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Aperiodic tilings and entropy

Bruno Durand, Guilhem Gamard, Anaël Grandjean
2017 Theoretical Computer Science  
Theorem (J.Kari and K. Culik, 1996) . .. . . . . . The Kari-Culik tileset is aperiodic. . Sketch of proof. . . . . . . . . Suppose there is a periodic tiling. Then each line has an average.  ...  If a substitutive square is found in any n × n-square of any tiling, then the entropy of the tiles is positive.  ...  Is there a sub-shift of finite type A, with positive entropy, such that any subshift of finite type ⊂ A also has positive entropy?  ... 
doi:10.1016/j.tcs.2016.12.013 fatcat:gk37isccqjhoxaq2ia7fo3ml3e

Aperiodic Tilings and Entropy [chapter]

Bruno Durand, Guilhem Gamard, Anaël Grandjean
2014 Lecture Notes in Computer Science  
In this paper we present a construction of Kari-Culik aperiodic tile set - the smallest known until now. With the help of this construction, we prove that this tileset has positive entropy.  ...  Aperiodic sequences and tilings A tile is an unit square with colored sides.  ...  All tilings with this tileset are aperiodic.  ... 
doi:10.1007/978-3-319-09698-8_15 fatcat:joqxmsoujjdx3blcvxznuyo32m

Towards Explicit Discrete Holography: Aperiodic Spin Chains from Hyperbolic Tilings [article]

Pablo Basteiro, Giuseppe Di Giulio, Johanna Erdmenger, Jonathan Karl, René Meyer, Zhuo-Yu Xian
2022 arXiv   pre-print
Moreover we compute the entanglement entropy in this setup in two different ways: a discretization of the Ryu-Takayanagi formula and a generalization of the standard computation for the boundary aperiodic  ...  tiles.  ...  Acknowledgements We are grateful to Rathindra Nath Das, Emmanuel Floratos, Haye Hinrichsen and Ronny Thomale for useful discussions.  ... 
arXiv:2205.05693v1 fatcat:pmdu4hj5ejhghpo7azi64mbixa

Page 54 of Mathematical Reviews Vol. , Issue 2001A [page]

2001 Mathematical Reviews  
Next, we show that for a second class of tilings, the octahdral and tetrahedral tilings have equal entropy.  ...  In this paper, Goodman-Strauss presents a new and remarkably small and simple set of aperiodic tiles: there are only two tiles and eight translation classes.  ... 

The phase problem for quasicrystals in reciprocal space

Janusz Wolny, Ireneusz Buganski, Maciej Chodyn, Bartlomiej Kozakowski, Pawel Kuczera, Radoslaw Strzalka
2014 Acta Crystallographica Section A: Foundations and Advances  
Decorated Fibonacci sequence, Penrose tiling and Ammann tiling are used as model structures for 1D, 2D, and 3D quasicrystals respectively.  ...  It has been shown, that diffraction patterns of aperiodic structures (quasicrystals and modulated structures) consists of periodic series of peaks [1,2].  ...  Decorated Fibonacci sequence, Penrose tiling and Ammann tiling are used as model structures for 1D, 2D, and 3D quasicrystals respectively.  ... 
doi:10.1107/s2053273314088020 fatcat:vhmbw3tq75fb5akq5343cxefny

Stability of quasicrystals: energy, entropy and phason modes

M. de Boissieu
2006 Philosophical Magazine  
-Experimental results on diffuse scattering and phason elasticity do not seem to be in complete agreement with an entropy term arising from a tile configuration alone.  ...  The two models which are frequently opposed are referred to as the energy and entropy stabilised QC.  ... 
doi:10.1080/14786430500419411 fatcat:yvfgriseczg7vl7dox7c7gjshy

Coverability in Two Dimensions [chapter]

Guilhem Gamard, Gwenaël Richomme
2015 Lecture Notes in Computer Science  
We show that such words have zero entropy. However, contrarily to the one-dimensional case, they may not be uniformly recurrent.  ...  This notion was previously studied in the domains of text algorithms and combinatorics of right infinite words. We extend several results to two dimensions.  ...  We will now prove that µ(w) is aperiodic for any aperiodic bidimensional word w. First, we need a technical lemma about our tiles. Figure 4 ).  ... 
doi:10.1007/978-3-319-15579-1_31 fatcat:a2nejzxrzvhfrbrsggrctm3hcy

Aperiodicity in Equilibrium Systems: Between Order and Disorder

A.C.D. van Enter
2014 Acta Physica Polonica. A  
Question 3: Aperiodic order as a form of disorder. What can ndings in the quasicrystal community teach us about theory of (spin-) glasses [3]? How (dis)ordered can tilings and tiling models be?  ...  In particular, we consider some spectral characterisations, and shortly review what is known about the occurrence of aperiodic order in lattice models at zero and nonzero temperatures.  ...  Acknowledgments I wish to thank the conference organisers of ICQ12 for inviting me, and my co-authors for all they taught me.  ... 
doi:10.12693/aphyspola.126.621 fatcat:77yful7245fx7cbrdo7aoxmz4i

Page 7581 of Mathematical Reviews Vol. , Issue 2003j [page]

2003 Mathematical Reviews  
There are very close the dynamical systems arising from projection method aperiodic connections between such sets and tilings of R?. The work of H. patterns.  ...  The metric Quas, Anthony (1-MEMP; Memphis, TN) arises in the usual way from considering the Hausdorff metric on Entropy along convex shapes, random tilings and shifts of finite type. increasing bounded  ... 

Aperiodicity in equilibrium systems: Between order and disorder [article]

Aernout C.D.van Enter
2013 arXiv   pre-print
Here we discuss some issues related to the notion and occurrence of aperiodic order in equilibrium statistical mechanics.  ...  In particular, we consider some spectral characterisations,and shortly review what is known about the occurrence of aperiodic order in lattice models at zero and non-zero temperatures.  ...  Acknowledgements: I wish to thank the conference organisers of ICQ12 for inviting me, and my coauthors for all they taught me.  ... 
arXiv:1310.0267v2 fatcat:ybbgczo3e5dzxd52c36pfesnxa

The Jewett-Krieger Construction for Tilings [article]

Ian Palmer, Jean Bellissard
2010 arXiv   pre-print
Given a random distribution of impurities on a periodic crystal, an equivalent uniquely ergodic tiling space is built, made of aperiodic, repetitive tilings with finite local complexity, and with configurational  ...  entropy close to the entropy of the impurity distribution.  ...  Kellendonk and L. Sadun for comments and useful new references.  ... 
arXiv:0906.2997v5 fatcat:x7gbtr2ptjbntaonocbzpklxey

Page 5071 of Mathematical Reviews Vol. , Issue 99g [page]

1999 Mathematical Reviews  
The authors consider a random tiling model whose tiles are rect- angles and triangles, the rectangles of side lengths | and 2 sin(z/5) and the triangles of side lengths 1, 1 and 2 sin(z/5) (there are natu  ...  As the temperature T goes to zero, the asymptotic behavior of the entropy and the specific heat are (Jo/2T) exp(—Jo/2T) and (Jj /4T*) exp(—Jo/2T ), respectively.”  ... 

Aperiodic Points in Z²-subshifts

Anael Grandjean, Benjamin Hellouin De Menibus, Pascal Vanier, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming  
We consider the structure of aperiodic points in Z 2 -subshifts, and in particular the positions at which they fail to be periodic.  ...  Another consequence is that Z 2 -subshifts with no aperiodic point have a very strong dynamical structure and are almost topologically conjugate to some Z-subshift.  ...  Acknowledgements The authors wish to thank anonymous reviewers for many helpful remarks and improvements.  ... 
doi:10.4230/lipics.icalp.2018.128 dblp:conf/icalp/GrandjeanMV18 fatcat:54xjss4fxrfglnspjn75yssbl4

Cluster models of decagonal tilings and quasicrystals

Franz Gähler, Michael Reichert
2002 Journal of Alloys and Compounds  
Two different relaxed versions of Gummelt's aperiodic cluster covering rules are considered.  ...  The entropy density of this covering ensemble is determined by Monte Carlo simulations, using entropic sampling techniques.  ...  Acknowledgments We would like to thank Petra Gummelt for useful discussions on the relationships between the different relaxed coverings and random tiling ensembles.  ... 
doi:10.1016/s0925-8388(02)00168-8 fatcat:ezvgff43o5h5tnh7mthyanel24

Fifty years of aperiodic crystals

T. Janssen
2012 Acta Crystallographica Section A Foundations of Crystallography  
Given a finite set of tiles, can one cover the plane with copies of these tiles such that there are no gaps and no overlaps? Are there sets that force the tiling to be aperiodic?  ...  Penrose, who constructed a set of two tiles together with rules to put the tiles together that produced such an aperiodic tiling, later called the Penrose tiling.  ... 
doi:10.1107/s0108767312033715 pmid:23075609 fatcat:jqhwbo7qgzfizjp5chtknido3i
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