# The Unexpected Fractal Signatures in Fibonacci Chains

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## Abstract

**:**

## 1. Introduction

## 2. The Fractal Signature of the Fibonacci Chain in Fourier Space

#### 2.1. Fractal Dimension

#### 2.2. Universality Near the Real Line

#### 2.3. Self Similarity

## 3. The Variations of the Fibonacci Chain in Fourier Space

#### 3.1. Variations by Cyclic Permutations

#### 3.2. Variations and the Generalized Mandelbrot Set

## 4. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The rescaled Fourier space representation of a Fibonacci chain is shown here using the substitution method with different iterations, starting with 20. The horizontal axis represents the real part and the vertical axis represents the imaginary part of the Fourier coefficients.

**Figure 2.**(

**a**) The Mandelbrot set, (

**b**) overlays of the Mandelbrot set and the Fourier space with matching cardioid, and (

**c**) the fractal structure in the Fourier space of a Fibonacci chain with 25 iterations.

**Figure 3.**Appearance of the fractal pattern at the 34th iteration of the Fibonacci chain: (

**a**) general view showing the cardioid; (

**b**) detail of the central part; (

**c**) zoom near the real axis, some lines are getting very close and perpendicular to the real axis, they are called ‘trunks’ in Figure 6.

**Figure 5.**The Julia sets on the real line, while sliding from the Golden julia set (on the top) where c = −1 to the limit of non fractality, c = −2 on the bottom, shows a sliding of the attractors and anti-attractors. For example the left end slides from −$\varphi $ to $-2$.

**Figure 6.**The Fourier space representation of a Fibonacci chain is shown here with the geodesics of Fibonacci steps: 5 in red, 8 in orange, 13 in yellow, 21 in green, 34 in cyan, 55 in blue, 89 in magenta and 144 in purple. The “seeds” are the point closest to the horizontal axis: 56 on the left, 22 spreading a green trunk, 14 with a yellow trunk and 35 with a cyan trunk. The other trees are omitted.

**Figure 7.**The Fourier space representation of a Fibonacci chain of 25th iteration, with (

**a**) the first segment removed; (

**b**) the first two segments removed; (

**c**) the first three segments removed; (

**d**) the first six segments removed, (

**e**) the first seven segments removed, (

**f**) the first segment length replaced with 0; (

**g**) no modification; (

**h**) the last segment removed, (

**i**) the last two segments remove; (

**j**) the last 46,367 segments removed; (

**k**) the last 46,368 segments removed and the Fibonacci chain of 25th iteration is truncated to the Fibonacci chain of 24th iteration; (

**l**) the 1st segment L replaced with S; (

**m**) the last two segments flipped order; (

**n**) the order of the last five segments scrambled; (

**o**) the order of the last ten segments scrambled; (

**p**) the order of the last 100 segments scrambled, (

**q**) the superposition of spectra of cyclic permutations of the 17th iteration and the comparison between (

**r**) the original Fibonacci chain of 27th iteration and (

**s**) the chain with modified $L/S$ ratio where $L/S=2$.

**Figure 8.**Curves on the complex plane, from main features of the generalised Mandelbrot sets, with their equations; (

**a**–

**d**) Epicycloid from the Mandelbrot set and its 3 first circular bulbs centered on the real line (

**e**–

**h**) Nephroid and its higher generalisations from the generalized Mandelbrot set for exponents 3 to 6.

**Figure 9.**(

**a**) A picture of a geometric chuck; (

**b**–

**d**) Cycloids generated by the geometric chuck with different settings. The following settings are used: $P=55$, ${V}_{1}=\frac{111}{55}=\frac{2P+1}{P}$, (

**a**): ${V}_{2}=55\phantom{\rule{3.33333pt}{0ex}}out$, $E{x}_{1}=30$, $E{x}_{2}=55$, $SR=27$, (

**b**): ${V}_{2}=55\phantom{\rule{3.33333pt}{0ex}}in$, $E{x}_{1}=35$, $E{x}_{2}=45$, $SR=30$, (

**c**): ${V}_{2}=\frac{55}{2}\phantom{\rule{3.33333pt}{0ex}}out$, $E{x}_{1}=30$, $E{x}_{2}=55$, $SR=27$, The equation is not given but the information in Reference [15] should help with finding it.

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**MDPI and ACS Style**

Fang, F.; Aschheim, R.; Irwin, K. The Unexpected Fractal Signatures in Fibonacci Chains. *Fractal Fract.* **2019**, *3*, 49.
https://doi.org/10.3390/fractalfract3040049

**AMA Style**

Fang F, Aschheim R, Irwin K. The Unexpected Fractal Signatures in Fibonacci Chains. *Fractal and Fractional*. 2019; 3(4):49.
https://doi.org/10.3390/fractalfract3040049

**Chicago/Turabian Style**

Fang, Fang, Raymond Aschheim, and Klee Irwin. 2019. "The Unexpected Fractal Signatures in Fibonacci Chains" *Fractal and Fractional* 3, no. 4: 49.
https://doi.org/10.3390/fractalfract3040049