Equivalence classes of Fibonacci lattices and their similarity properties

TL;DR

This paper explores the diffraction spectra of Fibonacci lattices, establishing equivalence classes through a composition rule, and demonstrates that these classes share characteristic diffraction patterns, supported by theoretical and experimental evidence.

Contribution

It introduces a novel composition rule to classify Fibonacci lattices into equivalence classes with shared diffraction properties, linking structure to spectral features.

Findings

Fibonacci lattices can be grouped into equivalence classes via a composition rule.

Each class's diffraction pattern is primarily determined by its generator sequence.

Experimental results confirm the theoretical predictions about diffraction spectra.

Abstract

We investigate, theoretically and experimentally,the properties of diffraction spectra of Fibonacci lattices with arbitrary spacings. We show that, by means of a suitable composition rule, a Fibonacci sequence can be mapped into another one with a different lattice spacing. In this way we are able to define equivalence classes of Fibonacci structures and their generators, namely the Fibonacci sequences from which all the others can be obtained by compostion rule. We show that each class can be characterized by a given diffraction pattern which is essentially the one of the generator, in the sense that the most prominent features of this spectrum are common to all the elements of the class. This theoretical prediction is in good agreement with experimental results.