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On translating one polyomino to tile the plane

1991
*
Discrete & Computational Geometry
*

If such a

doi:10.1007/bf02574705
fatcat:36dxnvrxoreebk27m7rh5zmtpy
*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]

2009
*
Lecture Notes in Computer Science
*

Deciding whether a finite set of

doi:10.1007/978-3-642-00982-2_54
fatcat:ndmu7zeplrhb5ej46ckwgzbkxm
*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

2007
*
RAIRO - Theoretical Informatics and Applications
*

For

doi:10.1051/ita:2007012
fatcat:3pz5wfvonjcypjknq2hjl4t7ay
*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]

2000
*
arXiv
*
pre-print

We define a convolution operation

arXiv:math/0012179v1
fatcat:2xhvgmctsndmbp7vbekavaxv4a
*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]

2014
*
Oligomerization of Chemical and Biological Compounds
*

Acknowledgements We would like

doi:10.5772/58577
fatcat:iuoq6abcqjerdhbyv422al3zby
*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

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

2001
*
Journal of combinatorial theory. Series A
*

We define a convolution operation

doi:10.1006/jcta.2000.3171
fatcat:ojqabiowyzbv7kqdpocfafwtla
*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

1999
*
Discrete & Computational Geometry
*

A polynomial time algorithm is given for deciding, for a given

doi:10.1007/pl00009442
fatcat:fof7cywpsjayblw4fort6oe74a
*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

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

1992
*
Journal of combinatorial theory. Series A
*

Klarner [2] defined

doi:10.1016/0097-3165(92)90058-3
fatcat:xyj5i44buvcnxbxbqutnrfsrwa
*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)

2018
*
Open Journal of Discrete Mathematics
*

In a recent paper, we revisited Golomb's hierarchy for

doi:10.4236/ojdm.2018.82005
fatcat:4idxh7rlobcapaajbqu7nceioi
*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

2007
*
Graphs and Combinatorics
*

We show a simple method

doi:10.1007/s00373-007-0719-y
fatcat:obdo6q425raybgvtzueqghayfu
*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]

2015
*
arXiv
*
pre-print

We give a O(n)-time algorithm for determining whether

arXiv:1504.07883v2
fatcat:p6nmw5yxafca7mvzx7ryyogb7m
*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

2007
*
European journal of combinatorics (Print)
*

We show that

doi:10.1016/j.ejc.2005.05.009
fatcat:7uhxbql6pnbr3kj5bb2apvkzmq
*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]

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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