Skip to main content

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

Theory of Computing Systems volume 54, pages 640–658 (2014)Cite this article

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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

References

  1. Berger, R.: The undecidability of the domino problem. Mem. Am. Math. Soc. 66, 72 (1966)

    Google Scholar 

  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. Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, vol. 138. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  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. 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. Fowler, J.J., Gethner, E.: Counting Escher’s m×m ribbon patterns. J. Geom. Graph. 10(1), 1–13 (2006)

    MATH  MathSciNet  Google Scholar 

  7. Gardner, M.: Penrose Tiles to Trapdoor Ciphers and the Return of Dr. Matrix. MAA Spectrum. Math. Assoc. of America, Washington (1997)

    MATH  Google Scholar 

  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. 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. Gethner, E., Schattschneider, D., Passiouras, S., Fowler, J.J.: Combinatorial enumeration of 2×2 ribbon patterns. Eur. J. Comb. 28(4), 1276–1311 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Itten, J.: The Art of Color: the Subjective Experience and Objective Rationale of Color. Wiley, New York (1974)

    Google Scholar 

  12. Mabry, R., Wagon, S., Schattschneider, D.: Automating Escher’s combinatorial patterns. Math. Educ. Res. J. 5, 38–52 (1996)

    Google Scholar 

  13. Osborne, H.: The Oxford Companion to Art. Clarendon, Oxford (1970)

    Google Scholar 

  14. Pisanski, T., Schattschneider, D., Servatius, B.: Applying Burnside’s lemma to a one-dimensional Escher problem. Math. Mag. 79(3), 167–180 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Robinson, R.M.: Undecidability and nonperiodicity for tilings of the plane. Invent. Math. 12, 177–209 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  17. Schattschneider, D.: M.C. Escher: Visions of Symmetry. Harry N. Abrams, New York (2004)

    Google Scholar 

  18. Sysło, M.M.: On cycle bases of a graph. Networks 9(2), 123–132 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Colorado Denver, Denver, CO, 80204, USA

    Ellen Gethner

  2. Department of Computer Science, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada

    David G. Kirkpatrick

  3. Department of Mathematics, Harvey Mudd College, Claremont, CA, 91711, USA

    Nicholas J. Pippenger

Authors
  1. Ellen Gethner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David G. Kirkpatrick
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nicholas J. Pippenger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ellen Gethner.

Additional information

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

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-013-9485-9

Keywords

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