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Sandpile Toppling on Penrose Tilings: Identity and Isotropic Dynamics

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Automata and Complexity

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 42))

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Abstract

We present experiments of sandpiles on grids (square, triangular, hexagonal) and Penrose tilings. The challenging part is to program such simulator; and our javacript code is available online, ready to play! We first present some identity elements of the sandpile group on these aperiodic structures, and then study the stabilization of the maximum stable configuration plus the identity, which lets a surprising circular shape appear. Roundness measurements reveal that the shapes are not approaching perfect circles, though they are close to be. We compare numerically this almost isotropic dynamical phenomenon on various tilings.

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Notes

  1. 1.

    The identity element of the sandpile group is unique.

  2. 2.

    Note that in order to enforce aperiodicity in P2 and P3, matching constraints should be added on tile edges, for example via notches, but the finite tiling generation methods we employ do not require such considerations.

  3. 3.

    Well, this is a bit disappointing, but we think that it is worth showing that it does not appear to be a fruitful research direction, or maybe a more insightful reader would encounter something out there....

  4. 4.

    This also takes place on other tilings, outside the scope of the present work.

  5. 5.

    They all remain stable with \(deg({v})-1\) grains until reaching m. Observe that any outer tile receiving some grain would topple, and that toppling any outer tile would result in toppling the whole maximum stable component it belongs to.

  6. 6.

    The difficulty may be to find constructions from non Wang tiles, because Wang tiles would lead to square grids for the sandpile model to play on (as we remove tile decorations).

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Acknowledgements

The authors are thankful to Valentin Darrigo for his contributions to JS-Sandpile, to Victor Poupet for stressing that the apparent isotropy of the \({(m+e)}^{\circ }\) process is surprising at the occasion of a talk given by KP during AUTOMATA’2014 in Himeji, to Thomas Fernique for sharing his expertise (and code!) regarding the cut and project method, and to Christophe Papazian for useful comments on quasi-periodicity functions. The work of JF was conducted while a Master student at Aix-Marseille Université, doing an internship at the LIS laboratory (UMR 7020), both in Marseille, France. The work of KP was funded mainly by his salary as a French State agent and therefore by French taxpayers’ taxes, affiliated to Aix-Marseille University, University de Toulon, CNRS, LIS, UMR 7020, Marseille, France and University Côte d’Azur, CNRS, I3S, UMR 7271, Sophia Antipolis, France. Secondary financial support came from ANR-18-CE40-0002 FANs project, ECOS-Sud C16E01 project, and STIC AmSud CoDANet 19-STIC-03 (Campus France 43478PD) project.

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Authors and Affiliations

  1. Université publique, Marseille, France

    Jérémy Fersula

  2. Cogitamus laboratory, Marseille, France

    Camille Noûs

  3. Université publique, Marseille, France

    Kévin Perrot

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  1. Jérémy Fersula
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  2. Camille Noûs
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  3. Kévin Perrot
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Correspondence to Kévin Perrot .

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  1. Unconventional Computing Centre, University of the West of England, Bristol, UK

    Andrew Adamatzky

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Fersula, J., Noûs, C., Perrot, K. (2022). Sandpile Toppling on Penrose Tilings: Identity and Isotropic Dynamics. In: Adamatzky, A. (eds) Automata and Complexity. Emergence, Complexity and Computation, vol 42. Springer, Cham. https://doi.org/10.1007/978-3-030-92551-2_10

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  • DOI: https://doi.org/10.1007/978-3-030-92551-2_10

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