Tilings and quasiperiodicity

Durand, Bruno

doi:10.1007/3-540-63165-8_165

Cited by 13 publications

(20 citation statements)

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“…This propensity for misassembly has been demonstrated experimentally (Rothemund, 2000). A similar argument for mistiling may be made for any Wang tile set that tiles the plane quasiperiodically, (meaning that finite patterns appear regularly in the plane (Durand, 1999)), and whose matching rules exclude periodic tiling. Many such tiles sets are known (Frank, 2008).…”

Section: Hierarchical Structure In Robinson and Other Quasiperiodic Pmentioning

confidence: 88%

“…The proof that every tiling enforced by the matching rules of these tiles is aperiodic follows from the necessity of arranging the tiles such that larger and larger squares are defined by the patterns on the tiles, (shown in gray in Figure 2). The emerging patterns may be described as quasiperiodic (Durand, 1999), self-similar (Lafitte & Weiss, 2008a), or hierarchical (Lafitte & Weiss, 2008a; Goodman-Strauss, 1999), and contain squares whose edges are defined by 2 n + 1 tiles at the n -th level in the hierarchy. Further experimentation with the tiles of Figure 1 demonstrates that, within the Wang tiling system, it is easy to make errors by adding tiles at random according to the edge matching rules, creating untilable regions.…”

Section: Hierarchical Structure In Robinson and Other Quasiperiodic Pmentioning

confidence: 99%

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Hierarchical self assembly of patterns from the Robinson tilings: DNA tile design in an enhanced Tile Assembly Model

2011

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We introduce a hierarchical self assembly algorithm that produces the quasiperiodic patterns found in the Robinson tilings and suggest a practical implementation of this algorithm using DNA origami tiles. We modify the abstract Tile Assembly Model, (aTAM), to include active signaling and glue activation in response to signals to coordinate the hierarchical assembly of Robinson patterns of arbitrary size from a small set of tiles according to the tile substitution algorithm that generates them. Enabling coordinated hierarchical assembly in the aTAM makes possible the efficient encoding of the recursive process of tile substitution.

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Section: Hierarchical Structure In Robinson and Other Quasiperiodic Pmentioning

confidence: 88%

Section: Hierarchical Structure In Robinson and Other Quasiperiodic Pmentioning

confidence: 99%

Hierarchical self assembly of patterns from the Robinson tilings: DNA tile design in an enhanced Tile Assembly Model

2011

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“…A quasiperiodic shift is a shift that contains only quasiperiodic configurations. Given a configuration x, a function of a quasiperiodicity for x is a mapping ϕ : N → N ∪ {∞} such that every finite pattern of size (diameter) n either never appears in x or appears in every cube of size ϕ(n) in x (see [8]). We assume ϕ(n) = ∞ if some pattern P of size n appears in x but there exist arbitrarily large areas in x that are free of P .…”

Section: Quasiperiodicity and Minimalitymentioning

confidence: 99%

The expressiveness of quasiperiodic and minimal shifts of finite type

Durand

Romashchenko

2020

Ergod. Th. Dynam. Sys.

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We study multidimensional minimal and quasiperiodic shifts of finite type. We prove for these classes several results that were previously known for the shifts of finite type in general, without restriction. We show that some quasiperiodic shifts of finite type admit only non-computable configurations; we characterize the classes of Turing degrees that can be represented by quasiperiodic shifts of finite type. We also transpose to the classes of minimal/quasiperiodic shifts of finite type some results on subdynamics previously known for the effective shifts without restrictions: every effective minimal (quasiperiodic) shift of dimension d can be represented as a projection of a subdynamics of a minimal (respectively, quasiperiodic) shift of finite type of dimension d + 1. Supported by the ANR grant Racaf ANR-15-CE40-0016-01. Preliminary versions of some of the presented results we published in conference papers on MFCS-2015 [23] (Theorems 3-4) and MFCS-2017 [26] (Theorems 6-7).

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“…A tiling that has a quasiperiodicity function defined for any integer n is said to be quasiperiodic. A result that has now acquired classical status states that if a tile set tiles the plane, then it generates at least one quasiperiodic tiling (Durand 1999).…”

Section: Basic Definitions and Notions Of Simulationmentioning

confidence: 99%

Tilings: simulation and universality

Lafitte¹,

Weiss²

2010

Math. Struct. Comp. Sci.

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Wang tiles are unit-size squares with coloured edges. In this paper, we approach one aspect of the study of tiling computability: the quest for a universal tile set. Using a complex construction, based on Robinson's classical construction and its different modifications, we build a tile set (pronounced ayin) that almost always simulates any tile set. By way of Banach-Mazur games on tilings topological spaces, we prove that the set of -tilings that do not satisfy the universality condition is meagre in the set of -tilings.

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Tilings and quasiperiodicity

Cited by 13 publications

References 14 publications

Hierarchical self assembly of patterns from the Robinson tilings: DNA tile design in an enhanced Tile Assembly Model

Hierarchical self assembly of patterns from the Robinson tilings: DNA tile design in an enhanced Tile Assembly Model

The expressiveness of quasiperiodic and minimal shifts of finite type

Tilings: simulation and universality

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