The Unexpected Fractal Signatures in Fibonacci chains

2016/09/01 by Fang Fang, Fang, Fang, Raymond Aschheim +3
Materials Science · Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #FOS: Mathematics #General Mathematics (math.GM) #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.1609.01159

openalex publication_date 2016/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Metallic Phase with Long-Range Orientational Order and No Translational Symmetry
• Exact eigenstates of tight-binding Hamiltonians on the Penrose tiling
• An Icosahedral Quasicrystal as a Golden Modification of the Icosagrid and its Connection to the E8 Lattice

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