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Aperiodic tiles

Robert Ammann, Branko Grünbaum, G. C. Shephard
1992 Discrete & Computational Geometry  
A set of tiles (dosed topological disks) is called aperiodic if there exist tilings of the plane by tiles congruent to those in the set, but no such tiling has any translational symmetry.  ...  We consider a number of aperiodic sets which were briefly described in the recent book Tilings and Patterns, but for which no proofs of their aperiodic character were given.  ...  This completes the proof of the aperiodicity for the set AS. The Aperiodic Set A3 Two variants of this set of three tiles are shown in Fig. 22 .  ... 
doi:10.1007/bf02293033 fatcat:ykxxqwa3c5c2bkoq5kjbammbmi

Aperiodic Hierarchical Tilings [chapter]

Chaim Goodman-Strauss
1999 Foams and Emulsions  
As an immediate corollary, in nite collections of forced aperiodic tilings are constructed. Here we give an expository account of the construction.  ...  Figure 1 .A substitution tiling  ...  But the unmarked L-tiles by themselves are not aperiodic since they can tile rectangles and hence can form periodic tilings. By themselves L-tiles are not forced to tile aperiodically.  ... 
doi:10.1007/978-94-015-9157-7_28 fatcat:rjz7h5mshzgqvc5ffocrwk23fm

Tiling the integers with aperiodic tiles

Franck Jedrzejewski
2009 Journal of Mathematics and Music - Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance  
In some cases, necessary and sufficient conditions for A to tile the integers are known.  ...  A finite subset A of integers tiles the discrete line Z if the integers can be written as a disjoint union of translates of A.  ...  β+γ+3 {1, p, ..., p α+2 } Some other tiles Lagarias and Szabó [10] proposed the following aperiodic canon as a counterexample to Tijdeman's conjecture.  ... 
doi:10.1080/17459730903040915 fatcat:ouqddozyrrfajoq6qtfgvecnry

Aperiodic tilings and entropy

Bruno Durand, Guilhem Gamard, Anaël Grandjean
2017 Theoretical Computer Science  
The Kari-Culik tileset is aperiodic. . Sketch of proof. . . . . . . . . Suppose there is a periodic tiling. Then each line has an average. The averages are periodic: contradiction.  ...  If a substitutive square is found in any n × n-square of any tiling, then the entropy of the tiles is positive.  ... 
doi:10.1016/j.tcs.2016.12.013 fatcat:gk37isccqjhoxaq2ia7fo3ml3e

Aperiodic tiling [computer graphics]

A. Glassner
1998 IEEE Computer Graphics and Applications  
From 20,000 to 2 The question of whether any aperiodic sets of tiles existed at all went unanswered until recently. In 1961 Hao Wang conjectured that there were no aperiodic tiles.  ...  That is, any set of tiles that could tile the plane aperiod- ically could also do so periodically.  ... 
doi:10.1109/38.674976 fatcat:t2qtpxfnfff4jlpm63yikjsfku

Conway and aperiodic tilings [article]

Charles Radin
2020 arXiv   pre-print
This is a brief introduction to the geometric aspects of aperiodic tiling and the collaboration of John Conway and the author in the decade 1990-2000.  ...  CONWAY AND APERIODIC TILINGS Figure 1. Kite and Dart tiling.  ...  He had gotten interested in the kite and dart tilings as the smallest known aperiodic set -two tiles -and had spent a lot of time trying to find a single planar tile which was aperiodic, without success  ... 
arXiv:2008.09085v1 fatcat:o32ffgv55zbh7dqsgk4m73ousi

Aperiodic Tilings: Breaking Translational Symmetry

Leonid A. Levin
2005 Computer journal  
Classical results on aperiodic tilings are rather complicated and not widely understood.  ...  ACKNOWLEDGEMENTS These remarks were developed in my attempts to understand the classical constructions of aperiodic tilings while working on [7] .  ...  A famous example of such mechanisms is aperiodic tiling: hierarchical selfsimilar constructions, first used for computational purposes in a classical-although rather complicated-work [1] .  ... 
doi:10.1093/comjnl/bxh124 fatcat:45navmzgpbbqniobelqle4qjpi

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.  ...  All tilings with this tileset are aperiodic.  ...  Aperiodic sequences and tilings A tile is an unit square with colored sides.  ... 
doi:10.1007/978-3-319-09698-8_15 fatcat:joqxmsoujjdx3blcvxznuyo32m

An aperiodic hexagonal tile

Joshua E.S. Socolar, Joan M. Taylor
2011 Journal of combinatorial theory. Series A  
A two-dimensional, hexagonal prototile with markings that enforce local matching rules is proven to be aperiodic by two independent methods.  ...  We show that a single prototile can fill space uniformly but not admit a periodic tiling.  ...  Direct proof of aperiodicity We prove the following theorem: Theorem 1.  ... 
doi:10.1016/j.jcta.2011.05.001 fatcat:ws5tav324jh6homn7dpzsunb5y

Deterministic Aperiodic Tile Sets

J. Kari, P. Papasoglu
1999 Geometric and Functional Analysis  
We construct an aperiodic set of Wang tiles that is strongly deterministic in the sense that any two adjacent edges of a tile determine the tile uniquely.  ...  Wang tiles are square tiles with colored edges.  ...  tile the plane only aperiodically.  ... 
doi:10.1007/s000390050090 fatcat:a6kaxbz4dvcabfdeczzcynjiyq

An aperiodic tiles machine

José Cáceres, Alberto Márquez
2002 Computational geometry  
The results we introduce in this work lead to get an algorithm which produces aperiodic sets of tiles using Voronoi diagrams. This algorithm runs in optimal worst-case time O(n log n).  ...  Since a wide range of new examples can be obtained, it could shed some new light on non-periodic tilings.  ...  Hence, we have added a new method to the two other known so far in order to produce aperiodic tiles (see [2] ).  ... 
doi:10.1016/s0925-7721(01)00060-8 fatcat:jn4mjik2pbde3gc6snzgx77bbe

Cyclotomic Aperiodic Substitution Tilings

Stefan Pautze
2017 Symmetry  
It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations.  ...  The class of Cyclotomic Aperiodic Substitution Tilings (CAST) is introduced. Its vertices are supported on the 2n-th cyclotomic field.  ...  The most common methods to generate aperiodic tilings are: • Matching rules, as introduced in the very first publication of an aperiodic tiling, the Wang-tiling by R. Berger [9] .  ... 
doi:10.3390/sym9020019 fatcat:4ew4udpgcjen5chhqosebvuxiq

Convex Polygons for Aperiodic Tiling [article]

Teruhisa Sugimoto
2017 arXiv   pre-print
If all tiles in a tiling are congruent, the tiling is called monohedral.  ...  In this paper, we prove that a convex polygon that can generate an edge-to-edge monohedral tiling must be able to generate a periodic tiling.  ...  Note that, although an aperiodic tiling is a nonperiodic tiling, a nonperiodic tiling is not necessarily an aperiodic tiling.)  ... 
arXiv:1602.06372v2 fatcat:jvr47g5jafhdljudntl7bn7v2m

An aperiodic set of 11 Wang tiles [article]

Emmanuel Jeandel
2020 arXiv   pre-print
We present a new aperiodic tileset containing 11 Wang tiles on 4 colors, and we show that this tileset is minimal, in the sense that no Wang set with either fewer than 11 tiles or fewer than 4 colors is  ...  aperiodic.  ...  The Wang set T is aperiodic if and only if T B is aperiodic. Suppose that T B is not aperiodic. We know that T , and thus T B tile the plane. Take a periodic tiling by T B .  ... 
arXiv:1506.06492v3 fatcat:jt3uhtggkbbh7aowaflgomkjw4

The Pegasus Tiles: an aperiodic pair of tiles [article]

Chaim Goodman-Strauss
2016 arXiv   pre-print
The Pegasus tiles are an aperiodic pair of tiles with "tip to tip" matching rules, first drawn in 1996. We present them here.  ...  In any case: Theorem: The Pegasus pair of tiles is aperiodic -that is, they do admit tilings, but only admit non-periodic tilings.  ...  At that time, I had already used "tip-to-tip" matching rules, in the Trilobite and Cross aperiodic pair of tiles [3] and realized that Penrose's construction could be recomposed into an aperiodic pair  ... 
arXiv:1608.07166v1 fatcat:jjn75kog2bcrfcqxsgw3c74v6q
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