# Visualization of Mandelbrot and Julia Sets of Möbius Transformations

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## Abstract

**:**

## 1. Introduction

## 2. Iteration Procedure

**Definition**

**1.**

**Definition**

**2.**

## 3. Affine Transformations

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

## 4. Inversion

**Definition**

**3.**

**Theorem**

**1.**

**Proof.**

**Conjecture**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

`classic[z_, l_] := z^2 + l`

`ManMobptbuild[{a_, b_, c_, d_}, l_, j_] :=`

`Module[{f, g}, f[z_] := (a classic[z, l] - b)/(c classic[z, l] - d);`

`g = NestList[f, 0, j];`

`Apply[Plus, Table[If[Abs[g[[k]]] > 2, 1, 0], {k, 1, Length[g]}]] //`

`Quiet]`

`ManMob2dpltbuild[{a_, b_, c_, d_}, j_, {xmin_, xmax_, xs_}, {ymin_, ymax_, ys_}] :=`

`ListDensityPlot[`

`Flatten[Table[`

`Table[{x, y, -ManMobptbuild[{a, b, c, d}, x + I y, j]}, {x, xmin,`

`xmax, xs}], {y, ymin, ymax, ys}], 1],`

`ColorFunction -> (Blend[{Red, Orange, Yellow, RGBColor[0, .8, 0],`

`Blue, Black}, #] &)] // Quiet`

`ManMob2dpltbuild[{0, .3851450, 1, 0}, 100, {-1, 1, .002}, {-1, 1, .002}]`

`toHP[Z_] := I (1 + Z)/(1 - Z)`

`toPD[Z_] := (Z - I)/(Z + I)`

`geo2[Z1_, Z2_] :=`

`Module[{uZ1, uZ2, mp, m, xin, rad, uhpgeo, geo}, uZ1 = toHP[Z1];`

`uZ2 = toHP[Z2]; mp = 1/2 (uZ1 + uZ2);`

`m = -(Re[uZ1] - Re[uZ2])/(Im[uZ1] - Im[uZ2]);`

`xin = (m Re[mp] - Im[mp])/m; rad = Abs[uZ1 - xin];`

`uhpgeo = rad Cos[Pi t] + xin + I rad Sin[Pi t];`

`geo = toPD[uhpgeo]]`

`geo2plt[Z1_, Z2_, opts_] :=`

`ParametricPlot[{Re[geo2[Z1, Z2]], Im[geo2[Z1, Z2]]}, {t, 0, 1},`

`Axes -> False, opts]`

`JSplotallclassicMobbuild[{a_, b_, c_, d_}, l_,`

`j_, {xmin_, xmax_, xs_}, {ymin_, ymax_, ys_}] :=`

`ListDensityPlot[`

`Flatten[Table[`

`Table[{x, y, JSMobptbuild[x + I y, {a, b, c, d}, l, j]}, {x,`

`xmin, xmax, xs}], {y, ymin, ymax, ys}], 1],`

`ColorFunction -> (Blend[{Black, Blue, Red, RGBColor[0, .8, 0],`

`Yellow, Orange, Red}, #] &), ColorFunctionScaling -> False,`

`PlotRange -> All] // Quiet`

`JSplotallclassicMobbuild[{0, .3851450, 1, 0}, -.45 + .05 I, 100,`

`{-2, 2, .04}, {-2, 2, .04}]`

## References

- Peitgen, H.-O.; Jürgens, H.; Saupe, D. Chaos and Fractals, 2nd ed.; Springer: New York, NY, USA, 2004. [Google Scholar]
- Falconer, K. Fractal Geometry: Mathematical Foundations and Applications, 2nd ed.; Wiley: Chichester, UK, 2004. [Google Scholar]
- Barnsley, M. Fractals Everywhere, 2nd ed.; Academic Press: Boston, MA, USA, 1993. [Google Scholar]
- Lyndon, R.C. Groups and Geometry; Cambridge University Press: Cambridge, UK, 1985. [Google Scholar]
- Beals, R.; Wong, R.S. Explorations in Complex Functions; Springer: New York, NY, USA, 2020. [Google Scholar]
- Katok, S. Fuchsian Groups; University Chicago Press: Chicago, IL, USA, 1992. [Google Scholar]
- Feijs, L.; Toeters, M. Exploring a taxicab-based Mandelbrot-like set. In Bridges 2019 Conference Proceedings; Johannes Kepler University: Linz, Austria, 2019. [Google Scholar]
- Teoters, M.; Feijs, L.M.G.; Van Loenhout, D.; Tieleman, C.; Virtala, N.; Jaakson, G.K. Algorithmic fashion aesthetics: Mandelbrot. In Proceedings of the ISWC ’19: 23rd International Symposium on Wearable Computers, London, UK, 11–13 September 2019. [Google Scholar]
- Yadav, A.; Rani, M.; Verma, V.K.; Mundra, A. A review on controlling of Julia and Mandelbrot sets. Int. J. Adv. Sci. Technol.
**2019**, 28, 213–223. [Google Scholar] - Xiang, Z.; Zhou, K.-Q.; Guo, Y. Gaussian mixture noised random fractals with adversarial learning for automated creation of visual objects. Fractals
**2020**, 28, 2050068. [Google Scholar] [CrossRef] - Kwun, K.C.; Tanveer, M.; Nazeer, W.; Gdawiec, K.; Kang, S.M. Mandelbrot and Julia Sets via Jungck-CR Iteration with s-Convexity. IEEE Access
**2019**, 7, 12167–12176. [Google Scholar] [CrossRef] - Kwun, K.C.; Tanveer, M.; Nazeer, W.; Abbas, M.; Kang, S.M. Fractal generation in modified Jungck-S orbit. IEEE Access
**2019**, 7, 35060–35071. [Google Scholar] [CrossRef] - Li, D.; Tanveer, M.; Nazer, W.; Guo, X. Boundaries of filled Julia Sets in generalized Jungck Mann orbit. IEEE Access
**2019**, 7, 76859–76867. [Google Scholar] [CrossRef] - Abbas, M.; Iqbal, H.; De la Sen, M. Generation of Julia and Mandelbrot Sets via Fixed Points. Symmetry
**2020**, 12, 1797. [Google Scholar] [CrossRef] - Tan, L. Similarity between the Mandelbrot Set and Julia Sets. Commun. Math. Phys.
**1990**, 134, 587–617. [Google Scholar] - Kawahira, T. Zalcman functions and similarity between the Mandelbrot set, Julia sets, and the tricorn. Anal. Math. Phys.
**2020**, 10, 16. [Google Scholar] [CrossRef] [Green Version] - Blankers, V.; Rendfrey, T.; Shukert, A.; Shipman, P.D. Julia and Mandelbrot Setsfor Dynamics over the Hyperbolic Numbers. Fractal Fract.
**2019**, 3, 6. [Google Scholar] [CrossRef] [Green Version] - Chaves, A.P.; Trojovský, P. A quadratic Diophantine equation involving generalized Fibonacci numbers. Mathematics
**2020**, 8, 1010. [Google Scholar] [CrossRef] - Drakopoulos, V. Comparing rendering methods for Julia sets. J. WSCG
**2002**, 10, 155–161. [Google Scholar] - Drakopoulos, V. Sequential visualisation methods for the Mandelbrot set. JCMSE
**2010**, 10, 37–45. [Google Scholar] [CrossRef] - Mork, L.K.; Vogt, T.; Sullivan, K.; Rutherford, D.; Ulness, D.J. Exploration of filled-in Julia sets arising from centered polygonal lacunary functions. Fractal Fract.
**2019**, 3, 42. [Google Scholar] [CrossRef] [Green Version] - Mork, L.K.; Sullivan, K.; Ulness, D.J. Lacunary Möbius fractals on the unit disk. Symmetry
**2021**, 13, 91. [Google Scholar] [CrossRef] - Mathematica 12; Wolfram Research: Champaign, IL, USA, 2020.

**Figure 1.**The j-averaged Mandelbrot set for simple inversion (left panel: $I=(0,1,1,0)$) and scaled inversion (middle panel: $A=(0,0.75,1,0)$ and right panel: $A=(0,1.25,1,0)$). A circle of radius $r=1.889090$ circumscribes the simple inversion Mandelbrot set. Scaling with a number less than 1 leads to a smoothing of the Mandelbrot set relative to the unscaled set. Conversely, scaling by a factor greater than 1 leads to greater roughness of the set.

**Figure 2.**Hyperbolic geometric features of ${}^{100}{h}^{\left(U\right)}\left(z\right)$. When the ideal regular triangle (white) is inscribed on the unit disk, the hyperbolic geodesics are tangent to the three larger orbiting domains. Figure 3 shows a blow-up of this region. The three pronounced yellow orbiting domains have an inward pointing mouth (see Figure 3) which form the vertices of the hyperbolic triangle shown in cyan (angles: $39.{621}^{\circ}$). The cyan geodesics are nearly but not tangent with the central domain (red) but they are tangent to the coronas of the inner green regions (see Figure 3).

**Figure 3.**Several blow-ups on one of three arms of the j-averaged Mandelbrot set in Figure 2. The top two panels reveal the tangency to the geodesics of the ideal regular triangle shown in white in Figure 2 and the cyan-colored hyperbolic triangle formed from the clefts of the yellow domains. The bottom left panel shows a blow-up of the cleft region of the yellow domain. The j-averaged Mandelbrot set is quite active in the region emerging from the nexus of the cleft. The bottom right panel shows two sequential blow-ups of the tail of the arm of the j-averaged Mandelbrot set.

**Figure 4.**Numerical fits for several features associated with the j-averaged Mandelbrot set for $A=U=(0,0.3851450,1,0)$. The top left graph shows a linear slice of the Mandelbrot set for $\lambda =x+0i$. The ordinate is a normalized scale representing the amount of times a point is not in the Mandelbrot set; 1 is the red domain in Figure 2, 0.667 is the yellow, 0.5 is the green, etc. The orange points are the values of $\lambda $ at the proximal cleft of each domain. The orange line is the fit of $C\sqrt{x+1}$ to these data (best fit parameter $C=0.920611$). The bottom left shows the position of the cleft (ordinate) versus the domain number (abscissa). The data are fit to $\frac{C}{x+{x}_{0}}+{C}_{0}$ (best fit parameters $C=0.851322$, ${x}_{0}=0.4812$, and ${C}_{0}=-1.05727$). The top right shows the first five hyperbolic triangles defined by the positions of the clefts. The bottom right graph shows the angle of the hyperbolic triangles (ordinate) versus position of the cleft (abscissa). The data are fit to $Cx+{x}_{0}$ (best fit parameters $C=76.7065$, ${x}_{0}=76.1939$).

**Figure 5.**Nature of the three-fold symmetry of the j-averaged Mandelbrot set on the j-averaged filled-in Julia sets. The top left panel shows the j-averaged filled-in Julia corresponding to $\lambda =-0.45+0.05i$. The top middle and right panels show $\lambda =(-0.45+0.05i){e}^{\frac{2\pi i}{3}}$ and $\lambda =(-0.45+0.05i){e}^{\frac{-2\pi i}{3}}$, respectively. The second row shows the j-averaged filled-in Julia sets corresponding to reflecting $\lambda $ across the respective dihedral mirrors. The bottom row shows the j-averaged filled-in Julia set coming from the simple inversion case ($A\equiv I(0,1,1,0)$). Each respective value of $\lambda $ is $1.889090$ times that from the top row. The j-averaged filled-in Julia sets look similar but not identical, and they are about 1.35 times bigger.

**Figure 6.**j-averaged filled-in Julia sets from points in the j-averaged Mandelbrot sets at positions of the orange data points in Figure 4. Domains 1 through 9 are shown. The appearance has significant similarities. One important difference is that the number of strands crossing from one side of an arm to the other is equal to the domain number.

**Figure 7.**A few intricate and impressive looking j-averaged filled-in Julia sets coming from the mouth regions of one of the three larger orbiting domains. For all plots $j=100$, sampled as a 1M pt square lattice. Top left $\lambda =-0.475+0.13i$, top middle, $\lambda =-0.46+0.135i$, top right $\lambda =-0.471+0.006i$. The bottom row simply shows the filled-in Julia set.

**Figure 8.**Effect of j on the appearance of the j-averaged filled-in Julia set associated with $\lambda =-0.475+0.13i$ for (left-to-right) $j=100$, $j=250$, and $j=500$. The set of points that are always in the filled-in Julia sets tighten up with increasing j.

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**MDPI and ACS Style**

Mork, L.K.; Ulness, D.J.
Visualization of Mandelbrot and Julia Sets of Möbius Transformations. *Fractal Fract.* **2021**, *5*, 73.
https://doi.org/10.3390/fractalfract5030073

**AMA Style**

Mork LK, Ulness DJ.
Visualization of Mandelbrot and Julia Sets of Möbius Transformations. *Fractal and Fractional*. 2021; 5(3):73.
https://doi.org/10.3390/fractalfract5030073

**Chicago/Turabian Style**

Mork, Leah K., and Darin J. Ulness.
2021. "Visualization of Mandelbrot and Julia Sets of Möbius Transformations" *Fractal and Fractional* 5, no. 3: 73.
https://doi.org/10.3390/fractalfract5030073