Some remarks on quasi-crystal structure

Abstract

The Fourier transform of a skeletal delta function that characterizes the most striking features of experimental quasicrystal diffraction patterns is evaluated. The result plays a role analogous to the Poisson summation formula for periodic delta functions that underlie classical crystallography. The real-space distribution can be interpreted in terms of a backbone comprising a system of intersecting equiangular spirals into which are inscribed (self-similar) gnomons of isosceles triangles with length-to-base ratio the golden mean τ. In addition to the vertices of these triangles, there is an infinite number of other points that may tile space in two or three dimensions. Other mathematical formulae of relevance are briefly discussed.