215 research outputs found

    A parallelogram tile fills the plane by translation in at most two distinct ways

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    We consider the tilings by translation of a single polyomino or tile on the square grid Z2 (Z exposant 2). It is well-known that there are two regular tilings of the plane, namely, parallelogram and hexagonal tilings. Although there exist tiles admitting an arbitrary number of distinct hexagon tilings, it has been conjectured that no polyomino admits more than two distinct parallelogram tilings. In this paper, we prove this conjecture

    On the shape of permutomino tiles

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    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

    La cadena fractal de Fibonacci y algunas generalizaciones

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    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®

    Introducing the Plaid Model

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    We introduce and prove some basic results about a combinatorial model which produces embedded polygons in the plane. The model is closely related to outer billiards on kites, and also is related to corner percolation, to Hooper's Truchet tile system, to self-similar tilings, and to polyhedron exchange transformations.Comment: 68 pages. This is an more polished version of the original submission. The connection to polytope exchanges is developed and one of the speculative sections has been eliminated. Otherwise, it has the same result

    The application of the principles of symmetry to the synthesis of multi-coloured counterchange patterns

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    Attention is focused on the theoretical principles governing the underlying geometry of motifs, border patterns and all-over patterns. The systematic classification and construction of two-dimensional periodic patterns and tilings is introduced, with particular relerence to two-colour and higher colour counterchange possibilities. An identification is made of the geometrical restraints encountered when introducing systematic interchange of colour. A wide ranging series of original patterns and tilings is constructed and fully illustrated; these designs have been printed in fabric form and are presented in the accompanying exhibition

    Topology and the Cosmic Microwave Background

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    Nature abhors an infinity. The limits of general relativity are often signaled by infinities: infinite curvature as in the center of a black hole, the infinite energy of the singular big bang. We might be inclined to add an infinite universe to the list of intolerable infinities. Theories that move beyond general relativity naturally treat space as finite. In this review we discuss the mathematics of finite spaces and our aspirations to observe the finite extent of the universe in the cosmic background radiation.Comment: Hilarioulsy forgot to remove comments to myself in previous version. Reference added. Submitted to Physics Report
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