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Asymptotical Unboundedness of the Heesch Number in \({\mathbb {E}}^d\) for \(d\rightarrow \infty \)

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Abstract

We solve d-dimensional Heesch’s problem in the asymptotic sense. Namely, we show that, if \(d\rightarrow \infty \), then there is no uniform upper bound on the set of all possible finite Heesch numbers in the space \({\mathbb {E}}^d\); in other words, given any nonnegative integer n, we can find a dimension d (depending on n) in which there exists a hypersolid whose Heesch number is finite and greater than n.

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References

  1. Bašić, B.: The Heesch number for multiple prototiles is unbounded. C. R. Math. Acad. Sci. Paris 353(8), 665–669 (2015)

    Article  MathSciNet  Google Scholar 

  2. Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W.H. Freeman and Company, New York (1987)

    MATH  Google Scholar 

  3. Heesch, H.: Aufbau der Ebene aus kongruenten Bereichen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Math. Phys. Kl. 1, 115–117 (1935)

    MATH  Google Scholar 

  4. Heesch, H.: Reguläres Parkettierungsproblem. Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, vol. 172. Westdeutscher Verlag, Cologne (1968)

    Google Scholar 

  5. Mann, C.E.: Heesch’s Problem and Other Tiling Problems. PhD thesis, University of Arkansas (2001)

  6. Mann, C.: Heesch’s tiling problem. Am. Math. Mon. 111(6), 509–517 (2004)

    Article  MathSciNet  Google Scholar 

  7. Myers, J.: Polyomino, polyhex and polyiamond tiling. https://www.polyomino.org.uk/mathematics/polyform-tiling/

  8. Reinhardt, K.: Zur Zerlegung der euklidischen Räume in kongruente Polytope. Sitzungsberichte der Preussischen Akademie der Wissenschaften. Phys. Math. Kl. 1928, 150–155 (1928)

    MATH  Google Scholar 

  9. Socolar, J.E.S.: The hexagonal parquet tiling: \(k\)-isohedral monotiles with arbitrarily large \(k\). Math. Intell. 29(2), 33–38 (2007)

    Article  MathSciNet  Google Scholar 

  10. Tarasov, A.S.: On the Heesch number for the hyperbolic plane. Math. Notes 88(1–2), 97–102 (2010)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the two anonymous reviewers for their devoted time and for careful reading of the article.

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  1. Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000, Novi Sad, Serbia

    Bojan Bašić & Anna Slivková

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  1. Bojan Bašić
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  2. Anna Slivková
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Correspondence to Bojan Bašić.

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The authors were supported by the Ministry of Education, Science and Technological Development of Serbia (grant no. 451-03-68/2020-14/200125).

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Bašić, B., Slivková, A. Asymptotical Unboundedness of the Heesch Number in \({\mathbb {E}}^d\) for \(d\rightarrow \infty \). Discrete Comput Geom 67, 328–337 (2022). https://doi.org/10.1007/s00454-020-00254-4

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  • DOI: https://doi.org/10.1007/s00454-020-00254-4

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