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Generation of Polyiamonds for p6 Tiling by the Reverse Search

  • Takashi Horiyama
  • Shogo Yamane
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7033)

Abstract

Polyiamonds are the two dimensional shapes made by connecting n unit triangles, joined along their edges. In this paper, we propose algorithms to generate polyiamonds for p6 tiling, i.e., those covering the plane by 6-fold rotations around two rotation centers (60 degrees rotations around the origin and 120 degrees rotations around the terminus). Our algorithm is based on the techniques of the reverse search: (1) No trial and error, since we design rules to generate the next. (2) No need to store already generated polyiamonds. According to these good properties and the proposed rule specific to p6 tiling, we have succeeded to generate 137,535 polyiamonds for p6 tiling up to n = 25, which include 2,246 polyiamonds up to n = 16 obtained by the conventional method.

Keywords

Equivalence Class Additional Rule Degree Rotation Rotation Center Family Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Avis, D., Fukuda, K.: Reverse Search for Enumeration. Discrete Appl. Math. 65, 21–46 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Fukuda, H., Mutoh, N., Nakamura, G., Schattschneider, D.: A Method to Generate Polyominoes and Polyiamonds for Tilings with Rotational Symmetry. Graphs and Combinatrics 23, 259–267 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Fukuda, H., Mutoh, N., Nakamura, G., Schattschneider, D.: Enumeration of Polyominoes, Polyiamonds and Polyhexes for Isohedral Tilings with Rotational Symmetry. In: Ito, H., Kano, M., Katoh, N., Uno, Y. (eds.) KyotoCGGT 2007. LNCS, vol. 4535, pp. 68–78. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Gürçay, H.: Introducing Iso-Tailer 3D: A 3D Tiling Visualizer. Hacettepe Journal of Mathematics and Statistics 37, 107–114 (2008)zbMATHGoogle Scholar
  5. 5.
    Horiyama, T., Samejima, M.: Enumeration of Polyominoes for p4 Tiling. In: Proc. of the 21st Canadian Conference on Computational Geometry, pp. 29–32 (2009)Google Scholar
  6. 6.
    Kaplan, C.S., Salesin, D.H.: Escherization. In: Proc. of SIGGRAPH, pp. 499–510 (2000)Google Scholar
  7. 7.
    Kaplan, C.S., Salesin, D.H.: Dihedral Escherization. In: Proc. of Graphics Interface, pp. 255–262. Canadian Human-Computer Communications Society (2004)Google Scholar
  8. 8.
    Katto, M., Yan, H., Kondo, S., Mitsuhashi, T., Kawanishi, T.: A Study of the Repetitive Pattern of Old Fabrics in Shoso-In —Analysis and Creation of the Pattern Formation Through the Mathematical Group Theory. In: Proc. of the 48th Annual Conference, pp. 396–397. Japanese Society for the Science of Design (2001)Google Scholar
  9. 9.
    Knuth, D.E.: The Art of Computer Programming. fascicle 1: Bitwise Tricks & Techniques; Binary Decision Diagrams, vol. 4. Addison-Wesley (2009)Google Scholar
  10. 10.
    Nakano, S., Uno, T.: Constant Time Generation of Trees with Specified Diameter. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 33–45. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Redelmeier, D.H.: Counting Polyominoes: Yet Another Attack. Discrete Mathematics 36/2, 191–203 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Schattschneider, D.: The Plane Symmetry Groups: Their Recognition and Notation. American Mathematical Monthly 85/6, 439–450 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yoshida, M.: Jink\(\bar{\textrm{o}}\)ki (1641) (First edition published in 1627)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Takashi Horiyama
    • 1
  • Shogo Yamane
    • 1
  1. 1.Graduate School of Science and EngineeringSaitama UniversityJapan

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