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No Weak Local Rules for the 4p-Fold Tilings

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Abstract

On the one hand, Socolar showed in 1990 that the n-fold planar tilings admit weak local rules when n is not divisible by 4 (the \(n=10\) case corresponds to the Penrose tilings and is known since 1974). On the other hand, Burkov showed in 1988 that the eightfold tilings do not admit weak local rules, and Le showed the same for the 12-fold tilings (unpublished). We here show that this is actually the case for all the 4p-fold tilings.

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Notes

  1. Note that it admits decorated local rules, as proved by Robert Ammann himself, see [1, 18]

  2. The first index is one.

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Acknowledgments

This work was supported by the ANR project QuasiCool (ANR-12-JS02-011-01).

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Authors and Affiliations

  1. Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453, Marseille, France

    Nicolas Bédaride

  2. Université Paris 13, CNRS, Sorbonne Paris Cité, UMR 7030, 93430, Villetaneuse, France

    Thomas Fernique

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  1. Nicolas Bédaride
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Correspondence to Thomas Fernique.

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Bédaride, N., Fernique, T. No Weak Local Rules for the 4p-Fold Tilings. Discrete Comput Geom 54, 980–992 (2015). https://doi.org/10.1007/s00454-015-9740-8

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