An Optimal Algorithm for Tiling the Plane with a Translated Polyomino

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 $\Theta(n)$ such tilings exist.Comment: In proceedings of ISAAC 201

La cadena fractal de Fibonacci y algunas generalizaciones

El objetivo de la charla es introducir la cadena de Fibonacci y mostrar sus propiedades geométricas y combinatorias. Esta cadena o palabra se puede generar a partir de la iteración de un homomorfismo entre lenguajes, además, se le puede asociar una curva a partir de unas reglas de dibujo análogas a las utilizadas en los L-sistemas, dicha curva lleva el nombre de curva fractal de Fibonacci. Asimismo, se presentará una familia de cadenas infinitas que generalizan la cadena de Fibonacci y su curva fractal. Finalmente, se asociará una familia de poliminós a estas cadenas, los cuales resultan ser poliminós cuadrados dobles, y se obtendrán algunos tapetes geométricos, los cuales están programados con el software Mathematica®

How many faces can polycubes of lattice tilings by translation of R3 have?

International audienceWe construct a class of polycubes that tile the space by translation in a lattice- periodic way and show that for this class the number of surrounding tiles cannot be bounded. The first construction is based on polycubes with an L-shape but with many distinct tilings of the space. Nevertheless, we are able to construct a class of more complicated polycubes such that each polycube tiles the space in a unique way and such that the number of faces is 4k + 8 where 2k + 1 is the volume of the polycube. This shows that the number of tiles that surround the surface of a space-filler cannot be bounded

Non lattice periodic tilings of R3 by single polycubes

International audienceIn this paper, we study a class of polycubes that tile the space by translation in a non lattice periodic way. More precisely, we construct a family of tiles indexed by integers with the property that Tk is a tile having k ≥ 2 has anisohedral number. That is k copies of Tk are assembled by translation in order to form a metatile. We prove that this metatile is lattice periodic while Tk is not a lattice periodic tile

Non lattice periodic tilings of R3 by single polycubes

International audienceIn this paper, we study a class of polycubes that tile the space by translation in a non lattice periodic way. More precisely, we construct a family of tiles indexed by integers with the property that Tk is a tile having k ≥ 2 has anisohedral number. That is k copies of Tk are assembled by translation in order to form a metatile. We prove that this metatile is lattice periodic while Tk is not a lattice periodic tile

Combinatorial aspects of Escher tilings

International audienceIn the late 30's, Maurits Cornelis Escher astonished the artistic world by producing some puzzling drawings. In particular, the tesselations of the plane obtained by using a single tile appear to be a major concern in his work, drawing attention from the mathematical community. Since a tile in the continuous world can be approximated by a path on a sufficiently small square grid - a widely used method in applications using computer displays - the natural combinatorial object that models the tiles is the polyomino. As polyominoes are encoded by paths on a four letter alphabet coding their contours, the use of combinatorics on words for the study of tiling properties becomes relevant. In this paper we present several results, ranging from recognition of these tiles to their generation, leading also to some surprising links with the well-known sequences of Fibonacci and Pell.Lorsque Maurits Cornelis Escher commença à la fin des années 30 à produire des pavages du plan avec des tuiles, il étonna le monde artistique par la singularité de ses dessins. En particulier, les pavages du plan obtenus avec des copies d'une seule tuile apparaissent souvent dans son œuvre et ont attiré peu à peu l'attention de la communauté mathématique. Puisqu'une tuile dans le monde continu peut être approximée par un chemin sur un réseau carré suffisamment fin - une méthode universellement utilisée dans les applications utilisant des écrans graphiques - l'objet combinatoire qui modèle adéquatement la tuile est le polyomino. Comme ceux-ci sont naturellement codés par des chemins sur un alphabet de quatre lettres, l'utilisation de la combinatoire des mots devient pertinente pour l'étude des propriétés des tuiles pavantes. Nous présentons dans ce papier plusieurs résultats, allant de la reconnaissance de ces tuiles à leur génération, conduisant à des liens surprenants avec les célèbres suites de Fibonacci et de Pell

On the Number of p4-Tilings by an n-Omino

A plane tiling by the copies of a polyomino is called isohedral if every pair of copies in the tiling has a symmetry of the tiling that maps one copy to the other. We show that, for every $n$-omino (i.e., polyomino consisting of n cells), the number of non-equivalent isohedral tilings generated by 90 degree rotations, so called p4-tilings or quarter-turn tilings, is bounded by a constant (independent of n). The proof relies on the analysis of the factorization of the boundary word of a polyomino

On the shape of permutomino tiles

International audienceIn this paper we explore the connections between two classes of polyominoes, namely the permu- tominoes and the pseudo-square polyominoes. A permutomino is a polyomino uniquely determined by a pair of permutations. Permutominoes, and in particular convex permutominoes, have been considered in various kinds of problems such as: enumeration, tomographical reconstruction, and algebraic characterization. On the other hand, pseudo-square polyominoes are a class of polyominoes tiling the the plane by translation. The characterization of such objects has been given by Beauquier and Nivat, who proved that a polyomino tiles the plane by translation if and only if it is a pseudo-square or a pseudo- hexagon. In particular, a polyomino is pseudo-square if its boundary word may be factorized as XY Xﰅ Yﰅ, where Xﰅ denotes the path X traveled in the opposite direction. In this paper we relate the two concepts by considering the pseudo-square polyominoes which are also convex permutominoes. By using the Beauquier-Nivat characterization we provide some geometrical and combinatorial properties of such objects, and we show for any fixed X, each word Y such that XYXﰅYﰅ is pseudo-square is prefix of an infinite word Y∞ with period 4|X|N |X|E. Also, we show that XY XﰅYﰅ are centrosymmetric, i.e. they are fixed by rotation of angle π. The proof of this fact is based on the concept of pseudoperiods, a natural generalization of periods

A Quasilinear-Time Algorithm for Tiling the Plane Isohedrally with a Polyomino

A plane tiling consisting of congruent copies of a shape is isohedral provided that for any pair of copies, there exists a symmetry of the tiling mapping one copy to the other. We give a O(n log2 n)-time algorithm for deciding if a polyomino with n edges can tile the plane isohedrally. This improves on the O(n18)-time algorithm of Keating and Vince and generalizes recent work by Brlek, Provençal, Fédou, and the second author.SCOPUS: cp.pinfo:eu-repo/semantics/publishe

Оценка числа решетчатых разбиений плоскости на полимино заданной площади

We study a problem of a number of lattice plane tilings by given area polyominoes. A polyomino is a connected plane geometric figure formed by joining edge to edge a finite number of unit squares. A tiling is a lattice tiling if each tile can be mapped to any other tile by translation which maps the whole tiling to itself. Let T(n) be a number of lattice plane tilings by given area polyominoes such that its translation lattice is a sublattice of Z². It is proved that 2n−3 + 2[ n−3 2 ] ≤ T(n) ≤ C(n + 1)3 (2.7)n+1. In the proof of a lower bound we give an explicit construction of required lattice plane tilings. The proof of an upper bound is based on a criterion of the existence of lattice plane tiling by polyomino and on the theory of self-avoiding walk. Also, it is proved that almost all polyominoes that give lattice plane tilings have sufficiently large perimeters.Рассматривается задача о числе решетчатых разбиений плоскости на полимино заданной площади. Полимино представляет собой связную фигуру на плоскости, составленную из конечного числа единичных квадратов, примыкающих друг к другу по сторонам. Разбиение называется решетчатым, если любую фигуру разбиения можно перевести в любую другую фигуру параллельным переносом, переводящим все разбиение в себя. Пусть T(n) – число решетчатых разбиений плоскости на полимино площади n, решетка периодов которых является подрешеткой решетки Z² . Доказано, что 2 n−3 + 2[ n−3 2 ] ≤ T(n) ≤ C(n + 1)3 (2.7)n+1. При доказательстве нижней оценки использована явная конструкция, позволяющая построить требуемое число решетчатых разбиений плоскости. Доказательство верхней оценки основано на одном критерии существования решетчатого разбиения плоскости на полимино, а также на теории самонепересекающихся блужданий на квадратной решетке. Также доказано, что почти все полимино, дающие решетчатые разбиения плоскости, имеют большой периметр