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Forcing Nonperiodicity with a Single Tile

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References

  1. D. P. DiVincenzo and P. J. Steinhardt. Quasicrystals: the state of the art, volume 16 of Direction in condensed matter physics. World Scientific, second edition, 1999.

  2. C. Janot. Quasicrystals: a primer. Oxford University Press, New York, 1997.

    Google Scholar 

  3. R. Berger. The undecidability of the domino problem. Mem. Amer. Math. Soc., 66:1–72, 1966.

    Google Scholar 

  4. R. Penrose. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl., 10:266–271, 1974.

    Google Scholar 

  5. M. Gardner. Mathematical games. Sci. Am., 236:110–121, 1977.

    Article  Google Scholar 

  6. J. E. S. Socolar and J. M. Taylor. An aperiodic hexagonal tile. Journal of Combinatorial Theory: Series A, 118:2207–2231, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Dekking, M. Mendès France, and A. van der Poorten. Folds. Mathematical Intelligencer, 4, no. 4:173–181, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  8. P. Gummelt. Penrose tilings as coverings of congruent decagons. Geometriae Dedicata, 62:1–17, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  9. P. J. Steinhardt and H. C. Jeong. A simpler approach to Penrose tiling with implications for quasicrystal formation. Nature, 382:431–433, 1996.

    Article  Google Scholar 

  10. L. Danzer. A family of 3D–spacefillers not permitting any periodic or quasiperiodic tiling, pages 11–17. In Aperiodic ’94 (G. Chapuis, Ed., World Scientific, Singapore), 1995.

  11. C. Goodman-Strauss. Open questions in tilings. Available at http://comp.uark.edu/~strauss/papers.

  12. A. R. Kortan, R. S. Becker, F. A. Thiel, and H. S. Chen. Real-space atomic structure of a two-dimensional decagonal quasicrystal. Physical Review Letters, 64:200–203, 1990.

    Article  Google Scholar 

  13. Branko Grünbaum and G. C. Shephard. Tilings and patterns. W. H. Freeman, New York, NY, USA, 1986.

    Google Scholar 

  14. J. E. S. Socolar. Hexagonal parquet tilings: k–isohedral monotiles with arbitrarily large k. Mathematical Intelligencer, 29, no. 2:33–38, 2007. Proper version available at http://arxiv.org/abs/0708.2663.

  15. D. Fletcher. A construction of a nonperiodic tiling with simple atlas matching rules and one prototile. Private communication, December 2009.

  16. D. Frettloh and E. Harriss. Tilings encyclopedia. http://tilings.math.uni-bielefeld.de/.

  17. R. Penrose. Remarks on tiling: Details of a \((1+\epsilon+\epsilon^{2})\) –aperiodic set, pages 467–497. In The mathematics of long-range aperiodic order (R. V. Moody, Ed., NATO ASI Series C, 489, Kluwer Academic Publishers, Dordrecht), 1997.

  18. Y. Araki. Mimicry beetles. http://www.gigapan.org/gigapans/49694/, 2010.

  19. F. Gähler and R. Klitzing. The diffraction pattern of self–similar tilings, pages 141–174. In The mathematics of long-range aperiodic order (R. V. Moody, Ed., NATO ASI Series C, 489, Kluwer Academic Publishers, Dordrecht), 1997.

  20. C. Godreche. The sphinx: a limit–periodic tiling of the plane. Journal of Physics A: Mathematical and General, 22:L1163, 1989.

    Article  MathSciNet  Google Scholar 

  21. J. Myers. Polyomino, polyhex and polyiamond tiling. http://www.srcf.ucam.org/jsm28/tiling/, 2009. See table of n–hexes with anisohedral number k.

  22. C. Mann. Heesch’s problem. http://math.uttyler.edu/cmann/math/heesch/heesch.htm.

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Authors and Affiliations

  1. Physics Department and Center for Nonlinear and Complex Systems, Duke University, Durham, NC, 27708, USA

    Joshua E. S. Socolar

  2. Post Office Box U91, Burnie, TAS, 7320, Australia

    Joan M. Taylor

Authors
  1. Joshua E. S. Socolar
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  2. Joan M. Taylor
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Correspondence to Joshua E. S. Socolar.

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Socolar, J.E.S., Taylor, J.M. Forcing Nonperiodicity with a Single Tile. Math Intelligencer 34, 18–28 (2012). https://doi.org/10.1007/s00283-011-9255-y

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