DEFECTIVE VERTEX CONFIGURATIONS IN QUASICRYSTALLINE STRUCTURES

Abstract

Defective vertex configurations are important for the whole range of models for quasicrystalline structures from quasiperiodic tilings through random tilings to polyhedral glasses. The combinatorially possible vertex configurations are enumerated for the 1D Fibonacci chain, for the 2D Penrose pattern with its generalizations, as well as for the Beenker pattern and the triangle pattern, and for the 3D simple icosahedral tiling. The methods for quantifying the deviation of vertex configurations from perfection are reviewed. The simple method of partial dual overlap provides a means to estimate the abundancy of vertex configurations within random tilings. More sophisticated is the method of the defectivity functional; it is particularly suitable to deal with nearly perfect tilings. Local configurations are formally classified by characteristic integers: degree, rank and order. Some possible applications are hinted at.