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On translating one polyomino to tile the plane

D. Beauquier, M. Nivat
1991 Discrete & Computational Geometry
If such a tiling exists the polyomino is called an exact polyomino. Further, every such tiling of the plane by translated copies of the polyomino is half-periodic.  ...  Given a polyomino, we prove that we can decide whether translated copies of the polyomino can tile the plane. Copies that are rotated, for example, are not allowed in the tilings we consider.  ...  Then the polyomino qAn, belongs to the tiling. ct' ~(x ' --+ [--7 A B C D A' B' C' D' A I_.1 On Translating One Polyomino To Tile the Plane 587 Now we are able to give the last main statement of this paper  ...

Tiling the Plane with a Fixed Number of Polyominoes [chapter]

Nicolas Ollinger
2009 Lecture Notes in Computer Science
Deciding whether a finite set of polyominoes tiles the plane is undecidable by reduction from the Domino problem.  ...  In the case of tiling by translations only, we prove that the problem is undecidable for sets of 11 polyominoes.  ...  Acknowledgement The author thanks Bruno Durand for challenging him with the decision problem of tiling the plane with a fixed number of polyominoes.  ...

An algorithm for deciding if a polyomino tiles the plane

Ian Gambini, Laurent Vuillon
2007 RAIRO - Theoretical Informatics and Applications
For polyominoes coded by their boundary word, we describe a quadratic O(n 2 ) algorithm in the boundary length n which improves the naive O(n 4 ) algorithm.  ...  Techniques used emanate from algorithmics, discrete geometry and combinatorics on words. Mathematics Subject Classification. 68R15, 52C20.  ...  A tiling by translations of P is a partition of the whole plane by translated images of P . A polyomino that tiles the plane by translation is called a tile.  ...

Polyomino convolutions and tiling problems [article]

Ali Ulas Ozgur Kisisel
2000 arXiv   pre-print
We define a convolution operation on the set of polyominoes and use it to obtain a criterion for a given polyomino not to tile the plane (rotations and translations allowed).  ...  We apply the criterion to several families of polyominoes, and show that the criterion detects some cases that are not detectable by generalized coloring arguments.  ...  Therefore it tiles the plane in many ways. The question for n ≥ 2 has the following answer: Proposition 1. D n tiles the plane iff n ≤ 3, translations and rotations allowed.  ...

From Tilings to Fibers – Bio-mathematical Aspects of Fold Plasticity [chapter]

Claire Lesieur, Laurent Vuillo
2014 Oligomerization of Chemical and Biological Compounds
Acknowledgements We would like to thank Claudia Billat who reads carefully a previous version of this article. Author details C. Lesieur 1 and L.  ...  Vuillon 2 * *Address all corespondence to: Laurent.Vuillon@univ-savoie.fr 1 Université Joseph Fourier, AGIM, Grenoble, France 2 Laboratoire de mathématiques, Université de Savoie, France  ...  In order to find a characterization of polyominoes that tile the plane by translation we focus on the boundary of a polyomino P.  ...

Tiling with sets of polyominoes

Solomon W. Golomb
1970 Journal of Combinatorial Theory
The problem of tiling the infinite plane with replicas of a finite set of polyominoes is proved to be logically equivalent to Wang's "domino problem," which is known to be algorithmically undecidable.  ...  The definitions and lattice hierarchy previously established for tiling regions with individual polyominoes are extended to finite sets of polyominoes.  ...  pairs (a, b) if the n-th set of polyominoes cannot be used to tile the plane; (ii) the "cell type number" r to be used at position (a, b) of the plane, in a specific tiling of the plane with the n-th  ...

Polyomino Convolutions and Tiling Problems

Ali Ulas Ozgur Kisisel
2001 Journal of combinatorial theory. Series A
We define a convolution operation on the set of polyominoes and use it to obtain a criterion for a given polyomino not to tile the plane (rotations and translations allowed).  ...  We apply the criterion to several families of polyominoes and show that the criterion detects some cases that are not detectable by generalized coloring arguments.  ...  The motivation for the main idea came from their minesweeper puzzle with clues modulo 2.  ...

Isohedral Polyomino Tiling of the Plane

K. Keating, A. Vince
1999 Discrete & Computational Geometry
A polynomial time algorithm is given for deciding, for a given polyomino P, whether there exists an isohedral tiling of the Euclidean plane by isometric copies of P.  ...  The decidability question for general tilings by copies of a single polyomino, or even periodic tilings by copies of a single polyomino, remains open.  ...  One of the first questions to arise in the subject was the following: Given a single polyomino P, can isometric copies of P tile the plane? It is known that every n-omino for n ≤ 6 tiles the plane.  ...

Polyomino tilings, cellular automata and codicity

Philippe Aigrain, Daniéle Beauquier
1995 Theoretical Computer Science
On the contrary, deciding whether a given set of polyominoes is a code has been shown to be undecidable (Beauquier and Nivat, 1993) .  ...  Recognizing whether a given polyomino can be tiled by translated copies of tiles taken from a given family of polyominoes is obviously decidable.  ...  Two tilings will be said to be distinct if there is a pair (u, c) belonging to one of the tilings and not to the other one.  ...

Polyominoes of order 3 do not exist

I.N Stewart, A Wormstein
1992 Journal of combinatorial theory. Series A
Klarner [2] defined the order of a polyomino P to be the minimum number of congruent copies of P (permitting reflections as well as rotations and translations) that can tile a rectangle.  ...  The order of a polyomino is the minimum number of congruent copies that can tile a rectangle. It is an open question whether any polyomino can have an odd order greater than one.  ...  The order of a polyomino is the minimum number of congruent copies that can tile a rectangle. It is an open question whether any polyomino can have an odd order greater than one.  ...

Revisiting a Tiling Hierarchy (II)

Viorel Nitica
2018 Open Journal of Discrete Mathematics
In a recent paper, we revisited Golomb's hierarchy for tiling capabilities of finite sets of polyominoes. We considered the case when only translations are allowed for the tiles.  ...  The goal of this note is to study the validity of all implications in these new tiling hierarchies if one replaces the simply connected regions by deficient ones.  ...  In [5] we revisited Golomb's tiling hierarchy for finite sets of polyominoes if only translations are allowed for polyominoes.  ...

A Method to Generate Polyominoes and Polyiamonds for Tilings with Rotational Symmetry

Hiroshi Fukuda, Nobuaki Mutoh, Gisaku Nakamura, Doris Schattschneider
2007 Graphs and Combinatorics
We show a simple method to generate polyominoes and polyiamonds that produce isohedral tilings with p3, p4 or p6 rotational symmetry by using n line segments between lattice points on a regular hexagonal  ...  We exhibit all possible tiles generated by this algorithm up to n = 9 for p3, n = 8 for p4, and n = 13 for p6.  ...  An isohedral tiling of the plane is one in which congruent copies of a single tile fill the plane without gaps or overlaps, and the symmetry group of the tiling acts transitively on the tiles.  ...

An Optimal Algorithm for Tiling the Plane with a Translated Polyomino [article]

Andrew Winslow
2015 arXiv   pre-print
We give a O(n)-time algorithm for determining whether translations of a polyomino with n edges can tile the plane.  ...  The algorithm is also a O(n)-time algorithm for enumerating all such tilings that are also regular, and we prove that at most Θ(n) such tilings exist.  ...  Acknowledgments The author thanks Stefan Langerman for fruitful discussions and comments that greatly improved the paper, and anonymous reviewers for pointing out an error in an earlier version of the  ...

Complexity of cutting words on regular tilings

Pascal Hubert, Laurent Vuillon
2007 European journal of combinatorics (Print)
We show that the complexity of a cutting word u in a regular tiling with a polyomino Q is equal to P n (u) = ( p + q − 1)n + 1 for all n ≥ 0, where P n (u) counts the number of distinct factors of length  ...  n in the infinite word u and where the boundary of Q is constructed of 2 p horizontal and 2q vertical unit segments.  ...  A tiling P of the plane by one polyomino is called regular if the vectors of translations form the lattice R (for each vector of translation v, there exist a and b such that v = at 1 + bt 2 , (a, b) ∈  ...

Tiling the Plane with Permutations [chapter]

Alexandre Blondin Massé, Andrea Frosini, Simone Rinaldi, Laurent Vuillon
2011 Lecture Notes in Computer Science
On the other side, Beauquier and Nivat [5] introduced and gave a characterization of the class of pseudo-square polyominoes, i.e. polyominoes that tile the plane by translation: a polyomino is pseudo-square  ...  In this paper we consider the pseudo-square polyominoes which are also convex permutominoes.  ...  Polyominoes that tile the plane In [5] , Beauquier and Nivat studied the class of exact polyominoes, i.e. polyominoes that tile the plane by translation.  ...
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