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More ways to tile with only one shape polygon

  • Article
  • The Hexagonal Parquet Tiling: K-Isohedral Monotiles with Arbitrarily Large K
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An Erratum to this article was published on 01 September 2007

Conclusion

I have exhibited several types of monotiles with matching rules that force the construction of a hexagonal parquet. The isohedral number of the resulting tiling can be made as large as desired by increasing the aspect ratio of the monotile. Aside from illustrating some elegant peculiarities of the hexagonal parquet tiling, the constructions demonstrate three points.

  1. 1.

    Monotiles with arbitrarily large isohedral number do exist;

  2. 2.

    The additional topological possibilities afforded in 3D allow construction of a simply connected monotile with a rule enforced by shape only, which is impossible for the hexagonal parquet in 2D;

  3. 3.

    The precise statement of the tiling problem matters— whether color matching rules are allowed; whether multiply connected shapes are allowed; whether spacefilling is required as opposed to just maximum density. So what about the quest for thek = ∞ monotile? Schmitt.

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Authors and Affiliations

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

    Joshua E. S. Socolar

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  1. Joshua E. S. Socolar
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Correspondence to Joshua E. S. Socolar.

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An erratum to this article is available at http://dx.doi.org/10.1007/BF02986167.

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Socolar, J.E.S. More ways to tile with only one shape polygon. The Mathematical Intelligencer 29, 33–38 (2007). https://doi.org/10.1007/BF02986203

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