This review focuses on the peculiarities of quasiperiodic order for the properties of photonic
and phononic (sonic) heterostructures. The most beneficial feature of quasiperiodicity is that …

Quasicrystals and Geometry brings together for the first time the many strands of
contemporary research in quasicrystal geometry and weaves them into a coherent whole …

In a recent paper, we introduced the concept of quasicrystals [Phys. Rev. Lett. 53, 2477
(1984)], a new class of ordered atomic structures. Quasicrystals have long-range …

Discovery of quasicrystals and other deterministic aperiodic structures initiated new fields of
research in solid-state photonics. In this review, we first make an overview of recently …

In this paper we examine the detailed relationship between the density-wave and unit-cell
descriptions of quasicrystals. We show that phonons, phasons, and dislocations correspond …

In this (second) paper, quasicrystalline configurations of unit cells are analyzed for the cases
of pentagonal and icosahedral orientational symmetry. We illustrate how two ideal …

Penrose tilings have become the canonical model for quasicrystal structure, primarily
because of their simplicity in comparison with other decagonally symmetric quasiperiodic …

The first quasicrystal (QC) structure was observed in 1984. QCs possess long-range
orientational and translational order while lacking the periodicity of crystals. An overview is …

We present a general method for the design of 2-dimensional nonlinear photonic
quasicrystals that can be utilized for the simultaneous phase matching of arbitrary optical …

This paper relates two mathematical concepts of long-range order of a set of atoms Λ, each
of which is based on restrictions on the set of interatomic distances Λ− Λ. A set Λ in ℝ n is a …