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Quasicrystalline alloys and their composites have been extensively studied due to their complex atomic structures, mechanical properties, and their unique tribological and thermal behaviors. However, technological applications of these materials have not yet come of age and still require additional developments. In this review, we discuss the recent advances that have been made in the last years toward optimizing fabrication processes and properties of Al-matrix composites reinforced with quasicrystals. We discuss in detail the high-strength rapid-solidified nanoquasicrystalline composites, the challenges involved in their manufacturing processes and their properties. We also bring the latest findings on the fabrication of Al-matrix composites reinforced with quasicrystals by powder metallurgy and by conventional metallurgical processes. We show that substantial developments were made over the last decade and discuss possible future studies that may result from these recent findings.

Quasicrystals are non-periodic solids that were discovered in 1982 by Dan Shechtman, Nobel Prize Laureate in Chemistry 2011. The underlying mathematics, known as the theory of aperiodic order, is the subject of this comprehensive multi-volume series. This first volume provides a graduate-level introduction to the many facets of this relatively new area of mathematics. Special attention is given to methods from algebra, discrete geometry and harmonic analysis, while the main focus is on topics motivated by physics and crystallography. In particular, the authors provide a systematic exposition of the mathematical theory of kinematic diffraction. Numerous illustrations and worked-out examples help the reader to bridge the gap between theory and application. The authors also point to more advanced topics to show how the theory interacts with other areas of pure and applied mathematics.

Quasicrystals occupy a unique position between periodic and disordered systems, where localization phenomena such as Anderson transitions and mobility edges (MEs) can emerge even in the absence of disorder. This distinctive behavior motivates the development of robust, all-optical diagnostic tools capable of probing the structural, topological, and dynamical properties of such systems. In this work, focusing on generalized Aubry–André–Harper models and on an incommensurate potential in the continuum limit, we demonstrate that high-harmonic generation (HHG) phenomenon serves as a powerful probe of localization transitions and MEs in quasicrystals. We introduce a new parameter–dipole mobility–which captures the impact of intraband dipole transitions and enables classification of nonlinear optical regimes, where excitation and HHG yield can differ by orders of magnitude. We show that the cutoff frequency of harmonics is strongly influenced by the position of the ME, providing a robust and experimentally accessible signature of localization transitions in quasicrystals.

Moiré lattices consist of two superimposed identical periodic structures with a relative rotation angle. Moiré lattices have several applications in everyday life, including artistic design, the textile industry, architecture, image processing, metrology and interferometry. For scientific studies, they have been produced using coupled graphene–hexagonal boron nitride monolayers 1 , 2 , graphene–graphene layers 3 , 4 and graphene quasicrystals on a silicon carbide surface 5 . The recent surge of interest in moiré lattices arises from the possibility of exploring many salient physical phenomena in such systems; examples include commensurable–incommensurable transitions and topological defects 2 , the emergence of insulating states owing to band flattening 3 , 6 , unconventional superconductivity 4 controlled by the rotation angle 7 , 8 , the quantum Hall effect 9 , the realization of non-Abelian gauge potentials 10 and the appearance of quasicrystals at special rotation angles 11 . A fundamental question that remains unexplored concerns the evolution of waves in the potentials defined by moiré lattices. Here we experimentally create two-dimensional photonic moiré lattices, which—unlike their material counterparts—have readily controllable parameters and symmetry, allowing us to explore transitions between structures with fundamentally different geometries (periodic, general aperiodic and quasicrystal). We observe localization of light in deterministic linear lattices that is based on flat-band physics 6 , in contrast to previous schemes based on light diffusion in optical quasicrystals 12 , where disorder is required 13 for the onset of Anderson localization 14 (that is, wave localization in random media). Using commensurable and incommensurable moiré patterns, we experimentally demonstrate the two-dimensional localization–delocalization transition of light. Moiré lattices may feature an almost arbitrary geometry that is consistent with the crystallographic symmetry groups of the sublattices, and therefore afford a powerful tool for controlling the properties of light patterns and exploring the physics of periodic–aperiodic phase transitions and two-dimensional wavepacket phenomena relevant to several areas of science, including optics, acoustics, condensed matter and atomic physics. A superposition of tunable photonic lattices is used to create optical moiré patterns and demonstrate the resulting localization of light waves through a mechanism based on flat-band physics.

Quasicrystalline alloys and their composites have been extensively studied due to their complex atomic structures, mechanical properties, and their unique tribological and thermal behaviors. However, technological applications of these materials have not yet come of age and still require additional developments. In this review, we discuss the recent advances that have been made in the last years toward optimizing fabrication processes and properties of Al‐matrix composites reinforced with quasicrystals. We discuss in detail the high‐strength rapid‐solidified nanoquasicrystalline composites, the challenges involved in their manufacturing processes and their properties. We also bring the latest findings on the fabrication of Al‐matrix composites reinforced with quasicrystals by powder metallurgy and by conventional metallurgical processes. We show that substantial developments were made over the last decade and discuss possible future studies that may result from these recent findings.

Electronic states in quasicrystals generally preclude a Bloch description 1 , rendering them fascinating and enigmatic. Owing to their complexity and scarcity, quasicrystals are underexplored relative to periodic and amorphous structures. Here we introduce a new type of highly tunable quasicrystal easily assembled from periodic components. By twisting three layers of graphene with two different twist angles, we form two mutually incommensurate moiré patterns. In contrast to many common atomic-scale quasicrystals 2 , 3 , the quasiperiodicity in our system is defined on moiré length scales of several nanometres. This ‘moiré quasicrystal’ allows us to tune the chemical potential and thus the electronic system between a periodic-like regime at low energies and a strongly quasiperiodic regime at higher energies, the latter hosting a large density of weakly dispersing states. Notably, in the quasiperiodic regime, we observe superconductivity near a flavour-symmetry-breaking phase transition 4 , 5 , the latter indicative of the important role that electronic interactions play in that regime. The prevalence of interacting phenomena in future systems with in situ tunability is not only useful for the study of quasiperiodic systems but may also provide insights into electronic ordering in related periodic moiré crystals 6 – 12 . We anticipate that extending this platform to engineer quasicrystals by varying the number of layers and twist angles, and by using different two-dimensional components, will lead to a new family of quantum materials to investigate the properties of strongly interacting quasicrystals. A moiré quasicrystal constructed by twisting three layers of graphene with two different twist angles shows high tunability between a periodic-like regime at low energies and a strongly quasiperiodic regime at higher energies alongside strong interactions and superconductivity.

Phase transitions connect different states of matter and are often concomitant with the spontaneous breaking of symmetries. An important category of phase transitions is mobility transitions, among which is the well known Anderson localization 1 , where increasing the randomness induces a metal–insulator transition. The introduction of topology in condensed-matter physics 2 – 4 lead to the discovery of topological phase transitions and materials as topological insulators 5 . Phase transitions in the symmetry of non-Hermitian systems describe the transition to on-average conserved energy 6 and new topological phases 7 – 9 . Bulk conductivity, topology and non-Hermitian symmetry breaking seemingly emerge from different physics and, thus, may appear as separable phenomena. However, in non-Hermitian quasicrystals, such transitions can be mutually interlinked by forming a triple phase transition 10 . Here we report the experimental observation of a triple phase transition, where changing a single parameter simultaneously gives rise to a localization (metal–insulator), a topological and parity–time symmetry-breaking (energy) phase transition. The physics is manifested in a temporally driven (Floquet) dissipative quasicrystal. We implement our ideas via photonic quantum walks in coupled optical fibre loops 11 . Our study highlights the intertwinement of topology, symmetry breaking and mobility phase transitions in non-Hermitian quasicrystalline synthetic matter. Our results may be applied in phase-change devices, in which the bulk and edge transport and the energy or particle exchange with the environment can be predicted and controlled.  A triple phase transition, where changing a single parameter simultaneously gives rise to metal–insulator, topological and a parity–time symmetry-breaking phase transitions, is observed in non-Hermitian Floquet quasicrystals.

Applying the mathematical concept of topology to the wave-vector space of photonics yields exciting opportunities for creating new states of light with useful properties such as unidirectional propagation and the ability to flow around imperfections. The application of topology, the mathematics of conserved properties under continuous deformations, is creating a range of new opportunities throughout photonics. This field was inspired by the discovery of topological insulators, in which interfacial electrons transport without dissipation, even in the presence of impurities. Similarly, the use of carefully designed wavevector-space topologies allows the creation of interfaces that support new states of light with useful and interesting properties. In particular, this suggests unidirectional waveguides that allow light to flow around large imperfections without back-reflection. This Review explains the underlying principles and highlights how topological effects can be realized in photonic crystals, coupled resonators, metamaterials and quasicrystals.

Quasicrystal lattices, which can have rotational order but lack translational symmetry, can be used to explore electronic properties of materials between crystals and disordered solids. Ahn et al. grew graphene bilayers rotated exactly 30° that have 12-fold rotational order. Electron diffraction and microscopy confirmed the formation of quasicrystals, and angle-resolved photoemission spectroscopy revealed anomalous interlayer electronic coupling that was quasi-periodic. The millimeter-scale layers can potentially be transferred to other substrates. Science , this issue p. 782 A Dirac fermion quasicrystal with 12-fold rotational symmetry results from twisted bilayer graphene rotated exactly 30°. Quantum states of quasiparticles in solids are dictated by symmetry. We have experimentally demonstrated quantum states of Dirac electrons in a two-dimensional quasicrystal without translational symmetry. A dodecagonal quasicrystalline order was realized by epitaxial growth of twisted bilayer graphene rotated exactly 30°. We grew the graphene quasicrystal up to a millimeter scale on a silicon carbide surface while maintaining the single rotation angle over an entire sample and successfully isolated the quasicrystal from a substrate, demonstrating its structural and chemical stability under ambient conditions. Multiple Dirac cones replicated with the 12-fold rotational symmetry were observed in angle-resolved photoemission spectra, which revealed anomalous strong interlayer coupling with quasi-periodicity. Our study provides a way to explore physical properties of relativistic fermions with controllable quasicrystalline orders.

For quasicrystals of cut-and-project type in \\(\\mathbb{R}^d\\), it was proved by Marklof and Str\"ombergsson that the limit local statistical properties of the directions to the points in the set are described by certain \\(\\operatorname{SL}_d(\\mathbb{R})\\)-invariant point processes. In the present paper we make a detailed study of the tail asymptotics of the limiting gap statistics of the directions, for certain specific classes of planar quasicrystals.