej Mostowski on his sixtieth birthday, VIII Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Ellen Gethner
    • 1
  • David G. Kirkpatrick
    • 2
  • Nicholas J. Pippenger
    • 3
  1. 1.Department of Computer ScienceUniversity of Colorado DenverDenverUSA
  2. 2.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada
  3. 3.Department of MathematicsHarvey Mudd CollegeClaremontUSA

Personalised recommendations