beta integers as natural counting systems for quasicrystals recently discrete sets of numbers the integers have been proposed as numbering tools in quasicrystalline studies indeed there exists a unique numeration system based on the irrational in which the integers are all real numbers with no fractional part these integers appear to be quite appropriate for describing some quasilattices relevant to quasicrystallography when precisely is equal to golden mean to or to i e when is one of the self similarity ratios observed in quasicrystalline structures as a matter of fact integers are natural candidates for coordinating quasicrystalline nodes and also the bragg peaks beyond a given intensity in corresponding diffraction patterns they could play the same role as ordinary integers do in crystallography in this paper we prove interesting algebraic properties of the sets when is a quadratic unit pv number a class of algebraic integers which includes the quasicrystallographic cases we completely characterize their respective meyer additive and multiplicative properties where f and g are finite sets and also their respective galois conjugate sets these properties allow one to develop a notion of a quasiring we hope that in this way we will initiate a sort of algebraic quasicrystallography in which we can understand quasilattices which be module on a quasiring in we give also some two dimensional examples with

the algebraic theory of quasicrystals with five fold symmetries an algebraic binary operation is introduced into quasicrystals admitting five fold symmetry in terms of this many quasicrystals displaying full pentagonal or icosahedral symmetry are seen to be finitely generated examples are given in dimensions 1 2 3 and 4 the operation of left quasicrystal addition is affine linear the monoid generated by these operators is discussed and a presentation for it given in the generic case

the birth of e8 out of the spinors of the icosahedron e8 is prominent in mathematics and theoretical physics and is generally viewed as an exceptional symmetry in an eight dimensional 8d space very different from the space we inhabit for instance the lie group e8 features heavily in 10d superstring theory contrary to that point of view here we show that the e8 root system can in fact be constructed from the icosahedron alone and can thus be viewed purely in terms of 3d geometry the 240 roots of e8 arise in the 8d clifford algebra of 3d space as a double cover of the 120 elements of the icosahedral group generated by the root system h3 as a by product by restricting to even products of root vectors spinors in the 4d even subalgebra of the clifford algebra one can show that each 3d root system induces a root system in 4d which turn out to also be exactly the exceptional 4d root systems the spinorial point of view explains their existence as well as their unusual automorphism groups this spinorial approach thus in fact allows one to construct all exceptional root systems within the geometry of three dimensions which opens up a novel interpretation of these phenomena in terms of spinorial geometry

substitution rules for aperiodic sequences of the cut and project type we consider one dimensional aperiodic sequences arising from a cut and project scheme with quadratic unitary pisot numbers a construction of the substitution rule is described under rather general assumptions it allows one to build a given cut and project sequence starting from its arbitrary point for a sequence with a convex acceptance window interval we prove that a substitution rule exists precisely if the boundary points of are in the corresponding quadratic field typically such a substitution has a reducible characteristic polynomial our main result is an algorithm for construction of such a substitution rule some examples are shown

voronoi and delaunay cells of root lattices classification of their faces and facets by coxeter dynkin diagrams the voronoi domains their duals delaunay domains and all their faces of any dimension are classified and described in terms of the weyl group action on a representative of each type of face the representative of a face type is specified by a decoration of the corresponding coxeter dynkin diagram the rules of domain description are uniform for root lattices of simple lie groups of all types an explicit description of the representatives of all faces is carried out for the domains of root lattices of the four classical series and for the five exceptional simple lie groups the coxeter dynkin diagrams required here are the diagrams extended by the highest short root each diagram is partitioned into two subdiagrams one describing completely a d face of the voronoi domain its complement completely describing the dual of the d face the applicability of the authors classification method to generalized kaleidoscopes is explained

local properties bialgebras and representations for one dimensional tilings for any one dimensional tiling we discuss finite dimensional standard modules for the associated tiling bialgebra we will note that such modules are completely reducible and we will parametrize finite dimensional irreducible ones using the set of all patches furthermore we will discuss the associated completed groups and iwasawa type decompositions we also characterize for one dimensional tilings to be locally nondistinguishable by the associated tiling bialgebra structures

octagonal quasicrystals and a formula for shelling octagonal quasicrystals will be realized in a cyclotomic field and a formula for shelling will be given using number theory

reduction of orbits of finite coxeter groups of non crystallographic type a reduction of orbits of finite reflection groups to their reflection subgroups is produced by means of projection matrices which transform points of the orbit of any group into points of the orbits of its subgroup projection matrices and branching rules for orbits of finite coxeter groups of non crystallographic type are presented the novelty in this paper is producing the branching rules that involve non crystallographic coxeter groups moreover these branching rules are relevant to any application of non crystallographic coxeter groups including molecular crystallography and encryption

classification of voronoi and delone tiles in quasicrystals i general method a new general method is presented which allows one to find all distinct voronoi and delone tiles in any quasicrystal from a large family this includes the tiles which may be present with arbitrarily low density 0 at all stages the method requires only consideration of a possibly large finite number of cases our method is applicable in principle to quasicrystals in any dimension and with any irrationality this is the first of three papers where the voronoi and delone tilings are studied two dimensional point sets quasicrystals arising from the a4 root lattice by means of the standard projection to a two dimensional plane with the irrationality 1 2 1 5 are considered in general we require that the acceptance window be bounded with non empty interior specific results are provided here for rhombic acceptance windows of any size oriented along the direction of simple roots of the coxeter group h2 within one quasicrystal the tiles are distinguished by their shape size and orientation the rhombic window case is indispensable for subsequent classification of voronoi and delone tiles in quasicrystals with general shape of the acceptance window voronoi and delone tiles of quasicrystals with circular and decagonal windows of any size are given in subsequent papers let vt denote the set of distinct voronoi tiles making up a quasicrystal with a given acceptance window there are three vt sets of the generic type and three of the singular type the latter occur for one precise value of the size of the acceptance window any other vt set is a uniform scaling of the tiles listed here similar results differing in detail are provided for the sets of distinct delone tiles dt altogether there are four different sets dt of delone tiles

an algebraic slice in the coadjoint space of the borel and the coxeter element abstract let g be a complex simple lie algebra and b a borel subalgebra the algebra y of polynomial semi invariants on the dual b of b is a polynomial algebra on rank g generators grothendieck and dieudonne 1965 1967 16 the analogy with the semisimple case suggests there exists an algebraic slice to coadjoint action that is an affine translate y v of a vector subspace of b such that the restriction map induces an isomorphism of y onto the algebra r y v of regular functions on y v this holds in type a and even extends to all biparabolic subalgebras joseph 2007 20 but the construction fails in general even with respect to the borel moreover already in type c 2 no algebraic slice exists very surprisingly the exception of type c 2 is itself an exception indeed an algebraic slice for the coadjoint action of the borel subalgebra is constructed for all simple lie algebras except those of types b 2 m c n and f 4 outside type a the slice obtained meets an open dense subset of regular orbits even though the special point y of the slice is not itself regular this explains the failure of our previous construction

a natural lattice basis problem with applications integer lattices have numerous important applications but some of them may have been overlooked because of the common assumption that a lattice basis is part of the problem instance this paper gives an application that requires finding a basis for a lattice defined in terms of linear constraints we show how to find such a basis efficiently

minimal distances in quasicrystals a general expression is derived for the minimal distance between points of a cut and project quasicrystal in with a convex acceptance window the study of minimal distances amounts to the study of one dimensional quasicrystals and their rescalings which occur in for an n dimensional ball as the exact value of is calculated for any radius for close to a ball a simple formula is given for all upper and lower bounds for are found the latter are easy to use even when is of a complicated shape

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

cohesive and non cohesive fracture by higher order enrichment of xfem summary a comprehensive study is performed on the use of higher order terms of the crack tip asymptotic fields as enriching functions for the extended fem xfem for both cohesive and traction free cracks for traction free cracks the williams asymptotic field is used to obtain highly accurate stress intensity factors sifs directly from the enriched degrees of freedom without any post processing the low accuracy of the results of the original research on this subject by liu et al int j numer meth engng 2004 59 1103 1118 is remedied here by appropriate modifications of the enrichment scheme the modifications are simple and can be easily included into an xfem computer code for cohesive cracks the relevant asymptotic field is used and two widely used criteria including the sifs criterion and the stress criterion are examined for the crack growth simulation both linear and nonlinear cohesive laws are used for the stress criterion averaging is avoided due to the highly accurate crack tip approximation because of the higher order enrichment then a modified stress criterion is proposed which is shown to be applicable to a wider class of problems several numerical examples including straight and curved cracks stationary and growing cracks single and multiple cracks and traction free and cohesive cracks are studied to investigate in detail the robustness and efficiency of the proposed enrichment scheme copyright 2012 john wiley sons ltd

maximal subgroups of the coxeter group w h4 and quaternions abstract the largest finite subgroup of o 4 is the non crystallographic coxeter group w h 4 of order 14 400 its derived subgroup is the largest finite subgroup w h 4 z 2 of so 4 of order 7200 moreover up to conjugacy it has five non normal maximal subgroups of orders 144 two 240 400 and 576 two groups w h 2 w h 2 z 4 and w h 3 z 2 possess non crystallographic structures with orders 400 and 240 respectively the groups of orders 144 240 and 576 are the extensions of the weyl groups of the root systems of su 3 su 3 su 5 and so 8 respectively we represent the maximal subgroups of w h 4 with sets of quaternion pairs acting on the quaternionic root systems

the e8 lattice and quasicrystals geometry number theory and quasicrystals the authors study the quasiperiodic structures which can be derived from e8 this lattice suitably oriented leads to a 4d quasicrystal which has 3 3 5 symmetry they develop a modified version of the out and projection method the high dimensional lattice is first foliated into successive shells surrounding a vertex the shells are embedded into 7d s7 spheres they then use the so called hopf fibration of s7 to gather families of e8 shell sites into s3 fibres which contain 24 sites or a multiple of 24 which are symmetrically disposed in s3 this has two advantages first in the selection process a whole fibre is either selected or rejected which reduces the computation more importantly the selection process for a fiber amounts to simple arithmetical criteria the whole process leads to a shell by shell analysis of the 4d quasicrystal which can be brought onto a 2d 1d algorithm similar to the fibonacci chain construction they propose an arithmetic formula which gives the number of points on the shells this 4d quasicrystal has the interesting feature that it can be sliced into lower dimensional quasicrystals for example with icosahedral and tetrahedral symmetry in three dimensions

grand antiprism and quaternions vertices of the four dimensional 4d semi regular polytope the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm it follows from the icosian representation of the e8 root system which decomposes into two copies of the root system of h4 the symmetry of the grand antiprism is a maximal subgroup of the coxeter group w h4 it is the group aut h2 h 2 which is constructed in terms of 20 quaternionic roots of the coxeter diagram h2 h 2 the root system of h4 represented by the binary icosahedral group i of order 120 constitutes the regular 4d polytope 600 cell when its 20 quaternionic vertices corresponding to the roots of the diagram h2 h 2 are removed from the vertices of the 600 cell the remaining 100 quaternions constitute the vertices of the grand antiprism we give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions the dual polytope of the grand antiprism has also been constructed

noncrystallographic coxeter group h4 in e8 the e8 lattice is constructed in terms of icosians by matching two sets of f4 lattices described by quaternions embedding the noncrystallographic group h4 into the weyl group w e8 has been described using matrix generators with an emphasis on the relevant coxeter elements the conjugacy classes of h4 in terms of quaternions and the characters of the two four dimensional irreducible representations are explicitly calculated

improved rounding for spline coefficients and knots when representing the coefficients and knots of a spline using only small integers independently rounding each infinite precision value is not the best strategy we show how to build an affine model for the error expanded about the optimal full precision free knot or parameterized spline then use the lovasz basis reduction algorithm to select a better rounding the technique could be used for other situations in which a quadratic error model can be computed

quaternionic representation of the coxeter group w h4 and the polyhedra the vertices of the four dimensional polytope 3 3 5 and its dual 5 3 3 admitting the symmetry of the non crystallographic coxeter group w h4 of order 14 400 are represented in terms of quaternions with unit norm where the polytope 3 3 5 is represented by the elements of the binaryicosahedral group of quaternions of order 120 we projected the polytopes to three dimensional euclidean space where the quaternionic vertices are the orbits of the coxeter group w h3 icosahedral group with inversion where w h3 z2 is one of the maximal subgroups of the coxeter group w h4 the orbits of the icosahedral group w h3 in the polytope 3 3 5 are the conjugacy classes of the binary icosahedral group and represent a number of icosahedrons dodecahedrons and one icosidodecahedron in three dimensions the 15 orbits of the icosahedral group w h3 in the polytope 5 3 3 represent the dodecahedrons icosidodecahedrons small rhombicosidodecahedrons and some convex solids possessing the icosahedral symmetry one of the convex solids with 60 vertices is very similar to the truncated icosahedron soccer ball but with two different edge lengths which can be taken as a realistic model of the c60 molecule at extreme temperature and pressure

modeling holes and inclusions by level sets in the extended finite element method a methodology to model arbitrary holes and material interfaces inclusions without meshing the internal boundaries is proposed the numerical method couples the level set method s osher j a sethian j comput phys 79 1 1988 12 to the extended finite element method x fem n moes j dolbow t belytschko int j numer methods engrg 46 1 1999 131 in the x fem the finite element approximation is enriched by additional functions through the notion of partition of unity the level set method is used for representing the location of holes and material interfaces and in addition the level set function is used to develop the local enrichment for material interfaces numerical examples in two dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique

a hierarchy of polynomial time lattice basis reduction algorithms we present a hierarchy of polynomial time lattice basis reduction algorithms that stretch from lenstra lenstra lovasz reduction to korkine zolotareff reduction let l be the length of a shortest nonzero element of a lattice l we present an algorithm which for k n finds a nonzero lattice vector b so that b 2 6k2 nk l 2 this algorithm uses o n2 kk o k n2 log b arithmetic operations on o n log b bit integers this holds provided that the given basis vectors b1 bn zn are integral and have the length bound b this algorithm successively applies korkine zolotareff reduction to blocks of length k of the lattice basis we also improve kannan s algorithm for korkine zolotareff reduction

quasicrystals and icosians a family of quasicrystals of dimensions 1 2 3 4 governed by the root lattice e8 is constructed the use of the icosian ring found in the quaternions with coefficients in q square root 5 allows simultaneous interpretation of the construction both in physical space and as a result of the standard cut and projection method in double dimension icosians are seen to provide a natural co ordination scheme for these quasicrystals nested sequences of quasicrystals form systems whose symmetries are all derivable from inflational and reflective symmetries directly related to the arithmetic of the icosians the use of coxeter diagrams clarifies the amazing relationship of e8 and quasicrystal symmetries and leads to the fundamental chain e8 contains implies d6 contains implies a4 contains implies a1 a1 that underlies five fold symmetry in quasicrystals decomposition of quasicrystals into concentric shells and a counting formula for the cardinalities of these shells is discussed

high order boundary and finite elements for 3d fracture propagation in brittle materials abstract the quasi static propagation of fracture in brittle materials was studied in several recent publications a variational formulation salvadori 2008 salvadori and carini 2011 salvadori and fantoni 2013 led to three dimensional crack tracking strategies salvadori and fantoni 2014 2 3 salvadori and fantoni 2016 one of the complexities of this new class of algorithms is the evaluation of high order terms of the expansion of the crack opening and sliding in this paper new types of finite and boundary elements are formulated that capture the near crack front asymptotical displacement behavior up to the order 3 2 the use of these elements with the quasi static crack propagation algorithms of the above references is demonstrated for a simple crack configuration

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

algorithms for piecewise polynomials and splines with free knots methode de resolution de certains systemes non lineaires garantissant la convergence lors de l initialisation avec des points arbitraires

a generalized finite element method for the simulation of three dimensional dynamic crack propagation this paper is aimed at presenting a partition of unity method for the simulation of three dimensional dynamic crack propagation the method is a variation of the partition of unity finite element method and hp cloud method in the context of crack simulation this method allows for modeling of arbitrary dynamic crack propagation without any remeshing of the domain in the proposed method the approximation spaces are constructed using a partition of unity pu and local enrichment functions the pu is provided by a combination of shepard and finite element partitions of unity this combination of pus allows the inclusion of arbitrary crack geometry in a model without any modification of the initial discretization it also avoids the problems associated with the integration of moving least squares or conventional shepard partitions of unity used in several meshless methods the local enrichment functions can be polynomials or customized functions these functions can efficiently approximate the singular fields around crack fronts the crack propagation is modeled by modifying the partition of unity along the crack surface and does not require continuous remeshings or mappings of solutions between consecutive meshes as the crack propagates in contrast with the boundary element method the proposed method can be applied to any class of problems solvable by the classical finite element method in addition the proposed method can be implemented into most finite element data bases several numerical examples demonstrating the main features and computational efficiency of the proposed method for dynamic crack propagation are presented

non crystallographic reduction of generalized calogero moser models we apply a recently introduced reduction procedure based on the embedding of non crystallographic coxeter groups into crystallographic ones to calogero moser systems for rational potentials the familiar generalized calogero hamiltonian is recovered for the hamiltonians of trigonometric hyperbolic and elliptic types we obtain novel integrable dynamical systems with a second potential term which is rescaled by the golden ratio we explicitly show for the simplest of these non crystallographic models how the corresponding classical equations of motion can be derived from a lie algebraic lax pair based on the larger crystallographic coxeter group

generating functions for the coxeter group generating functions are found for decomposition of the space of homogeneous polynomials of any degree in four variables into the direct sum of subspaces irreducible under the group the non crystallographic coxeter group of order 14 400 the four variables are coordinates of a vector from the defining four dimensional representation space of the group as the defining representation we consider any of the four non equivalent irreducible representations of of dimension four analogous generating functions for the binary icosahedral group of order 120 generated by reflections i e a subgroup of but not of and for the dihedral group of order 10 are also rederived and shown the groups are naturally related by

a geometrical approach of quasiperiodic tilings tilings provide generalized frames of coordinates and as such they are used in different areas of physics the aim of the present paper is to present a unified and systematic description of a class of tilings which have appeared in contexts as disconnected as crystallography and dynamical systems the tilings of this class show periodic or quasiperiodic ordering and the tiles are related to the unit cube through affine transformations we present a section procedure generating canonical quasiperiodic tilings and we prove that true tilings are indeed obtained moreover the procedure provides a direct and simple characterization of quasiperiodicity which is suitable for tilings but which does not refer to fourier transform

inflation centres of the cut and project quasicrystals a rather general form of the conventional cut and project scheme is used to define quasicrystals as point sets in real n dimensional euclidean space the inflation or equivalently the self similarity properties of such quasicrystals are studied here assuming only the convexity of the acceptance window our result is a description of inflation centres of all types in a quasicrystal and a proof that our description is complete there are no other inflation centres for any chosen quasicrystal point internal inflation centre u its inflation properties are given as a set of scaling factors it turns out that the scaling factors form a one dimensional quasicrystal with a u dependent acceptance window scaling window the intersection of the scaling windows associated with all points of a quasicrystal is the one dimensional quasicrystal of universal internal scaling symmetries its acceptance window is the interval external inflation centres of a cut and project quasicrystal are those which are not among quasicrystal points their complete description is given analogically to the description of the internal ones imposing some additional requirements on the scaling factors between any two adjacent quasicrystal points one finds a countable infinity of external inflation centres the scaling factors belonging to any such centre u form an infinite u dependent subset of points of the quasicrystal with acceptance window containing

a certain algebraic construction of quasicrystals and their isomorphism classes certain quasicrystals will be realized in cyclotomic fields and their isomorphism structures will be given in the case of seven fold or 30 fold symmetry

affine extension of noncrystallographic coxeter groups and quasicrystals unique affine extensions haff2 haff3 and haff4 are determined for the noncrystallographic coxeter groups h2 h3 and h4 they are used for the construction of new mathematical models for quasicrystal fragments with tenfold symmetry the case of haff2 corresponding to planar point sets is discussed in detail in contrast to the cut and project scheme we obtain by construction finite point sets which grow with a model specific growth parameter

hybrid analytical and extended finite element method hax fem a new enrichment procedure for cracked solids a new numerical method is proposed to estimate directly stress intensity factors at the tip of a crack in an elastic body in the same spirit as for the extended finite element method the approximation of the displacement field is enriched in the vicinity of the crack tip yet the method proposed herein differs by the way the enrichment is introduced instead of using partition of unity concepts a two description formulation is implemented the first one uses standard finite elements while the second one in the vicinity of the crack tip resorts to a few purely analytical expressions these two descriptions are then coupled by partitioning the energy with an overlapping zone the performance of this new enrichment technique is illustrated and compared with existing techniques by means of two examples

root lattices and quasicrystals it is shown that root lattices and their reciprocals might serve as the right pool for the construction of quasicrystalline structure models all noncrystallographic symmetries observed so far are covered in minimal embedding with maximal symmetry

a direct analytical method to extract mixed mode components of strain energy release rates from irwin s integral using extended finite element method summary a new analytical approach within the extended finite element framework is proposed to compute mixed mode components of strain energy release rates directly from irwin s integral crack tip enrichment functions in extended fem allow for evaluation of integral quantities in closed form for some crack configurations studied and therefore resulting in a simple and accurate method several benchmark examples on pure and mixed mode problems are studied in particular we analyze the effects of high order enrichments mesh refinement and the integration limits of irwin s integral the results indicate that high order enrichment functions have significant effect on the convergence in particular when the integral limits are finite when the integral limits tend to zero simpler strain energy release rate expressions are obtained and high order terms vanish nonetheless these terms contribute indirectly via coefficients of first order terms the numerical results show that high accuracy can be achieved with high order enrichment terms and mesh refinement however the effect of the integral limits remains an open question with finite integration intervals chosen as h 2 tending to give more accurate results copyright 2013 john wiley sons ltd

non planar 3d crack growth by the extended finite element and level sets part i mechanical model a methodology for solving three dimensional crack problems with geometries that are independent of the mesh is described the method is based on the extended finite element method in which the crack discontinuity is introduced as a heaviside step function via a partition of unity in addition branch functions are introduced for all elements containing the crack front the branch functions include asymptotic near tip fields that improve the accuracy of the method the crack geometry is described by two signed distance functions which in turn can be defined by nodal values consequently no explicit representation of the crack is needed examples for three dimensional elastostatic problems are given and compared to analytic and benchmark solutions the method is readily extendable to inelastic fracture problems copyright 2002 john wiley sons ltd

a highly symmetric four dimensional quasicrystal a quasiperiodic pattern or quasicrystal is constructed in real four dimensional euclidean space having the non crystallographic reflection group 3 3 5 of order 14 400 as its point group it is obtained as a projection of the eight dimensional lattice e8 and has as a cross section a three dimensional quasicrystal with icosahedral symmetry

classification of voronoi and delone tiles of quasicrystals ii circular acceptance window of arbitrary size in this paper we describe all the voronoi and delone tiles arising in tilings of point sets quasicrystals built by the standard projection of the root lattice of type a4 onto a two dimensional plane spanned by the roots of the coxeter group h2 dihedral group of order 10 the acceptance window for is a disc of any radius 0 r there are 22 distinct sets vtj j 1 22 of voronoi tiles and eight sets dtk k 1 8 of delone tiles up to a uniform scaling by a factor n where 1 2 1 5 is the golden ratio and n

self similar delone sets and quasicrystals in this paper we answer the question whether any delone set invariant under quasiaddition of berman and moody can be identified with a cut and project quasicrystal for any such set we find an acceptance window which is bounded but has only convex interior the cut and project quasicrystal is then identified with an affine image of constructive methods used in the paper allow one in principle to put bounds on from a given fragment of a delone set

wavelet multiresolutions for the fibonacci chain most of the discrete wavelet families are periodic dyadic they live on the multiresolution sequence 2j j of lattice backgrounds at different scales we present here a construction of wavelet family based on a sequence of aperiodic discretizations of at a given scale this discretization is the point set of nodes of the fibonacci chain a well known stone inflation cut and project model for one dimensional quasicrystals corresponding multiresolution analysis and the elementary example of the haar system are presented

hyperbolic weyl groups and the four normed division algebras we study the weyl groups of hyperbolic kac moody algebras of over extended type and ranks 3 4 6 and 10 which are intimately linked with the four normed division algebras k r c h o respectively a crucial role is played by integral lattices of the division algebras and associated discrete matrix groups our findings can be summarized by saying that the even subgroups w of the kac moody weyl groups w are isomorphic to generalized modular groups over k for the simply laced algebras and to certain finite extensions thereof for the non simply laced algebras this hints at an extended theory of modular forms and functions

group theoretical analysis of 600 cell and 120 cell 4d polytopes with quaternions cell 3 3 5 and 120 cell 5 3 3 four dimensional dual polytopes relevant to quasicrystallography have been studied with the quaternionic representation of the coxeter group w h4 the maximal subgroups w su 5 z2 and w h3 z2 of w h4 play important roles in the analysis of cell structures of the dual polytopes in particular the weyl coxeter group w su 4 is used to determine the tetrahedral cells of the polytope 3 3 5 and the coxeter group w h3 is used for the dodecahedral cells of 5 3 3 using the lie algebraic techniques in terms of quaternions we explicitly construct cell structures forming the vertices of the 4d polytopes

matching rules and growth rules for pentagonal quasicrystal tilings preuve de l existence de ces regles pour un ensemble infini de pavages pentagonaux 2d autres que le pavage de peurose

densities minimal distances and coverings of quasicrystals quasicrystals are generalizaions of lattices in this spirit we consider here three kinds of geometric properties of mathematical models of quasicrystals minimal distances between quasicrystal points covering radii of quasicrystals and the densities of quasicrystal points

proving the deterministic period breaking of linear congruential generators using two tile quasicrystals we describe the design of a family of aperiodic prngs aprngs we show how a one dimensional two tile cut and project quasicrystal 2tqc used in conjunction with lcgs in an aprng generates an infinite aperiodic pseudorandom sequence in the suggested design any 2tqc corresponding to unitary quadratic pisot number combined with either one or two different lcgs can be used

the rings of n dimensional polytopes points of an orbit of a finite coxeter group g generated by n reflections starting from a single seed point are considered as vertices of a polytope g polytope centered at the origin of a real n dimensional euclidean space a general efficient method is recalled for the geometric description of g polytopes their faces of all dimensions and their adjacencies products and symmetrized powers of g polytopes are introduced and their decomposition into the sums of g polytopes is described several invariants of g polytopes are found namely the analogs of dynkin indices of degrees 2 and 4 anomaly numbers and congruence classes of the polytopes the definitions apply to crystallographic and non crystallographic coxeter groups examples and applications are shown

dualisation of voronoi domains and klotz construction a general method for the generation of proper space fillings the authors present a general method for the generation of proper space fillings of euclidean spaces with convex polytopes by projection from higher dimensional spaces as a special case quasiperiodic tilings are obtained by the projection of periodic structures the concept is general enough for the treatment of defects in quasiperiodic tilings by placing defects in high dimensional translation lattices

canonical substitutions tilings of ammann beenker type we present a large class of parallelogram tilings that have two important features in common with the ammann beenker tiling they can be constructed by canonical projection from z4 r4 to a plane and they admit substitution rules