on the thermally induced bistability of composite cylindrical shells for morphing structures abstract the multistability of composite thin structures has shown potential for morphing applications the present work combines a ritz model with path following algorithms to study bistable cylindrical panels polynomial discretisations of the displacements field are used to predict stable shapes geometry and other aspects of the nonlinear structural behaviour in order to improve the inherently poor conditioning properties of ritz approximations of slender structures a non dimensional shell lamination theory with sanders nonlinear strains is developed and presented an investigation on the relative importance of different nonlinear strain terms is shown to provide useful insight into the applicability of common assumptions about shell kinematics in the current approach we continue numerical solutions in parameter space that is we path follow equilibrium configurations as the control parameter varies find stable and unstable configurations and identify bifurcations the numerics is carried out using a set of in house m atlab routines for numerical continuation results are compared with detailed finite elements analysis throughout the course of the paper

singular continuous spectrum for palindromic schrodinger operators we give new examples of discrete schrodinger operators with potentials taking finitely many values that have purely singular continuous spectrum if the hullx of the potential is strictly ergodic then the existence of just one potentialx inx for which the operator has no eigenvalues implies that there is a generic set inx for which the operator has purely singular continuous spectrum a sufficient condition for the existence of such anx is that there is az x that contains arbitrarily long palindromes thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inx the class includes well known substitutions like fibonacci thue morse period doubling binary non pisot and ternary non pisot we also show that the operator has no absolutely continuous spectrum for allx x ifx derives from a primitive substitution for potentials defined by circle maps x n 1 j 0 n we show that the operator has purely singular continuous spectrum for a generic subset inx for all irrational and every half open intervalj

on the least number of palindromes in two dimensional words abstract we investigate the least number of distinct palindromic sub arrays in two dimensional words over a finite alphabet a 1 a 2 a q for a given alphabet size q we discuss the case for both periodic as well as aperiodic words

growth rate of binary words avoiding xxxr consider the set of those binary words with no non empty factors of the form x x x r du mousavi schaeffer and shallit asked whether this set of words grows polynomially or exponentially with length in this paper we demonstrate the existence of upper and lower bounds of the form n lg n o lg n on the number of such words of length n where lg n denotes the base 2 logarithm of n

decision algorithms for fibonacci automatic words ii we use a decision procedure for the fibonacci automatic words to solve problems about a number of different sequences in particular we prove that there exists an aperiodic infinite binary word avoiding the pattern x x x r this is the first avoidability result concerning a nonuniform morphism proven purely mechanically

the infinite volume limit of dissipative abelian sandpiles we construct the thermodynamic limit of the stationary measures of the bak tang wiesenfeld sandpile model with a dissipative toppling matrix sand grains may disappear at each toppling we prove uniqueness and mixing properties of this measure and we obtain an infinite volume ergodic markov process leaving it invariant we show how to extend the dhar formalism of the abelian group of toppling operators to infinite volume in order to obtain a compact abelian group with a unique haar measure representing the uniform distribution over the recurrent configurations that create finite avalanches

a compactness property of the k abelian monoids abstract the k abelian equivalence of words counting the numbers of occurrences of factors of length at most k has been analyzed in recent years from several different directions we continue this analysis the k abelian equivalence classes are known to constitute a monoid hence equations over these monoids are well defined we show that these monoids satisfy a compactness property each system of equations with a finite number of unknowns is equivalent to some of its finite subsystems we give two proofs for this compactness result one is based the fact that the monoid can be embedded into the multiplicative monoid of matrices and the other directly on linear algebra the former method allows the application of hilbert s basis theorem the latter one in turn allows to conclude an upper bound for the size of the finite subsystem

generalization of automatic sequences for numeration systems on a regular language let l be an infinite regular language on a totally ordered alphabet feeding a finite deterministic automaton with output with the words of l enumerated lexicographically with respect to leads to an infinite sequence over the output alphabet of the automaton this process generalizes the concept of k automatic sequence for abstract numeration systems on a regular language instead of systems in base k here we study the first properties of these sequences and their relations with numeration systems

reversals and palindromes in continued fractions several results on continued fractions expansions are on indirect consequences of the mirror formula we survey occurrences of this formula for sturmian real numbers for simultaneous diophantine approximation and for formal power series

multiparameter actuation for shape control of bistable composite plates this paper studies the stable equilibrium shapes of free multilayered orthotropic plates loaded by inelastic deformations induced by thermal and piezoelectric effects starting with a von karman plate kinematics and an energetic formulation a discrete intrinsic nonlinear model in terms of curvatures only is deduced the model has 3 degrees of freedom namely the components of the symmetric curvature tensor which is supposed to be uniform in space despite of this rough assumption the analytical results about the equilibrium shapes and their stability show a good agreement with the finite element simulations performed with a commercial code literature results about the bistable behavior of isotropic plates under a single parameter loading are extended to the orthotropic case with two loading parameters in light of a global stability analysis and a phase portrait as a function of the inelastic curvatures we study possible actuation techniques for controlling the transition of the plate between its two stable configurations we show that with a suitable two parameter actuation it is possible to get a controlled quasi static transition avoiding any instability phenomenon numerical simulations on a realistic case study support the technological feasibility of the proposed actuation technique when using active piezoelectric layers to control the inelastic curvature

abelian sandpiles and the harmonic model we present a construction of an entropy preserving equivariant surjective map from the d dimensional critical sandpile model to a certain closed shift invariant subgroup of mathbb t mathbb z d the harmonic model a similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the bernoulli property of the measure of maximal entropy for that model

multistable composite plates with piecewise variation of lay up in the planform abstract the non linear out of plane displacements of partially unsymmetric laminates are modelled using both finite element analysis and an analytical method attention is focused on the effects that thermal stresses have on the potential multiple shapes of a composite structure the paper extends previous analytical models which could only take into account free free boundary conditions the shape functions that model the out of plane displacements are modified to include variations of the curvatures within the domain the new analytical formulation is compared with literature and finite element analysis for a square plate and then it is tested for laminates with piecewise variation of lay up in the planform the results are validated against finite element analysis and experimental tests and a good correlation is obtained finally a parametric study is made on the effect of changing the fibres orientation and the laminate thickness the results confirm that it is possible to introduce bi stable composites within structures to obtain systems that are both flexible and stiff depending on the loading environment

a new characteristic property of rich words originally introduced and studied by the third and fourth authors together with j justin and s widmer 2008 rich words constitute a new class of finite and infinite words characterized by containing the maximal number of distinct palindromes several characterizations of rich words have already been established a particularly nice characteristic property is that all complete returns to palindromes are palindromes in this note we prove that rich words are also characterized by the property that each factor is uniquely determined by its longest palindromic prefix and its longest palindromic suffix

factor versus palindromic complexity of uniformly recurrent infinite words we study the relation between the palindromic and factor complexity of infinite words we show that for uniformly recurrent words one has p n p n 1 email protected n 2 for all email protected n for a large class of words this is a better estimate of the palindromic complexity in terms of the factor complexity than the one presented in j p allouche m baake j cassaigne d damanik palindrome complexity theoret comput sci 292 2003 9 31 we provide several examples of infinite words for which our estimate reaches its upper bound in particular we derive an explicit prescription for the palindromic complexity of infinite words coding r interval exchange transformations if the permutation p connected with the transformation is given by p k r 1 k for all k then there is exactly one palindrome of every even length and exactly r palindromes of every odd length

sturmian words structure combinatorics and their arithmetics abstract we prove some new results concerning the structure the combinatorics and the arithmetics of the set per of all the words w having two periods p and q p q which are coprimes and such that w p q 2 a basic theorem relating per with the set of finite standard sturmian words was proved in de luca and mignosi 1994 the main result of this paper is the following simple inductive definition of per the empty word belongs to per if w is an already constructed word of per then also aw and bw belong to per where denotes the operator of palindrome left closure i e it associates to each word u the smallest palindrome word u having u as a suffix we show that by this result one can construct in a simple way all finite and infinite standard sturmian words we prove also that up to the automorphism which interchanges the letter a with the letter b any element of per can be codified by the irreducible fraction p q this allows us to construct for any n 0 a natural bijection that we call farey correspondence of the set of the farey series of order n 1 and the set of special elements of length n of the set st of all finite sturmian words finally we introduce the concepts of farey tree and farey monoid this latter is obtained by defining a suitable product operation on the developments in continued fractions of the set of all irreducible fractions p q

avoiding three consecutive blocks of the same size and same sum we show that there exists an infinite word over the alphabet 0 1 3 4 containing no three consecutive blocks of the same size and the same sum this answers an open problem of pirillo and varricchio from 1994

on the aperiodic avoidability of binary patterns with variables and reversals in this work we present a characterisation of the avoidability of all unary and binary patterns that do not only contain variables but also reversals of their instances with respect to aperiodic infinite words these types of patterns were studied recently in either more general or particular cases

abelian properties of parry words abelian complexity of a word u is a function that counts the number of pairwise non abelian equivalent factors of u of length n we prove that for any c balanced parry word u the values of the abelian complexity function can be computed by a finite state automaton the proof is based on the notion of relative parikh vectors the approach works for any function f n that can be expressed in terms of the set of relative parikh vectors corresponding to the length n for example we show that the balance function of a c balanced parry word is computable by a finite state automaton as well

avoidance bases for formulas with reversal abstract in the interest of studying formulas with reversal of high avoidability index we find n avoidance bases for formulas with reversal for n 1 2 3 we demonstrate that there is a unique formula with reversal in each of these three bases of highest avoidability index n 2 these formulas are xx x y x y r and x y z x y r z r which belong to an infinite family of formulas with reversal that has been the subject of recent study by the authors

on the least number of palindromes contained in an infinite word we investigate the least number of palindromic factors in an infinite word we first consider general alphabets and give answers to this problem for periodic and non periodic words closed or not under reversal of factors we then investigate the same problem when the alphabet has size two

tristability of thin orthotropic shells with uniform initial curvature composite shells show a rich multistable behaviour of interest for the design of shape changing morphing structures previous studies have investigated how the initial shape determines the shell

approximation to real numbers by cubic algebraic integers i in 1969 h davenport and w m schmidt studied the problem of approximation to a real number xi by algebraic integers of degree at most 3 they did so using geometry of numbers by resorting to the dual problem of finding simultaneous approximations to xi and xi 2 by rational numbers with the same denominator in this paper we show that their measure of approximation for the dual problem is optimal and that it is realized for a countable set of real numbers xi we give several properties of these numbers including measures of approximation by rational numbers by quadratic real numbers and by algebraic integers of degree at most 3

on k abelian palindromes abstract a word is called a palindrome if it is equal to its reversal in the paper we consider a k abelian modification of this notion two words are called k abelian equivalent if they contain the same number of occurrences of each factor of length at most k we say that a word is a k abelian palindrome if it is k abelian equivalent to its reversal a question we deal with is the following how many distinct palindromes can a word contain it is well known that a word of length n can contain at most n 1 distinct palindromes as its factors such words are called rich on the other hand there exist infinite words containing only finitely many distinct palindromes as their factors such words are called poor we show that in the k abelian case there exist infinite words containing finitely many distinct k abelian palindromic factors for rich words we show that there exist finite words of length n containing n 2 distinct k abelian palindromes as their factors

bistable plates for morphing structures a refined analytical approach with high order polynomials abstract the multistability of composite thin structures has shown potential for morphing applications the present work combines a ritz model with path following algorithms to study bistable plates behaviour classic low order ritz models predict stable shapes geometry with reasonable accuracy however they may fail when modelling other aspects of the elastic structural behaviour a refined higher order model is here presented in order to improve the inherently poor conditioning properties of ritz approximations of slender structures a non dimensional version of classical plate lamination theory with von karman nonlinear strains is developed and presented in the current approach we continue numerical solutions in parameter space that is we path follow equilibrium configurations as the control parameter varies find stable and unstable configurations and identify bifurcations the numerics are carried out using a set of in house matlab routines for numerical continuation the increased degrees of freedom within the model are shown to accurately reflect buckling loads and provide useful insight into the relative importance of different aspects of nonlinear behaviour finally the complex experimentally observed snap through geometry is captured analytically for the first time results are validated against finite elements analysis throughout the course of the paper

episturmian words and some constructions of de luca and rauzy abstract in this paper we study infinite episturmian words which are a natural generalization of sturmian words to an arbitrary alphabet a characteristic property is they are closed under reversal and have at most one right special factor of each length they are first obtained by a construction due to de luca which utilizes the palindrome closure they can also be obtained by the way of extended rauzy rules

fibonacci morphisms and sturmian words abstract marton kosa 1987 has stated five conjectures devoted to the study of some binary morphisms and the factors of infinite words these morphisms generate positive answers to the first four of these conjectures are given in this paper the main result being the characterization of a class of morphisms whose stationary words are sturmian

sandpile models and random walkers on finite lattices abelian sandpile models both deterministic such as the bak tang wiesenfeld btw model p bak c tang and k wiesenfeld phys rev lett 59 381 1987 and stochastic such as the manna model s s manna j phys a 24 l363 1991 are studied on finite square lattices with open boundaries the avalanche size distribution p l n is calculated for a range of system sizes l the first few moments of this distribution are evaluated numerically and their dependence on the system size is examined the sandpile models are conservative in the sense that grains are conserved in the bulk and can leave the system only through the boundaries it is shown that the conservation law provides an interesting connection between the sandpile models and random walk models using this connection it is shown that the average avalanche sizes l for the btw and manna models are equal to each other and both are equal to the average path length of a random walker starting from a random initial site on the same lattice of size l this is in spite of the fact that the sandpile models with deterministic btw and stochastic manna toppling rules exhibit different critical exponents indicating that they belong to different universality classes

analytical investigation of structurally stable configurations in shape memory alloy actuated plates strains produced by active materials embedded in plates have been extensively used to manipulate the shape of surface like engineering structures shape memory alloys smas are active materials that provide a significant amount of strain under large stresses a characteristic of great utility in such morphing structures in this work an analytical approach to approximate the deformation of plates with sma constituents is developed via the rayleigh ritz method an additive set of kinematically admissible displacement fields with unknown coefficients is used to describe the plate displacement field the total potential energy is then calculated using the displacement field loading conditions and constitutive relations for the plate layer s composed of sma wire meshes dense sma films and or elastic material the unknown coefficients are then found via minimization of the total potential energy this approach provides closed form expressions for the approximate deformation of the plates including multistable configurations the response of circular sma based plates is studied herein the results show that temperature fields with a linear variation in the radial direction induce multistable configurations in which the plate gaussian curvature is determined by the direction of the temperature gradient an alternative inversion of the proposed approach is used to directly compute the temperature field required to morph a plate towards a prescribed goal shape the obtained closed form expressions show good agreement with detailed non linear finite element analysis simulations the proposed approach provides analysts with a low computational cost and relatively simple implementation to assess the potentially stable configurations of sma based plates under given loading conditions knowledge of such stable configurations is very valuable in the design of sma based morphing structures c 2015 elsevier ltd all rights reserved

dissipative abelian sandpiles and random walks we show that the dissipative abelian sandpile on a graph mathcal l can be related to a random walk on a graph that consists of mathcal l extended with a trapping site from this relation it can be shown using exact results and a scaling assumption that the correlation length exponent ensuremath nu of the dissipative sandpiles always equals 1 d w where d w is the fractal dimension of the random walker this leads to a new understanding of the known result that ensuremath nu 1 2 on any euclidean lattice our result is however more general and as an example we also present exact data for finite sierpinski gaskets which fully confirm our predictions

discrete approximations of the foppl von karman shell model from coarse to more refined models abstract the problem of deducing from the foppl von karman energy functional a sequence of reduced discrete models having few degrees of freedom is analyzed similar discrete models have been recently intensively studied to analyze the multistable behavior of shallow shells the bifurcations of composite laminates under temperature loads or the wrinkling in soft tissues in particular three relevant examples are discussed and compared among them where the curvature is assumed uniform linearly and quadratically varying through the shell while the uniform curvature assumption dates back to mansfield 1962 linear variations of the shell curvatures can describe smooth transitions between everted configurations while quadratic variations can account for the usually disregarded bending boundary conditions for their deduction we revisit the maxwell mohr method accordingly a sequence of auxiliary elliptic problems of plane elasticity is solved to determine the statically unknown membranal stresses this is a key ingredient for the presented models to compare extremely well with finite element approximations or with literature models with far more degrees of freedom

rich sturmian and trapezoidal words in this paper we explore various interconnections between rich words sturmian words and trapezoidal words rich words first introduced by the second and third authors together with j justin and s widmer constitute a new class of finite and infinite words characterized by having the maximal number of palindromic factors every finite sturmian word is rich but not conversely trapezoidal words were first introduced by the first author in studying the behavior of the subword complexity of finite sturmian words unfortunately this property does not characterize finite sturmian words in this note we show that the only trapezoidal palindromes are sturmian more generally we show that sturmian palindromes can be characterized either in terms of their subword complexity the trapezoidal property or in terms of their palindromic complexity we also obtain a similar characterization of rich palindromes in terms of a relation between palindromic complexity and subword complexity

growth problems for avoidable words abstract a word w is said to avoid a word u if no block subword factor of w is the image of u under a homomorphism of free semigroups without unit the theory of words avoiding xx square free words has been much studied the word u is said to be avoidable on n letters if there are arbitrarily long words on an n letter alphabet that avoid u if u is avoidable on n letters for some n let u be the minimum possible such n we show that u has a linear bound in terms of the alphabet size of u we further show that there exists a word that is avoidable on four letters but not on three letters moreover if u is this word the number of words of length l on a u letter alphabet that avoid u has a polynomial bound in terms of l so that the question of the existence of such an example is resolved in the affirmative in contrast for xx the bound is known to be exponential

approximation to real numbers by cubic algebraic integers ii it has been conjectured for some time that for any integer n 2 any real number e 0 and any transcendental real number there would exist infinitely many algebraic integers a of degree at most n with the property that h n e where h denotes the height of a although this is true for n 2 we show here that for n 3 the optimal exponent of approximation is not 3 but 3 5 2 2 618

a generalization of the fibonacci word fractal and the fibonacci snowflake in this paper we introduce a family of infinite words that generalize the fibonacci word and we study their combinatorial properties we associate with this family of words a family of curves that are like the fibonacci word fractal and reveal some fractal features finally we describe an infinite family of polyominoes stems from the generalized fibonacci words and we study some of their geometric properties such as perimeter and area these last polyominoes generalize the fibonacci snowflake and they are double squares polyominoes i e tile the plane by translation in exactly two distinct ways

substitutions and b systems of numbers resume dans son fameux article cobham a montre que la suite des etats d un k automate etait le point fixe d une substitution de longueur constante cobham 1972 apres avoir introduit de nouvelles notions substitution de longueur suite automatique nous donnons un theoreme analogue pour les systemes de numeration associes a des nombres

a generalization of automatic sequences abstract we generalize the uniform tag sequences of cobham which arise as images of fixed points of k uniform homomorphisms to the case where the homomorphism is not necessarily uniform but rather satisfies the analogue of an algebraic equation we show that these sequences coincide with 1 a the class of sequences accepted by a finite automaton with generalized digits as input and 2 b generalizations of the locally catenative formula of rozenberg and lindenmayer examples include the infinite fibonacci word which is generated as the fixed point of the homomorphism a ab b a and sequences of rauzy and de bruijn

problems in between words and abelian words k abelian avoidability we consider a recently defined notion of k abelian equivalence of words in connection with avoidability problems this equivalence relation for a fixed natural number k takes into account the numbers of occurrences of the different factors of length k and the prefix and the suffix of length k 1 we search for the smallest alphabet in which k abelian squares and cubes can be avoided respectively for 2 abelian squares this is four as in the case of abelian words while for 2 abelian cubes we have only strong evidence that the size is two as it is in the case of words however we are able to prove this optimal value only for 8 abelian cubes

mirror substitutions and palindromic sequences by introducing the notion of mirror substitution we show that given a substitutive sequence over two letters it is palindromic that is it contains arbitrarily long palindromes if and only if its language is mirror invariant that is it is closed under the mirror image map then we solve a question of hof knill and simon in the 2 letter case and also by constructing a counterexample we give a negative answer to the open problem 4 in the webpage http iml univ mrs fr bernat openquestions html

last cases of dejean s conjecture dejean conjectured that the repetition threshold for a k letter alphabet is kk 1 when k 5 dejean s conjecture has already been proved for k 14 and for k 27 we present here a proof for 8 k 38 the same technique is also applied to prove ochem s stronger version of the conjecture for 9 k 38

on abelian squares and substitutions we study substitutions preserving abelian square free words and the more general notion of substitutions with bounded abelian squares in particular we prove the existence of algorithms deciding whether a substitution mapping each letter into a set of commutatively equivalent words belongs to one of these classes

schrodinger operators with rudin shapiro potentials are not palindromic we prove a conjecture of a hof o knill and b simon commun math phys 174 149 159 1995 by showing that the rudin shapiro sequence is not palindromic i e does not contain arbitrarily long palindromes we prove actually this property for all paperfolding sequences and all rudin shapiro sequences deduced from paperfolding sequences as a consequence and as guessed by the above authors their method cannot be used for establishing that discrete schrodinger operators with rudin shapiro potentials have a purely singular continuous spectrum

palindromes and sturmian words abstract an infinite word x over the alphabet a is sturmian if and only if g x n n 1 for any integer n where g x n is the number of distinct words of length n occurring in x a palindrome is a word that can be read indistinctly from left to right or from right to left we prove that x is sturmian if and only if h x n 1 n mod 2 for any integer n where h x n is the number of palindromes of length n occurring in x an infinite word x over the alphabet a is generated by a morphism f if there exists a letter c a such that lim n f n c x we prove the existence of a morphism that generates the palindromes of any infinite sturmian word generated by a morphism

palindrome complexity we study the palindrome complexity of infinite sequences on finite alphabets i e the number of palindromic factors blocks of given length occurring in a given sequence we survey the known results and obtain new results for some sequences in particular for rote sequences and for fixed points of primitive morphisms of constant length belonging to class p of hof knill simon we also give an upper bound for the palindrome complexity of a sequence in terms of its block complexity

avoiding 2 binomial squares and cubes two finite words u v are 2 binomially equivalent if for all words x of length at most 2 the number of occurrences of x as a scattered subword of u is equal to the number of occurrences of x in v this notion is a refinement of the usual abelian equivalence a 2 binomial square is a word uv where u and v are 2 binomially equivalent in this paper considering pure morphic words we prove that 2 binomial squares resp cubes are avoidable over a 3 letter resp 2 letter alphabet the sizes of the alphabets are optimal

uniformly growing k th power free homomorphisms a string is called kth power free if it does not have xk as a nonempty substring for all nonnegative rational numbers k kth power free strings and kth power free homomorphisms are investigated and the shortest uniformly growing square free k 2 and cube free k 3 homomorphisms mapping into least alphabets with three and two letters are introduced it is shown that there exist exponentially many square free and cube free strings of each length over these alphabets sharpening the kth power freeness to the repetitive threshold rt n of n letter alphabets we provide arguments for the nonexistence of various rt n th power free homomorphisms

numeration systems linear recurrences and regular sets a numeration system based on a strictly increasing sequence of positive integers u0 1 u1u2 expresses a non negative integer n as a sum n j 0 i ajuj in this case we say the string a i a i 1 a1 a0 is a representation for n

on some generalizations of abelian power avoidability we prove that 2 abelian cubes are avoidable over a binary alphabet and that 3 abelian squares are avoidable over a ternary alphabet answering positively to two questions of karhumaki et al we also show the existence of infinite additive cube free words on several ternary alphabets to achieve this we give sufficient conditions for a morphism to be k abelian n power free resp additive n power free and then we give several morphisms which respect these conditions additionally all our constructions show that the number of such words grows exponen tially as a corollary we get a new lower bound of 3 1 19 1 059526 for the growth rate of abelian cube free words

arbitrary versus periodic storage schemes and tessellations of the plane using one type of polyomino given n distinct memory modules the elements of an infinite array in storage are distributed such that any set of n elements arranged according to a given data template t can be accessed rapidly in parallel array embeddings that allow for this are called skewing schemes and have been studied in connection with vector processing and simd machines in 1975 shapiro ieee trans comput c 27 1978 421 428 proved that there exists a valid skewing scheme for a template t if and only if t tessellates the plane a conjecture of shapiro is settled and it is proved that for polyominos p a valid skewing scheme exists if and only if there exists a valid periodic skewing scheme periodicity implies a rapid technique to locate data elements the proof shows that when a polyomino p tessellates the plane without rotations or reflections then it can tessellate the plane periodically i e with the instances of p arranged in a lattice it is also proved that there is a polynomial time algorithm to decide whether a polyomino tessellates the plane assuming the polyominos in the tessellation should all have an equal orientation

two infinite families of polyominoes that tile the plane by translation in two distinct ways it has been proved that among the polyominoes that tile the plane by translation the so called squares tile the plane in at most two distinct ways in this paper we focus on double squares that is the polyominoes that tile the plane in exactly two distinct ways our approach is based on solving equations on words which allows us to exhibit properties about their shape moreover we describe two infinite families of double squares the first one is directly linked to christoffel words and may be interpreted as segments of thick straight lines the second one stems from the fibonacci sequence and reveals some fractal features