Eigenmodes of quasicrystals

Abstract

Recently, a new state of matter, referred to as quasicrystalline, was discovered. Previously, solids could be classified as crystalline, with periodic lattice spacing, or as glassy, with random site spacing. The new quasicrystal structures appear to have long‐range order, showing sharp peaks in the Fourier transform space as in a periodic system, but they also have properties that are impossible for any periodic structure, such as fivefold rotational symmetry. For periodic systems, Bloch's theorem may be used to understand physical properties such as wave transmission, and it is of current interest to learn if any such symmetry theorems apply to quasiperiodic systems. Rigorous theorems for one‐dimensional quasiperiodic patterns based on a Fibonacci sequence suggest that such a pattern may be useful in control of vibration transmission in rib‐stiffened plates. However, in two and higher dimensions, little is known about the consequences of quasiperiodic structure. Recently, acoustic measurements have been ...