MERLIN SCIENCE — Fourier Transform Bridges Lattice Geometry and Fibonacci Quasiperiodic — E8 Intelligence Research

The finding is this: the Fourier transform of a Fibonacci chain produces sharp, self-similar Bragg peaks indexed by the ring of integers in the golden field, and this is the precise mathematical bridge between lattice geometry and quasiperiodic diffraction. For context, the problem has been with us since Shechtman's quasicrystals forced a rewrite of crystallography. A crystal diffracts because it repeats; a quasicrystal diffracts despite not repeating. The question is what replaces translational symmetry. The answer is not disorder, but a different kind of order, and the Fibonacci chain is the cleanest laboratory for seeing it. Here is the mechanism. For a periodic lattice, intensity is the squared modulus of the Fourier transform of the density, and peaks appear only at reciprocal lattice vectors. For the Fibonacci chain, built by the inflation rule A to AB and B to A, the Fourier transform gives peaks at wavevectors q equals two pi times m plus n tau, all over a tau, where tau is t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com