# The Symmetry in the Noise-Perturbed Mandelbrot Set

^{*}

## Abstract

**:**

## 1. Introduction

^{th}Century, French mathematician Gaston Julia [1] focused on the following simple map:

- (1)
- Application of the “symmetry criterion” method [30] to investigate the Mandelbrot set. To the best of our knowledge, the work in [30] was the first study that addressed the quantization of the symmetry destruction of the noise-perturbed Julia set. In this work, we applied it to the research of another fractal set, namely the Mandelbrot set.
- (2)
- The proposition of a new “visual symmetry criterion” method. It is noted that the current $SC$ method’s principle of calculating the quantized ratio of the symmetric region to the whole M-set is not very effective for measuring some visually-asymmetric sets. Thus, by adding a weight to the “symmetry index”, we modified the $SC$ method to a novel one named the “visual symmetry criterion” method, which is more in line with visual habits.

## 2. Preliminaries

**Definition**

**1**

**.**The Mandelbrot set of System (1), denoted as $M\left(f\right)$, is the set of all values of parameters $({c}_{1},{c}_{2})$ under the conditions of $({x}_{0},{y}_{0})=(0,0)$ such that:

^{th}iteration of the initial point $({x}_{0},{y}_{0})$.

## 3. Methods

#### 3.1. The “Symmetry Criterion” Method

- (1)
- For both ${w}_{n}\sim \mathcal{U}(0,1)$ and ${w}_{n}\sim \mathcal{N}(0,1)$, the symmetry of $M\left({f}^{m}\right)$ decreases significantly when $m<0,01$, and it tends to be stable on the whole. This observation is in accordance with the conclusion for 3D Julia sets [30].
- (2)
- For both ${w}_{n}\sim \mathcal{U}(0,1)$ and ${w}_{n}\sim \mathcal{N}(0,1)$, the symmetry of $M\left({f}^{a}\right)$ decreases with the increase in a, and it continues this trend on the whole.
- (3)
- For both ${w}_{n}\sim \mathcal{U}(0,1)$ and ${w}_{n}\sim \mathcal{N}(0,1)$, it can be seen that $S{C}_{{c}_{1}}\left({f}^{a}\right)$ is larger than $S{C}_{{c}_{1}}\left({f}^{m}\right)$ with the same noise strength $\alpha $. This supports the conclusion in Figure 2.
- (4)
- Figure 4a,b seems to have roughly the same trend. That is, with the same noise strength, the noises ${w}_{n}\sim \mathcal{U}(0,1)$ and ${w}_{n}\sim \mathcal{N}(0,1)$ have the same quantized symmetry criterion. A small difference in the details is that the fluctuation of $S{C}_{{c}_{1}}\left({f}_{n}^{a}\right)$ is bigger than that of $S{C}_{{c}_{1}}\left({f}_{u}^{a}\right)$ when $a>0.7$. The visual evidence of this fluctuation can be seen from Figure 2i, in which $M\left({f}_{n}^{a=0.8}\right)$ has almost lost its fractal structure.
- (5)
- Both $S{C}_{{c}_{1}}\left({f}_{u}\right)$ and $S{C}_{{c}_{1}}\left({f}_{n}\right)$ remain at a value close to one. That is, the visually-asymmetric noise-perturbed Mandelbrot set may have a high quantified symmetry index.

#### 3.2. The Modified “Visual Symmetry Criterion” Method

## 4. Discussion

- (1)
- “Symmetry criterion”: The $SC$ method [30] focuses on the essence of the symmetry distribution. It can be seen in Figure 5 that the region with the dense distribution has a greater effect on the $SC$ results than the region with the sparse distribution. Thus, we can also title the $SC$ method the “virtual symmetry criterion”. It can be applied to engineering problems that require high data accuracy.
- (2)
- “Visual symmetry criterion”: The $VSC$ method focuses more on the sense of vision. That is, the farther away from the symmetry axis, the greater the effect on the visual sense. Thus, the “visual symmetry criterion” can be applied to some pattern recognition fields that focus on the senses of human beings.

- (1)
- References [21,22,25,26,27,29] mainly discussed the deviation distance or the structural damages in noise-perturbed fractals sets. Some scholars have examined the symmetry property of noise-perturbed fractal sets [23,24,28], but quantitative investigations of symmetry in a noise-perturbed fractal set are limited to [30] and this work.
- (2)
- We introduce more types of noise, ${w}_{n}\sim \mathcal{U}(0,1)$ and ${w}_{n}\sim \mathcal{N}(0,1)$, into such kinds of research.
- (3)
- A more effective method, the “visual symmetry criterion” method, for classifying the visually-asymmetric set is proposed in this work.

- (1)
- The underlying mathematical proof of the graphical and algorithm observations is difficult to provide. We intend to investigate this topic in future work.
- (2)
- The calculation quantity of the “visual symmetry criterion” is large because it needs to calculate a weight w for each step. Thus, it is hard to extend this method to the spatial case.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The noise-perturbed Mandelbrot set: (

**a**) $M\left({f}_{u}^{a=0.01}\right)$; (

**b**) $M\left({f}_{u}^{a=0.2}\right)$; (

**c**) $M\left({f}_{u}^{a=0.8}\right)$; (

**d**) $M\left({f}_{u}^{m=0.01}\right)$; (

**e**) $M\left({f}_{u}^{m=0.2}\right)$; (

**f**) $M\left({f}_{u}^{m=0.8}\right)$; (

**g**) $M\left({f}_{n}^{a=0.01}\right)$; (

**h**) $M\left({f}_{n}^{a=0.2}\right)$; (

**i**) $M\left({f}_{n}^{a=0.8}\right)$; (

**j**) $M\left({f}_{n}^{m=0.01}\right)$; (

**k**) $M\left({f}_{n}^{m=0.2}\right)$; (

**l**) $M\left({f}_{n}^{m=0.8}\right)$.

**Figure 4.**(

**a**) The $S{C}_{{c}_{1}}\left(f\right)$ curve with ${w}_{n}\sim \mathcal{U}(0,1)$ as the strength of noise increases: additive noise is represented by cool colors, and multiplicative noise is represented by warm colors. (

**b**) The $S{C}_{{c}_{1}}\left(f\right)$ curve with ${w}_{n}\sim \mathcal{N}(0,1)$ as the strength of noise increases: additive noise is represented by cool colors, and multiplicative noise is represented by warm colors.

**Figure 6.**(

**a**) The $S{C}_{{c}_{1}}^{Vis}\left(f\right)$ curve with ${w}_{n}\sim \mathcal{U}(0,1)$ as the strength of the noise increases: additive noise is represented by cool colors, and multiplicative noise is represented by warm colors. (

**b**) The $S{C}_{{c}_{1}}^{Vis}\left(f\right)$ curve with ${w}_{n}\sim \mathcal{N}(0,1)$ as the strength of the noise increases: additive noise is represented by cool colors, and multiplicative noise is represented by warm colors.

Map (3) | Map (4) | ||
---|---|---|---|

Noise Type | Symbol | Noise Type | Symbol |

${w}_{n}\phantom{\rule{3.33333pt}{0ex}}\sim \phantom{\rule{3.33333pt}{0ex}}\mathcal{U}(0,1)$ | ${J}_{u}^{a}$ | ${w}_{n}\phantom{\rule{3.33333pt}{0ex}}\sim \phantom{\rule{3.33333pt}{0ex}}\mathcal{U}(0,1)$ | ${J}_{u}^{m}$ |

${w}_{n}\phantom{\rule{3.33333pt}{0ex}}\sim \phantom{\rule{3.33333pt}{0ex}}\mathcal{N}(0,1)$ | ${J}_{n}^{a}$ | ${w}_{n}\phantom{\rule{3.33333pt}{0ex}}\sim \phantom{\rule{3.33333pt}{0ex}}\mathcal{N}(0,1)$ | ${J}_{n}^{m}$ |

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**MDPI and ACS Style**

Sun, T.; Wang, D.
The Symmetry in the Noise-Perturbed Mandelbrot Set. *Symmetry* **2019**, *11*, 577.
https://doi.org/10.3390/sym11040577

**AMA Style**

Sun T, Wang D.
The Symmetry in the Noise-Perturbed Mandelbrot Set. *Symmetry*. 2019; 11(4):577.
https://doi.org/10.3390/sym11040577

**Chicago/Turabian Style**

Sun, Tianwen, and Da Wang.
2019. "The Symmetry in the Noise-Perturbed Mandelbrot Set" *Symmetry* 11, no. 4: 577.
https://doi.org/10.3390/sym11040577