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On self-assembly of planar octagonal tilings of finite type

Abstract : Self-assembly is the process in which the components of a system, whether molecules, polymers, or macroscopic particles, are organized into ordered structures as a result of local interactions between the components themselves, without exterior guidance. This thesis is devoted to the self-assembly of aperiodic tilings. Aperiodic tilings serve as a mathematical model for quasicrystals - crystals that do not have any translational symmetry. Because of the specific atomic arrangement of these crystals, the question of how they grow still remains open. Our aim is to develop a growth algorithm for a particular class of aperiodic tilings - octagonal tilings of finite type. In order to mimic the growth of real-world quasicrystals, we demand the algorithm be local: the tiles must be added one by one, using only the local information and no data must be stored between the steps. Simulations strongly support the evidence that this algorithm grows aperiodic tilings, up to an unavoidable but neglectable proportion of missing tiles.
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Ilya Galanov. On self-assembly of planar octagonal tilings of finite type. Computational Geometry [cs.CG]. Université Paris-Nord - Paris XIII, 2019. English. ⟨NNT : 2019PA131018⟩. ⟨tel-02982366⟩

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