Quasicrystals are fractal because they are scale invariant and self similar. 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 variations such as truncation from the head or tail, scrambling the orders of the sequence, changeing 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 the change in the Fibonacci order but not to the L/S ratio.
fractal and fractional
Article
The Unexpected Fractal Signatures in
Fibonacci Chains
Fang Fang
*, Raymond Aschheim and Klee Irwin
Quantum Gravity Research, Los Angeles, CA 90290, USA; raymond@quantumgravityresearch.org (R.A.);
Klee@quantumgravityresearch.org (K.I.)
* Correspondence: Fang@quantumgravityresearch.org
Received: 2 August 2019; Accepted: 29 October 2019; Published: 6 November 2019
Abstract: In this paper, a new 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 = φ.
The corresponding pointwise dimension is 1.7. 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.
Keywords: Fibonacci chain; fractal signature; Fourier space
1. Introduction
Quasicrystals possess exotic and sometimes anomalous properties that have interested the
scientific community since their discovery by Shechtman in 1982 [1]. Of particular interest in this
manuscript is the self similar property of quasicrystals that links them to fractal. Historically, research
on the fractal aspect of quasicrystalline properties has revolved around spectral and wave function
analysis [2–5]. Mathematical investigation [6–9] of the geometric structure of quasicrystals is less
represented in the literature than experimental work.
In this paper, a new framework for analyzing the fractal nature of quasicrystals is introduced.
Specifically, the fractal properties of a one-dimensional Fibonacci chain and its variations are studied in
the complex Fourier space. The results may also be found in two and three dimensional quasicrystals
that can be constructed using a network of one dimensional Fibonacci chains [10].
2. The Fractal Signature of the Fibonacci