Skip to main content
SpringerLink
Log in

Escher-like Tilings with Weights

  • Conference paper
  • First Online:
Discrete and Computational Geometry and Graphs (JCDCGG 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9943))

Included in the following conference series:

Abstract

A tiling of the plane is a set of figures, called tiles, that cover the plane without gaps or overlaps. On tiling we consider “Escherization problem": Given a closed figure in the plane, find a new closed figure that is similar to the original and can tile the plane. In this study, we give a new formulation of the problem with the weighted Procrustes distance and an algorithm to solve the problem optimally. We conduct computational experiments with animal shape tiles to confirm the effectiveness of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Escher, M.C.: The Graphic Work, Taschen America Llc, Special Edition (2008)

    Google Scholar 

  2. Escher, M.C.: The Official Website. http://www.mcescher.com/

  3. Grünbaum, B., Shephard, G.C.: Tiling and Patterns. W. H. Freeman, New York (1987)

    MATH  Google Scholar 

  4. Imahori, S., Sakai, S.: A local-search based algorithm for the Escherization problem. In: The IEEE International Conference on Industrial Engineering and Engineering Management, pp. 151–155 (2012)

    Google Scholar 

  5. Kaplan, C.S.: Introductory Tiling Theory for Computer Graphics. Morgan & Claypool Publishers, San Rafael (2009)

    MATH  Google Scholar 

  6. Kaplan, C.S., Salesin, D.H.: Escherization. In: Proceedings of SIGGRAPH, pp. 499–510 (2000)

    Google Scholar 

  7. Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16, 81–121 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Koizumi, H., Sugihara, K.: Maximum eigenvalue problem for Escherization. Graphs Comb. 27, 431–439 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Koschat, M.A., Swayne, D.F.: A weighted Procrustes criterion. Psychometrika 56, 229–239 (1991)

    Article  Google Scholar 

  10. Mooijaart, A., Commandeur, J.J.F.: A general solution of the weighted orthonormal Procrustes problem. Psychometrika 55, 657–663 (1990)

    Article  MATH  Google Scholar 

  11. Ruppert, J.: A Delaunay refinement algorithm for quality 2-dimensional mesh generation. J. Algorithms 18, 548–585 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Werman, M., Weinshall, D.: Similarity and affine invariant distances between 2D point sets. IEEE Trans. Pattern Anal. Mach. Intell. 17, 810–814 (1995)

    Article  Google Scholar 

Download references

Acknowledgments

This work was partly supported by JSPS Grant-in-Aid for Scientific Research (B) (No. 24360039) and (C) (No. 25330024). The authors would like to thank the anonymous reviewers for their valuable comments.

Author information

Authors and Affiliations

  1. Department of Information and System Engineering, Faculty of Science and Engineering, Chuo University, Tokyo, Japan

    Shinji Imahori

  2. Graduate School of Engineering, Nagoya University, Nagoya, Japan

    Shizuka Kawade

  3. Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan

    Yoko Yamakata

Authors
  1. Shinji Imahori
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shizuka Kawade
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yoko Yamakata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shinji Imahori .

Editor information

Editors and Affiliations

  1. Tokyo University of Science , Tokyo, Japan

    Jin Akiyama

  2. The University of Electro-Communications , Tokyo, Japan

    Hiro Ito

  3. Tokai University , Tokyo, Japan

    Toshinori Sakai

  4. Osaka Prefecture University , Sakai, Japan

    Yushi Uno

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing AG

About this paper

Cite this paper

Imahori, S., Kawade, S., Yamakata, Y. (2016). Escher-like Tilings with Weights. In: Akiyama, J., Ito, H., Sakai, T., Uno, Y. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2015. Lecture Notes in Computer Science(), vol 9943. Springer, Cham. https://doi.org/10.1007/978-3-319-48532-4_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-48532-4_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-48531-7

  • Online ISBN: 978-3-319-48532-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation