Quasicrystals

A logical physical ontology is code theory, wherein reality is neither deterministic nor random. In light of Conway and Kochens' free will theorem and strong free will theorem, we discuss the plausibility of a third axiomatic option-geometric language; the code theoretic axiom. We suggest free will choices at the syntactically free steps of a geometric language of space-time form the code theoretic substrate upon which particle and gravitational physics emerge.

Un cours pour débutants sur la géométrie cristalline. Le diamant et le quartz sont des cristaux typiques par leur transparence pure et leurs formes géométriques. La géométrie extérieure est la manifestation d'une géométrie intérieure. Les cristaux sont construits par l'empilement ordonné périodique de petites briques atomiques élémentaires, les mailles. Lorsque cet ordre fait défaut, la matière est dite amorphe ou vitreuse. La cristallisation est une transformation extraordinaire qui fait passer du désordre liquide à l'ordre compact. Les roches, et toute matière y compris les matériaux industriels (métaux, silicium, polymères), contiennent des domaines cristallins et amorphes. Récemment on a reconnu l'existence de domaines cristallins aux géométries plus complexes (les quasi-cristaux), dont la périodicité n'est détectable que dans un espace à dimension supérieure. Les cristaux fournissent des exemples de la façon dont l'espace peut être pavé par des polyèdres tels que cube, octaèdre, dodécaèdre. Cet article vise à révéler la beauté, la précision et l'intelligence des formes géométriques cristallines.

In this study, new compositions of Tsai-type 1/1 quasicrystal approximants Gd(14)Au(70)Si(16), Gd(14)Au(67)Ge(19), Tb(14)Au(70)Si(16) and Yb(16)Au(65)Ge(19) are synthesized using both self-flux and arc-meltingannealing techniques. Both syntheses routes resulted single phase samples. The crystal structures of the compounds are determined by collecting single crystal X-ray and/or powder X-ray and powder neutron diffraction intensities.  The atomic structure refinements indicated that the compounds are essentially iso-structural with the prototype Tsai-type 1/1 approximant crystal, YbCd6. However, there are subtle structural variations at their cluster centers and in the so-called cubic interstices which affects some of their physical properties. Thermoelectric and magnetic properties of the compounds are investigated. Significant differences are observed in the thermoelectric properties of Gd(14)Au(70)Si(16), Gd(14)Au(67)Ge(19) and Yb(16)Au(65)Ge(19) compounds which are explained on the bases of their crystal structures and chemical compositions. Magnetic susceptibility and specific heat measurements revealed ferromagnetic transitions at low temperatures, Tc ≈ 22.5 K for Gd(14)Au(70)Si(16) and Tc ≈ 13.1 K for Gd(14)Au(67)Ge(19), whereas, for the Tb(14)Au(70)Si(16) compound a ferrimagnetic-like transition is observed at Tc ≈ 9 K. Moreover, a re-entrant spin-glass transition is observed at TRSG ≈ 3.3 K for Gd(14)Au(67)Ge(19) compound. Finally, the magnetic structure of the Tb(14)Au(70)Si(16) compound was determined from powder neutron diffraction data which is to our knowledge the first magnetic structure refinement report in the family of quasicrystals and approximants.

Trois textes réunis : Contre la méthode, Soleil de minuit 2015-2017 et Entrelacs quasi-cristallins, 2016

Ce mémoire décrit en détail le travail de recherche qui a mené à la création de deux projets d'œuvres publiques monumentales. Seule une des deux a effectivement été réalisée par la suite.

Texte / images / mise en page : Adrien Lucca
Édition Jeunesse et Arts Plastiques, Bruxelles, 2017

Inspired by the Hilbert–Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly nontrivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large [Formula: see text]) region. The ordinates [Formula: see text] are the positive imaginary parts of the nontrivial zeta zeroes in the critical line :[Formula: see text]. The latter results are consistent with the validity of the Bohr–Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly nontrivial functional form of the potential is found via the Bohr–Sommerfeld quantization formula using the full-fledged Riemann–von Mangoldt count...

Based on the Cayley-Dickson process, a sequence of multidimensional structured natural numbers (infinions) creates a path from quantum information to quantum gravity. Octonionic structure, exceptional Jordan algebra, and e8 Lie algebra are encoded on a graph with E9 connectivity, decorated by integral matrices. With the magic star, a toy model for a quantum gravity is presented with its naturally emergent quasicrystalline projective compactification.

We introduce the 2-dimensional square Fibonacci tiling and its generalization to higher dimensions as models for quasicrystals without 'forbidden' symmetries. We derive some of the basic mathematical properties of the tiling as well as calculate its diffraction pattern. We discuss the relevance of the Fibonacci tiling for quasicrystal research and for applications in other fields. 

Given that the two-parameter $ p, q$ quantum-calculus deformations  of the integers $ [ n ]_{ p, q}  = (p^n - q^n)/ ( p - q) =  F_n $  coincide
precisely with the Fibonacci numbers (integers), as a result of Binet's formula when
$ p = \tau  = { 1  +  \sqrt 5 \over 2}$, $ q  =  { \tilde \tau}  =  { 1 - \sqrt 5 \over 2 }$ (Galois-conjugate pairs),  we extend this result to 
the $generalized$ Binet's formula (corresponding to generalized Fibonacci sequences) studied by  Whitford.  Consequently,   
the Galois-conjugate pairs  $ (p, q = \tilde p ) = { 1\over 2} ( 1 \pm \sqrt m ) $,  in the very special case when $ m = 4 k + 1$ and square-free, 
generalize Binet's formula  $ [ n ]_{ p, q} = G_n$ generating integer-values for the generalized Fibonacci numbers $G_n$'s.  For these reasons, we expect that the two-parameter $ (p, q = \tilde p)$ quantum calculus should play an important role  in the physics of quasicrystals with $4k+1$-fold rotational symmetry.

Achieving high mechanical strength and ductility in age-hardenable Al7000 series (Al-Zn-Mg) alloys fabricated by selective laser melting (SLM) remains challenging. Here, we show that crack-free AlZnMgCuScZr alloys with an unprecedented strength-ductility synergy can be fabricated via SLM and heat treatment. The as-built samples had an architectured microstructure consisting of a multimodal grain structure and a hierarchical phase morphology. It consisted of primary Al 3 (Sc x ,Zr 1Àx) particles which act as inoculants for ultrafine grains, preventing crack formation. The metastable Mg-, Zn-, and Cu-rich icosahedral quasicrystals (I-phase) ubiquitously dispersed inside the grains and aligned as a filigree skeleton along the grain boundaries. The heat treated SLM-produced AlZnMgCuScZr alloy exhibited tunable mechanical behaviors through trade-off among the hierarchical features, including the dual-nanoprecipitation, viz, g 0 phase, and secondary (Al,Zn) 3 (Sc 9 Zr), and grain coarsening. Less coarsening of grains and (Al,Zn) 3 (Sc 9 Zr) particles, due to a reduced solution treatment temperature and time, could overwhelm the more complete dissolution of I-phase (triggering more g 0 phase), resulting in higher yield strength. Optimal combination of the hierarchical features yields the highest yield strength ($647 MPa) among all reported SLM-produced Al alloys to date with appreciable ductility ($11.6%). The successful fabrication of high-strength Al7000 series alloys with an adjustable hierarchical microstructure paves the way for designing and fine-tuning SLM-produced aluminum engineering components exposed to high mechanical loads.

Two-dimensional aperiodic tilings are collections of polygons, devoid of any translational symmetries, capable of covering a plane without gaps and overlaps. Although aperiodic, these structures can exhibit order and symmetry in an extended sense. In this paper, we study the radiation properties of planar antenna arrays based on certain categories of two-dimensional aperiodic tilings that illustrate diverse aspects of aperiodic order. Background material on aperiodic tilings and their known electromagnetic properties is reviewed. Results are illustrated to highlight the effects of aperiodic order in the antenna array radiation properties. Potential applications are also envisaged.

A rigorous approach for the analysis of diffraction from quasicrystalline gratings is presented. Previous methods for determining the diffraction properties of quasicrystalline gratings have relied on periodic supercell approximations. Our method exploits the cut-and-project method, which constructs quasicrystals as irrational slices of higher-dimensional periodic structures onto the physical space. The periodicity in the higher-dimensional space allows for the application of Floquet's theorem. The solutions can then be obtained by solving Maxwell's equations in the higher-dimensional space and projecting the results to the lower dimensional physical space. As an example, the method is applied to a one-dimensional aperiodic grating based on a Fibonacci quasicrystal (QC) where the results that were generated are shown to be in near-perfect agreement with those obtained using the supercell approximations.

It is argued that the prevailing definition of quasicrystals, requiring them to contain an axis of symmetry that is forbidden in periodic crystals, is inadequate. This definition is too restrictive in that it excludes an important and interesting collection of structures that exhibit all the well-known properties of quasicrystals without possessing any forbidden symmetries.

The irreversible transformation from an icosahedral quasicrystal (i-QC) CaAu4.39Al1.61 to its cubic 2/1 crystalline approximant (CA) Ca13Au56.31(3)Al21.69 (CaAu4.33(1)Al1.67, Pa3̅ (No. 205); Pearson symbol: cP728; a = 23.8934(4)), starting at ∼570 °C and complete by ∼650 °C, is discovered from in situ, high-energy, variable-temperature powder X-ray diffraction (PXRD), thereby providing direct experimental evidence for the relationship between QCs and their associated CAs. The new cubic phase crystallizes in a Tsai-type approximant structure under the broader classification of polar intermetallic compounds, in which atoms of different electronegativities, viz., electronegative Au + Al vs electropositive Ca, are arranged in concentric shells. From a structural chemical perspective, the outermost shell of this cubic approximant may be described as interpenetrating and edge-sharing icosahedra, a perspective that is obtained by splitting the traditional structural description of this shell as a 92-atom rhombic triacontahedron into an 80-vertex cage of primarily Au [Au59.86(2)Al17.14□3.00] and an icosahedral shell of only Al [Al10.5□1.5]. Following the proposal that the cubic 2/1 CA approximates the structure of the i-QC and on the basis of the observed transformation, an atomic site analysis of the 2/1 CA, which shows a preference to maximize the number of heteroatomic Au–Al nearest neighbor contacts over homoatomic Al–Al contacts, implies a similar outcome for the i-QC structure. Analysis of the most intense reflections in the diffraction pattern of the cubic 2/1 CA that changed during the phase transformation shows correlations with icosahedral symmetry, and the stability of this cubic phase is assessed using valence electron counts. According to electronic structure calculations, a cubic 1/1 CA, “Ca24Au88Al64” (CaAu3.67Al2.67) is proposed.

Abstract: We analyze transmission electron microscopy (TEM) images of self-assembled quasicrystals, composed of binary systems of nanoparticles. We use an automated procedure that identifies the positions of dislocations and determines their topological character. To achieve this we decompose the quasicrystal into its individual density modes, or Fourier components, and identify their topological winding numbers for every dislocation.

In light of the self-simulation hypothesis, a simple form implementation of the principle of efficient language is discussed in a self-referential geometric quasicrystalline state sum model in three dimensions. Emergence is discussed in context of geometric state sum models.

We present an icosahedral quasicrystal, a Fibonacci icosagrid, obtained by spacing the parallel planes in an icosagrid with the Fibonacci sequence. This quasicrystal can also be thought as a golden composition of five sets of Fibonacci tetragrids. We found that this quasicrystal embeds the quasicrystals that are golden compositions of the three-dimensional tetrahedral cross-sections of the Elser-Sloane quasicrystal, which is a four-dimensional cut-and-project of the E8 lattice. These compound quasicrystals are subsets of the Fibonacci icosagrid, and they can be enriched to form the Fibonacci icosagrid. This creates a mapping between the Fibonacci icosagrid and the E−8 lattice. It is known that the combined structure and dynamics of all gravitational and Standard Model particle fields, including fermions, are part of the E8 Lie algebra. Because of this, the Fibonacci icosagrid is a good candidates, for representing states and interactions between particles and fields in quantum mechanics. We coin the name Quasicrystalline Spin-Network (QSN) for this quasicrystalline structure.

A theoretical study of optical reflectance and reflection bands of 1-D photonic quasi-crystals (Fibonacci type arrangement) composed of exponentially graded material is presented. The proposed structures consist of two different layers, one of them is of constant refractive index (L) and the other one is of exponentially graded refractive index (S) dielectric materials. Four different generations (2 nd , 3 rd , 4 th and 5 th ) of the Fibonacci sequence for 10 periods in one dimension (1-D) are considered and compared in view of their optical reflectance and bandgaps for both TE and TM polarisations. Also, we proposed some heterostructures made by the combination of different Fibonacci generations and their periods to obtain suitable omnidirectional reflection band. We used the transfer matrix method (TMM) to obtain the reflectance, bandgaps and omnidirectional reflectional bandgaps (ODR) of such structures in near infrared spectrum (800-2200 nm) at different angles of incidence. We show that ODR exists in these types of structures. The number of ODRs and total bandgap depend on the Fibonacci generations. Extraordinary ODR bandgaps are obtained in the case of heterostructures formed by the combination of different generations of the Fibonacci sequence. The ODR for these structures is similar to the ODR of photonic crystals containing left-handed materials. This work would be useful to study the Fibonacci type photonic crystals having graded index materials and also it will open new window to design several photonic crystal devices like sensors, reflectors, etc. in the infrared region.

The orthorhombic -Al 5 Mg 11 Zn 4 phase is known to be related to quasicrystalline phases, but the exact relationship has not been shown yet. In this study, the relationship of this phase to the icosahedral quasicrystalline phase is explored through analysis of the electron diffraction patterns. It is shown that icosahedral coordinations in three orientations occur in the unit cell -one with three mutually perpendicular twofold axes along 1 0 0 of the unit cell, and two with a twofold axis along [1 0 0] and a fivefold along [0 0 1]. In this, this phase is similar to aluminum and zinc based hexagonal phases which are related to quasicrystals.

The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal's vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and Voronoi cell complexes of the lattice being projected. We describe a new method for calculating empires (forced tiles) which also borrows from the dualisation formalism and which generalizes to tilings generated projections of non-cubic lattices. These techniques were used to compute the vertex configurations, frequencies and empires of icosahedral quasicrystals obtained as a projections of the D 6 and Z 6 lattices to R 3 and we present our analyses. We discuss the implications of this new generalization.

An overview of recent experimental and theoretical investigations on undercooled liquids is given, indicating developing short-range order with undercooling. Our in situ synchrotron X-ray diffraction studies of electrostatically levitated droplets of elemental Ti, Zr, and Ni liquids and coordinated undercooling and structural studies of a Ti-Zr-Ni liquid that forms an icosahedral quasicrystal are discussed in the context of these investigations. Icosahedral order is demonstrated to be a dominant component to the structures of all undercooled liquids studied, although it becomes significantly distorted with an increasing importance of the d-band bonding character. The liquid short-range order couples with the nucleation barrier, as predicted by Frank over a half-century ago.

A rigorous approach for the analysis of diffraction from quasicrystalline gratings is presented. Previous methods for determining the diffraction properties of quasicrystalline gratings have relied on periodic supercell approximations. Our method exploits the cut-and-project method, which constructs quasicrystals as irrational slices of higher-dimensional periodic structures onto the physical space. The periodicity in the higher-dimensional space allows for the application of Floquet's theorem. The solutions can then be obtained by solving Maxwell's equations in the higher-dimensional space and projecting the results to the lower dimensional physical space. As an example, the method is applied to a one-dimensional aperiodic grating based on a Fibonacci quasicrystal (QC) where the results that were generated are shown to be in near-perfect agreement with those obtained using the supercell approximations.

In a previous paper we outlined a series of historical touchpoints between classical aether theories and modern theoretical physics which showed a shared conceptual lineage for the modern tools and methods of the most common interpretations and fluid based "Hydrodynamic" treatments of an electromagnetic medium. It was proposed that, though the weight of modern experimentation leaves an extremely narrow and convoluted window for even a reconceptualization of a medium, all of modern physics recognizes a plethora of behaviors and attributes for free space and these physics are interchangeable with modern methods for treating superfluid-like continuums. Thus the mathematical equivalence of the methods do not comprise alternative physics but an alternative interpretation of the same physics. Though many individual components describing a "neo-aether" or "quintessence" are available, an overarching structural outline of how these tools can work together to provide an alternative working overview of modern physics has remained undefined. This paper will propose a set of introductory concepts in the first outline of a toy model which will later connect the alternative tools and conceptualizations with their modern counterparts. This introductory paper provides the simpler "100-miles out" overview of the whole of physics from this perspective, in an easily comprehensible, familiar and intuitive, informal dialog fashion. While this paper grants the largest and loosest introductory overview, subsequent papers in this series will address the finite connections between modern physics and this hydrodynamic view.

In quasicrystals, any given local patch-called an emperor-forces at all distances the existence of accompanying tiles-called the empire-revealing thus their inherent nonlocality. In this chapter, we review and compare the methods currently used for generating the empires, with a focus on the cut-and-project method, which can be generalized to calculate empires for any quasicrystals that are projections of cubic lattices. Projections of non-cubic lattices are more restrictive and some modifications to the cut-and-project method must be made in order to correctly compute the tilings and their empires. Interactions between empires have been modeled in a game-of-life approach governed by nonlocal rules and will be discussed in 2D and 3D quasicrystals. These nonlocal properties and the consequent dynamical evolution have many applications in quasicrystals research, and we will explore the connections with current material science experimental research.

This article presents a novel Hamiltonian architecture based on vertex types and empires for demonstrating the emergence of aperiodic order (quasicrystal growth) in one dimension by a suitable prescription for breaking translation symmetry. At the outset, the paper presents different algorithmic, geometrical, and algebraic methods of constructing empires of vertex configurations of a given quasi-lattice. These empires have non-local scope and form the building blocks of the new lattice model. This model is tested via Monte Carlo simulations beginning with randomly arranged N tiles. The simulations clearly establish the Fibonacci configuration, which is a one dimensional quasicrystal of length N , as the final relaxed state of the system. The Hamiltonian is promoted to a matrix operator form by performing dyadic tensor products of pairs of interacting empire vectors followed by a summation over all permissible configurations. A spectral analysis of the Hamiltonian matrix is performed and a theoretical method is presented to find the exact solution of the attractor configuration that is given by the Fibonacci chain as predicted by the simulations. Finally, a precise theoretical explanation is provided which shows that the Fibonacci chain is the most probable ground state. The proposed Hamiltonian is a one dimensional model of quasicrystal growth.

The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal's vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and Voronoi cell complexes of the lattice being projected. We describe a new method for calculating empires (forced tiles) which also borrows from the dualisation formalism and which generalizes to tilings generated projections of non-cubic lattices. These techniques were used to compute the vertex configurations, frequencies and empires of icosahedral quasicrystals obtained as a projections of the D 6 and Z 6 lattices to R 3 and we present our analyses. We discuss the implications of this new generalization.

Nano-quasicrystalline Al-Fe-Cr based alloys produced by rapid solidification processes exhibit high strength at elevated temperatures. Nevertheless, the quasicrystalline particles in these systems become unstable at high temperature limiting the industrial applications. In early works, it was observed that the use of Nb or Ta increases the stability of the Al-Fe-Cr quasicrystalline phase delaying the microstructural transformation to higher temperatures. Thus, these nano-quasicrystalline Al-based alloys have become promising new high strength material to be used at elevated temperatures in the automotive and aeronautical industries. In previous works, nano-quasicrystalline Al-Fe-Cr-Nb based alloys were obtained by rapid solidification using the melt-spinning technique. In order to obtain bulk alloys for industrial applications other fabrication routes such as powder production by gas atomization followed by compaction and extrusion are required. In the present work, the production of Al-Fe-Cr-Nb based alloys by powder atomization at laboratory scale was investigated. The powders obtained were sieved in different ranges of sizes and the microstructures were characterised by means of X-ray diffraction, scanning and transmission electron microscopy, and energy dispersive of X-ray analysis. Mechanical properties have been measured by compression tests at room temperature and at 250°C. It was observed that a very high temperature is required to produce these alloys by gas atomization; the icosahedral quasicrystalline phase can be retained after the atomization in powder sizes typically under 75 lm, and also after the extrusion at 375°C. The extruded bars were able to retain a very high strength at elevated temperature, around 60% of the yield stress at room temperature, in contrast with the 10-30% typically obtained for many commercial Al alloys.

New crystal structure of an orthorhombic phase formed at the alloy composition Cu 48.1 Al 36.4 Sc 15.5 has been analyzed by means of Rietveld method using synchrotron radiation diffraction data. The starting model for this analysis was constructed theoretically using so called "cell constrained melt-quenching technique". The space group of the final model is Cmmm, and the unit cell includes 47.0 Cu, 39.0 Al and 16.0 Sc atoms. The lattice parameters are a = 8.337 (4) Å, b = 22.02 (1) Å and c = 8.305 Å, which are related to the 6-dimensional lattice parameter, a 6D = 6.959 Å, of the corresponding Cu-Al-Sc icosahedral quasicrystal as 1/0, 2/1 and 1/0 approximation, respectively. The characteristics of the structure can be understood as a framework consisting of Sc atoms, which is regarded as tiling of five local structural units; the largest is an icosahedron similar to that included in the Tsai-type quasicrystal. The second exhibits structural similarity to Mg 2 Zn 11 -type crystal, and the third is an octahedron.

A simple generalization of the Swift-Hohenberg equation is proposed as a model for the pattern- forming dynamics of a two-dimensional field with two unstable length scales. The equation is used to study the dynamics of surface waves in a fluid driven by a linear combination of two frequencies. The model exhibits steady-state solutions with 2-, 4-, 6-, and 12-fold symmetric patterns, similar to the periodic and quasiperiodic patterns observed in recent experiments.

As solar radiation is the most plentiful energy source on earth, thermoelectric energy harvesting emerges as an interesting solution for the Internet of Things (IoTs) in outdoor applications, particularly using semiconductor thermoelectric generators (TEGs) to power IoT devices. However, when a TEG is under solar radiation, the temperature gradient through TEG is minor, meaning that the TEG is useless. A method to keep a significant temperature gradient on a TEG is by using a solar absorber on one side for heating and a heat sink on the other side. In this paper, a compact TEG-based energy harvester that features a solar absorber based on a new class of solid matter, the so-called quasicrystal (QC), is presented. In addition, a water-cooled heat sink to improve the temperature gradient on the TEG is also proposed. The harvester is connected to a power management circuit that can provide an output voltage of 3 V and store up to 1.38 J in a supercapacitor per day. An experimental eval...

In this paper, we explore the effects of misfit strain fields on both heterogeneous nucleation behavior and anisotropic growth of islands at submonolayer coverages and compositional patterning at complete monolayer coverage via simulations of a phase-field crystal model. In particular, deposition on top of a herringbone structure and quasicrystalline (QC) substrate are considered, the former representing a system with spatially periodic misfit strain fields arising from the presence of surface dislocations, and the latter representing a system which inherently possesses a wide range of local, aperiodic misfit patterns. In the case of single-component systems, we demonstrate that misfit strain fields lead to heterogeneous nucleation behavior and anisotropic island growth. In the case of QC substrate, a wide range of morphologies, such as coexistence of locally hexagonally ordered atomic clusters within a larger scale arrangement with overall QC symmetry and so-called "starfish" aggregates, is observed in a pure system at submonolayer coverages when the adlayer-substrate interaction strength and lattice mismatch are tuned. In the case of bulk-immiscible binary systems at complete monolayer coverage, strain-stabilized compositional domains emerge at low line tension values for both substrates. Interestingly, the compositional domains on the QC substrate inherit their symmetries at sufficiently low line tension values, while at larger line tension values, the domain structure begins to resemble the classical spinodal microstructure. Such domain structures should be readily observable in colloidal systems in which attractive interparticle and particle-substrate interactions can be tuned.

In this paper, we will investigate the optical waveguiding properties of the quasi-periodic Band Gap structures. In this case the Fibonacci quasi-crystals are used and the Eigenmodes and eigenvalues of the electromagnetic wave propagating through these structures are obtained. Also, the band structure of the system is obtained and the guiding modes distribution is demonstrated. Finally these structures can be used as optical filters with multi reflection peaks, which can be used in optical signal processing and DWDM systems.

W e present an icosahedral quasicrystal, a Fibonacci icosagrid, obtained by spacing the parallel planes in an icosagrid with the Fibonacci sequence. This quasicrystal can also be thought as a golden composition of five sets of Fibonacci tetragrids. We found that this quasicrystal embeds the quasicrystals that are golden compositions of the three-dimensional tetrahedral cross-sections of the Elser-Sloane quasicrystal, which is a four-dimensional cut-and-project of the E8 lattice. These compound quasicrystals are subsets of the Fibonacci icosagrid, and they can be enriched to form the Fibonacci icosagrid. This creates a mapping between the Fibonacci icosagrid and the E−8 lattice. It is known that the combined structure and dynamics of all gravitational and Standard Model particle fields, including fermions, are part of the E8 Lie algebra. Because of this, the Fibonacci icosagrid is a good candidates, for representing states and interactions between particles and fields in quantum mech...

The fairly recent discovery of "quasicrystals", whose X-ray diffraction patterns reveal certain peculiar features which do not conform with spatial periodicity, has motivated studies of the wave-dynamical implications of "aperiodic order". Within the context of the radiation properties of antenna arrays, an instructive novel (canonical) example of wave interactions with quasiperiodic order is illustrated here for one-dimensional (1-D) array configurations based on the "modified-Fibonacci" sequence, with utilization of a two-scale generalization of the standard Poisson summation formula for periodic arrays. This allows for a "quasi-Floquet" analytic parameterization of the radiated field, which provides instructive insights into some of the basic wave mechanisms associated with quasiperiodic order, highlighting similarities and differences with the periodic case. Examples are shown for quasiperiodic infinite and spatially-truncated arrays, with brief discussion of computational issues and potential applications.

Os quase-cristais apresentam simetrias de rotação impossíveis em cristalografia que estão associadas aos números metálicos. Os seus átomos distam uns dos outros de modo diferente, embora permitam gerar padrões regulares não repetitivos.
Com vista à apresentação de algumas aplicações dos números metálicos em estruturas dos quase-cristais, evidencia-se previamente alguns resultados inerentes aos números metálicos e estabelece-se sinteticamente uma classificação dos grupos cristalográficos tridimensionais, seguida de uma caracterização das estruturas dos quase-cristais. 

The quasi-crystals have rotational symmetries impossible in crystallography that are associated with Metallic Means Numbers. Its atoms are distant from each other differently, although they can produce non-repeating regular patterns.
In order to present some applications of Metallic Means Numbers in the quasi-crystal structures, we note some results of the Metallic Means Numbers and we established, synthetically, a classification of three-dimensional crystallographic groups, followed by a characterization of quasi-crystals structures.

Lubricants can affect quasicrystalline coatings surfaces by modifying commensurability of the interfaces. We report results of the first computer simulation studies of physically adsorbed hydrocarbons on a quasicrystalline surface: methane, propane, and benzene on decagonal Al-Ni-Co. The grand canonical Monte Carlo method is employed, using novel Embedded Atom Method potentials generated from it ab initio calculations, and standard hydrocarbon interactions. The resulting adsorption isotherms and calculated structures show the films' evolution from submonolayer to condensation. We discover the presence and absence of the 5- to 6-fold topological transition, for benzene and methane, respectively, in agreement with a previsouly formulated phenomenological rule based on adsorbate-substrate size mismatch.

Room temperature adsorption of C 60 on the¯at quasicrystalline surface of Al 70 Pd 21 Mn 9 has been investigated using scanning tunnelling microscopy. A dispersed overlayer is formed at low coverage, with avoidance of step-edges. There is no evidence of island formation or clustering. As the coverage is increased, a higher density layer is formed with no evidence of the formation of hexagonal ordered adsorbate structures seen on other substrates. This is followed by the onset of second layer formation. A range of bonding sites for C 60 molecules is implied from measurements of apparent molecular heights and from thermal eects. Detailed analysis of the surface at a low coverage ($0.065 ML) provides evidence of adsorbate local order, with Fibonacci (s-scaling) relationships between the C 60 molecules. Where this occurs, the preferred adsorption site is tentatively identi®ed as the pentagonal hollow. These local correlations however are not found to extend over larger regions of the surface. Ó

The compound Ca 13 Cd 76 has been synthesized and its structure has been solved from single crystal X-ray diffraction data. The discovery of the stable binary icosahedral quasicrystals YbCd 5.7 and CaCd 5.7 opens the doors to new possibilities for understanding quasicrystalline structures. A stable quasicrystalline phase has the obvious advantage that large, single-grain crystals can be grown, which allow detailed investigations of physical properties in the bulk, structural studies with neutron radiation, etc. Most stable quasicrystals are ternary or multinary, and this leads to uncertainties in the occupancy of different atomic sites in the structure. A binary quasicrystal thus offers the additional advantage of simplicity in its structural description. The uniqueness of the MCd 5.7 (M Ca, Yb) quasicrystals lies in the combination of these features. Although icosahedral approximants are ubiquitous, the MCd 5.7 compounds are the only known icosahedral quasicrystals that exhibit both these characteristics.

Yb-Cd (Tsai-type) quasicrystals constitute the largest icosahedral quasicrystal family where Yb can be replaced by other rare earth elements (RE) and Cd by pairs of p-and d-block elements. YbCd 6 is a prototype 1/1 Tsai-type approximant phase which has a similar local structure to the Yb-Cd quasicrystal. In this study, the syntheses of Yb 15.Si 15.94 Tsai-type 1/1 quasicrystal approximants are optimized using the self-flux technique. The crystal structures of the compounds are refined by collecting single crystal x-ray diffraction data. The structural refinements indicated that the compounds are essentially isostructural with some differences at their cluster centers. The basic polyhedral cluster unit in all the three compounds can be described by concentric shells of icosahedra symmetry and of disordered tetrahedra and/or a rare earth atom at the cluster center. Furthermore, the thermoelectric properties of the compounds are probed and their dimensionless figures of merit are calculated at different temperatures. A significant difference is observed in their thermoelectric properties, which could arise due to the slight difference in their crystal structure and chemical composition, as we move from Ge to Si and/or Gd to Yb. Therefore, this study shows the systematic effect of the chemical substitution of structurally similar materials on their thermoelectric properties.

We present an icosahedral quasicrystal as a modification of the icosagrid, a multigrid with 10 plane sets that are arranged with icosahedral symmetry. We use the Fibonacci chain to space the planes, thereby obtaining a quasicrystal with icosahedral symmetry. It has a surprising correlation to the Elser-Sloane quasicrystal1, a 4D cut-and-project of the E8 lattice. We call this quasicrystal the Fibonacci modified icosagrid quasicrystal. We found that this structure totally embeds another quasicrystal that is a compound of 20 3D slices of the Elser-Sloane quasicrystal. The slices, which contain only regular tetrahedra, are put together by a certain golden ratio based rotation2. Interesting 20-tetrahedron clusters arranged with the golden ratio based rotation appear repetitively in the structure. They are arranged with icosahedral symmetry. It turns out that this rotation is the dihedral angle of the 600-cell (the super-cell of the Elser-Sloane quasicrystal) and the angle between the tetr...