The Unexpected Fractal Signatures in Fibonacci chains

TL;DR

This paper reveals a novel fractal signature in the Fourier space of Fibonacci chains, showing sensitivity to sequence modifications but not to length ratios, and introduces a new cycloidal fractal pattern related to the Mandelbrot set.

Contribution

It presents a new cycloidal fractal signature in the Fourier space of Fibonacci chains and analyzes its sensitivity to various modifications.

Findings

Fractal signature is sensitive to Fibonacci sequence modifications.

The signature is robust to changes in the L/S ratio.

A new cardioid-shaped fractal pattern is identified in Fourier space.

Abstract

Quasicrystals are fractal due to their self similar property. In this paper, a new cycloidal fractal signature possessing the cardioid shape in the Mandelbrot set is presented in the Fourier space of a Fibonacci chain with two lengths, L and S, where L/S = \{phi}. The corresponding pointwise dimension is 0.7. Various modifications, such as truncation from the head or tail, scrambling the orders of the sequence, and changing the ratio of the L and S, are done on the Fibonacci chain. The resulting patterns in the Fourier space show that that the fractal signature is very sensitive to changes in the Fibonacci order but not to the L/S ratio.