The Mathematical Intelligencer

, Volume 29, Issue 2, pp 33–38 | Cite as

More ways to tile with only one shape polygon

  • Joshua E. S. SocolarEmail author
Article The Hexagonal Parquet Tiling: K-Isohedral Monotiles with Arbitrarily Large K


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.



Matching Rule Density Tiling Black Edge Mathematical Intelligencer Figure Tile Edge 
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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Physics Department and Centers for Nonlinear and Complex SystemsDuke UniversityDurhamUSA

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