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Aperiodic tiles
1992
Discrete & Computational Geometry
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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]
1999
Foams and Emulsions
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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
2009
Journal of Mathematics and Music - Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance
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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
2017
Theoretical Computer Science
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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]
1998
IEEE Computer Graphics and Applications
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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]
2020
arXiv
pre-print
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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
2005
Computer journal
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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]
2014
Lecture Notes in Computer Science
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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
2011
Journal of combinatorial theory. Series A
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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
1999
Geometric and Functional Analysis
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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
2002
Computational geometry
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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
2017
Symmetry
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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]
2017
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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]
2016
arXiv
pre-print
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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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