Effects of quasiperiodic (Penrose tile) symmetry on the eigenvalues and eigenfunctions of the wave equation

In addition to the basic crystalline and amorphous structures for solids, it is possible that solids may also form with a quasiperiodic, or Penrose tile, structure. A current problem in condensed-matter physics is to determine how this structure affects the various physical properties of a material. A fundamental question involves the consequences of quasiperiodic symmetry in the eigenvalue spectrum and eigenfunctions of a wave equation. While rigorous theorems have been derived for one-dimensional systems, there is currently no known “quasi-Bloch theorem” for two and three dimensions. To gain insight into this problem, an acoustic experiment has been used to study a two-dimensional wave system with a Penrose tile symmetry. The results show an eigenvalue spectrum containing bands and gaps with widths which are in the ratio of the Golden Mean, (√5 + 1)/2.