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Summary

Compare with

• 
  DEM and sobel
• 
  LCM/J; ER=1000
• 
  LCM/J but better algorithm, ER = 2
• 
  LSM/J B&W; ER = 1000
• 
  LSM/J colour ( probably made with Fractint ); ER = 2

See also:

• implicit polynomial curves that converge to the boundary of the Mandelbrot set. from desmos

Long description

  " instead of iterating a point through a nice fractal-generating function until it exits the containing circle, I'm starting with the containing circle's function (2cos(t),2sin(t)) and iterating that circle function through the inverse of the fractal-generating function." Axis Angels[1]

Few lemniscates of Mandelbrot set^[2]. They are boundaries of Level Sets of escape time ( LSM/M ^[3]).

They are in parameter plane (c-plane, complex plane ).

Definition :

${\displaystyle L_{n}=\{c:\operatorname {abs} (z_{n})=ER\}\,}$

where

${\displaystyle ER\,}$ is Escape Radius, bailout value, radius of circle which is used to measure if orbit of ${\displaystyle z_{0}\,}$ is bounded; it is integer number

${\displaystyle z,c\,}$ are complex numbers (points of 2-D planes )

${\displaystyle z\,}$ is point of dynamical plane ( z-plane)

${\displaystyle c\,}$ is point of parameter plane ( c-plane)

${\displaystyle c=x+y*i\,}$

${\displaystyle z_{n+1}=f_{c}(z_{n})\,}$

${\displaystyle f_{c}(z)=z^{2}+c\,}$

${\displaystyle z_{0}=0\,}$ critical point of ${\displaystyle f_{c}\,}$

One can compute first few iterations :

${\displaystyle z_{1}=0*0+c=c\,}$

${\displaystyle z_{2}=z_{1}*z_{1}+c=c^{2}+c\,}$

${\displaystyle z_{3}=z_{2}*z_{2}+c=(c^{2}+c)^{2}+c\,}$

and so on .

Then :

${\displaystyle L_{1}=\{c:\operatorname {abs} (c)=ER\}=\{(x+y*i):\operatorname {sqrt} (x^{2}+y^{2})=ER\}\,}$

${\displaystyle L_{2}=\{c:\operatorname {abs} (c^{2}+c)=ER\}=\{(x+y*i):\operatorname {sqrt} ((-y^{2}+x^{2}+x)^{2}+(2*x*y+y)^{2})=ER\}\,}$

${\displaystyle L_{3}=\{c:\operatorname {abs} ((c^{2}+c)^{2}+c)=ER\}=\{(x+y*i):\operatorname {sqrt} ((y^{4}-6*x^{2}*y^{2}-6*x*y^{2}-y^{2}+x^{4}+2*x^{3}+x^{2}+x)^{2}+(-4*x*y^{3}-2*y^{3}+4*x^{3}*y+6*x^{2}*y+2*x*y+y)^{2})=ER\}\,}$

...

${\displaystyle L_{1}\,}$ is a circle,

${\displaystyle L_{2}\,}$ is an Cassini oval,

${\displaystyle L_{3}\,}$ is a pear curve^[4][5].

These curves tend to boundary of Mandelbrot set as n goes to infinity.

If ER < 2 they are inside Mandelbrot set^[6].

If ER = 2 curves meet together ( have common point) c = −2. Thus they can't be equipotential lines.

If ER ≥ 2 they are outside of Mandelbrot set. They can also be drawn using Level Curves Method.

If ER >> 2 they aproximate equipotential lines ( level curves of real potential , see CPM/M ).

Maxima source code

 /* based on the code by Jaime Villate */
 load(implicit_plot); /* package by Andrej Vodopivec */

 c: x+%i*y;

 ER:2; /* Escape Radius = bailout value it should be >=2 */

 f[n](c) := if n=1 then c else (f[n-1](c)^2 + c);

 ip_grid:[100,100];  /* sets the grid for the first sampling in implicit plots. Default value: [50, 50] */
 ip_grid_in:[15,15]; /* sets the grid for the second sampling in implicit plots. Default value: [5, 5] */

 my_preamble: "set zeroaxis; set title 'Boundaries of level sets of escape time of Mandelbrot set'; set xlabel 'Re(c)';  set ylabel 'Im(c)'";

 implicit_plot(makelist(abs(ev(f[n](c)))=ER,n,1,6), [x,-2.5,2.5],[y,-2.5,2.5],[gnuplot_preamble, my_preamble],
 [gnuplot_term,"png   size  1000,1000"],[gnuplot_out_file, "lemniscates6.png"]);

For curves 1-5 it works, but for curve number 6 this program fails( also Mathematica program^[7]), because of floating point error.

One have to change the method of computing lemniscates . Here is the code and explanation by Andrej Vodopivec" "You can trick implicit_plot to do computations in higher precision. Implicit_draw will draw the boundary of the region where the function has negative value. You can define a function f6 which computes the sign of f[6] using bigfloats and then plot f6."

/* based on the code by Jaime Villate and Andrej Vodopivec*/
c: x+%i*y;
ER:2;
f[n](c) := if n=1 then c else (f[n-1](c)^2 + c);
F(x,y):=block([x:bfloat(x), y:bfloat(y)],if abs((f[6](c)))>ER then 1 else -1); 
fpprec:32;
load(implicit_plot); /* package by Andrej Vodopivec */ 
ip_grid:[100,100];
ip_grid_in:[15,15];
implicit_plot(append(makelist(abs(ev(f[n](c)))=ER,n,1,5), ['(F(x,y))]),[x,-2.5,2.5],[y,-2.5,2.5]);

Questions

• What is mathemathical description of these curves ?

Rerferences

1. ↑ You tube video
2. ↑ lemniscates at Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo
3. ↑ LSM/M
4. ↑ Weisstein, Eric W. "Pear Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PearCurve.html
5. ↑ Mandelbrot lemniscate at 2DCurves by Jan Wassenaar
6. ↑ Polynomial_lemniscate
7. ↑ | Weisstein, Eric W. "Mandelbrot Set Lemniscate." From MathWorld--A Wolfram Web Resource.

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