grammars for icosahedral danzer tilings we have recently interpreted 2d quasi periodic patterns in terms of substitutions in this paper we extend this interpretation to 3d in particular we describe danzer tilings in terms of word sequences of l systems

curvature metric and parametrization of origami tessellations theory and application to the eggbox pattern origami tessellations are particular textured morphing shell structures their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape curvature and elongation on a global scale the existence of these global deformation modes allows for origami tessellations to fit non trivial surfaces thus inspiring applications across a wide range of domains including structural engineering architectural design and aerospace engineering the present paper suggests a homogenization type two scale asymptotic method which combined with standard tools from differential geometry of surfaces yields a macroscopic continuous characterization of the global deformation modes of origami tessellations and other similar periodic pin jointed trusses the outcome of the method is a set of nonlinear differential equations governing the parametrization metric and curvature of surfaces that the initially discrete structure can fit the theory is presented through a case study of a fairly generic example the eggbox pattern the proposed continuous model predicts correctly the existence of various fittings that are subsequently constructed and illustrated

algebraic structures for 1d quasiperiodic systems the structure of certain quasiperiodic systems is analysed and interpreted by algebraic means all systems considered have a free monoid and semigroup structure with n generators as treated in the field of combinatorics on words an extension from free monoids to free groups and their automorphisms as treated by combinatorial group theory applies to a class of iterative systems the group structures yield algebraic relations and conserved quantities examples encompass the fibonacci system and its generalizations motivated by the applications to the physics of these systems homomorphisms from the free groups into linear operators of su 2 and su 1 1 are studied the systems are also viewed in a geometry due to fricke and klein the homomorphism to the group su 1 1 is the basis for the determination of 1d electronic spectra the class types their multiplication and the commutator properties are given the physics of these systems corresponds to the continuous schrodinger equation the quasiperiodic structure gives rise to exact relations between continuous and discrete systems

localization problem in one dimension mapping and escape a one dimensional schr odinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices when the potential is periodic the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength this critical index is also evaluated as the solution to an escape rate problem for the recursion relations

remarks on the spectral properties of tight binding and kronig penney models with substitution sequences we comment on some recent investigations on the electronic properties of models associated with the thue morse chain point out that their conclusions contradict rigorously proven theorems and indicate some of the sources of these misinterpretations we briefly review and explain the current status of mathematical results in this field and discuss some conjectures and open problems

on the trace map for products of matrices associated with substitutive sequences in a recent article m kola and m k ali study the polynomial trace map for products of matrices associated with substitutive sequences on a two letter alphabet the existence of which has been proved by j p allouche and j peyriere computer calculations led them to conjecture some divisibility properties of the involved polynomials the present work explains mathematically why it is so

generators and subgroups for aut f3 the automorphisms of the free group f3 with three generators form the group aut f3 new involutive generators and relations for aut f3 are given which explain the relations given by nielsen finite coxeter groups and normal subgroups are derived which admit generalizations to n or 3

metabelian p groups and coclass theory abstract we show that the coclass tree associated with the metabelian p groups of a fixed coclass is virtually periodic the proof of this result is obtained by showing that the metabelian p groups of a fixed coclass consist of finitely many coclass families and finitely many other groups this also implies that the metabelian p groups of a fixed coclass can be classified in the sense that they can be described by finitely many parametrised presentations

fibonacci orbits and su 2 dynamics motivated by an interpretation of the fibonacci sequence as a subset of the free group f2 we describe orbits on finite subgroups of su 2 a prerequisite for the calculation of these orbits is the knowledge of all generating sets of finite subgroups of su 2 in order to facilitate this problem we introduce an equivalence relation between generating sets which can be extended to another one between fibonacci orbits additionally we describe a possible experiment with quasiperiodic dynamics

modelling the folding of paper into three dimensions using affine transformations abstract we model the folding of ordinary paper via piecewise isometries r 2 r 3 the collection of crease lines and vertices in the unfolded paper is called the crease pattern our results generalize the previously known necessity conditions from the more restrictive case of folding paper flat into r 2 if the crease pattern is foldable then the product in a non intuitive order of the associated rotational matrices is the identity matrix this condition holds locally in a multiple vertex crease pattern and can be adapted to a global condition sufficiency conditions are significantly harder and are not known except in the two dimensional single vertex case

classifying 2 groups by coclass now that the conjectures of leedham green and newrnan have been proved we probe deeper into the classification of p groups using coclass we determine the pro 2 groups of coclass at most 3 and use these to classify the 2 groups of coclass at most 3 into families using extensive computational evidence we make some detailed conjectures about the structure of these families we also conjecture that the 2 groups of arbitrary fixed coclass exhibit similar behaviour

modelling rigid origami with quaternions and dual quaternions this paper examines the mathematical modelling of rigid origami a type of origami where all the panels are rigid and can only rotate about crease lines the rotating vector model is proposed which establishes the loop closure conditions among a group of characteristic vectors by building up an explicit relationship between the single vertex origami and the spherical linkage mechanism the rotating vector model can conveniently and directly describe arbitrary three dimensional configurations and can detect some self intersection quaternion and dual quaternion are then employed to represent the origami model based on which two numerical methods have been developed through examples it has been shown that the first method can effectively track the entire rigid folding procedure of an initially flat or a non flat pattern with a single vertex or multiple vertices and thereby provide judgment for its rigid foldability and flat foldability furthermore its ability to rule out some self intersecting configurations during folding is illustrated in detail leading to its ability of checking rigid foldability in a more or less sufficient way the second method is especially for analysing the multi vertex origami it can also effectively track the trajectories of multiple vertices during folding

automorphism groups of 2 groups abstract a well known conjecture on p groups states that every non abelian p group g has the property that g divides aut g we exhibit periodic patterns in the automorphism group orders of the 2 groups of fixed coclass and we use this to show that for every positive integer r there are at most finitely many counterexamples to the conjecture among the 2 groups of coclass r

a free energy bound for the hopfield model the author gives a simple upper and lower bound on the free energy density of the hopfield model of size n with p stored patterns in the limit where n and p tend to infinity with p n to alpha 1 the two bounds coincide for alpha 0

involutive generators and actions for the group phi 2 the automorphisms of the free group f2 form the group phi 2 three involutions are shown to generate phi 2 the homomorphism h2 phi 2 to gl 2 z induced by the abelianization h1 f2 to z2 is generated by three affine reflections r3 to r3 the actions of phi 2 induced by the homomorphisms f2 to su 2 su 1 1 sl 2 c are generated by algebraic involutions and geometric reflections

a natural class of generalized fibonacci chains in this paper we propose a class of substitution rules that generate quasiperiodic chains sharing their typid properties with the quasiperiodic fibonacci chain for a subclass we explicitly con vct the atomic surface moreover scaling properties of the energy specmm are discussed in relation to the dynamics of trace maps

beyond bragg mirrors the design of aperiodic omnidirectional multilayer reflectors a new method that enables omnidirectional reflectivity for all polarizations of incident light over a wide selectable range of wavelengths was used to design aperiodic multilayer mirrors choosing the materials and with the desired threshold value of the reflection coefficient also being given the resulting reflectors work for all polarizations in a predetermined range of incidence angles and wavelengths they are 1d photonic aperiodic crystals composed of a stack of layers arranged according to a deterministic aperiodic substitutive sequence appropriately determined in the calculation of the optical multilayer properties with the rayleigh abeles matrices the use of the computationally efficient building block recurrence or trace antitrace map techniques avoids numerical instability and strongly reduces computational time experimental measurements of a prototype satisfying predetermined requirements fabricated after the sequence generated by a b bba bbba show excellent agreement

design of three dimensional origami structures based on a vertex approach origami geometric design is fundamental to many engineering applications of origami structures this paper presents a new method for the design of three dimensional 3d origami structures suitable

design and simulation of origami structures with smooth folds origami has enabled new approaches to the fabrication and functionality of multiple structures current methods for origami design are restricted to the idealization of folds as creases of zeroth o