Aperiodic Tilings and Entropy

  title={Aperiodic Tilings and Entropy},
  author={Bruno Durand and Guilhem Gamard and Ana{\"e}l Grandjean},
  journal={Theor. Comput. Sci.},
In this paper we present a construction of Kari-Culik aperiodic tile set, the smallest known until now. Our construction is self-contained and organized to allow reasoning on properties of the resulting sets of tilings. With the help of this construction, we prove that this tileset has positive entropy. We also explain why this result was not expected. 
A minimal subsystem of the Kari–Culik tilings
The Kari–Culik tilings are formed from a set of 13 Wang tiles that tile the plane only aperiodically. They are the smallest known set of Wang tiles to do so and are not as well understood as other
Pattern Complexity of Aperiodic Substitutive Subshifts
It is proved that the recent bound of Kari and Moutot, showing that any aperiodic subshift has pattern complexity at least mn+ 1, is optimal for fixed m and n, and a quadratic lower bound is proved on their pattern complexity.
Around the Domino Problem - Combinatorial Structures and Algebraic Tools. (Autour du problème du Domino - Structures combinatoires et outils algébriques)
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Hexagonal Run-Length Zero Capacity Region—Part I: Analytical Proofs
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An aperiodic set of 13 Wang tiles
A small aperiodic set of Wang tiles
Fixed-point tile sets and their applications
Complex tilings
All tilings have complexity n/r(n), given any unbounded computable monotone r, which adds a quantitative angle to classical results on non-recursivity of tilings -- that also develops in terms of Turing degrees of unsolvability.
Undecidability and nonperiodicity for tilings of the plane
This paper is related to the work of Hao Wang and others growing out of a problem which he proposed in [8], w 4.1. Suppose that we are given a finite set of unit squares with colored edges, placed
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A complete self-contained elementary construction of an aperiodic tile set is given and how to use this tile set to elementary prove the undecidability of the classical Domino Problem is sketched.
Entropic commensurate-incommensurate transition.
The equilibrium properties of a minimal tiling model are investigated and it is found that the transition from the ground state to the high temperature disordered phase proceeds through a sequence of periodic arrangements of rows, in analogy with the commensurate-incommensurate transition.
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This year saw the development of both the Go$ del–Herbrand–Kleene theory of recursive functions and, independently, the equivalent notion of machine computable sets of words by Turing.