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File:Lemniscates5.png

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Summary[edit]

Description

6 lemniscates of Mandelbrot set. Computed using implicit equations.
Date
Source self-made with help of many people, using free CAS Maxima, Gnuplot and implicit_plot package (by Andrej Vodopivec)
Author Adam majewski

Compare with[edit]

  • 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

Long description[edit]

Few lemniscates of Mandelbrot set[1]. They are boundaries of Level Sets of escape time ( LSM/M [2]).

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

Definition :

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

where

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

z,c\, are complex numbers (points of 2-D planes )

z\, is point of dynamical plane ( z-plane)

c\, is point of parameter plane ( c-plane)

c=x+y*i\,

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

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

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

One can compute first few iterations :

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

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

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

and so on .

Then :

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

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

L_{3}=\{c:abs((c^{2}+c)^{2}+c)=ER\}=\{(x+y*i):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\}\,

...

L_{1}\, is a circle,

L_{2}\, is an Cassini oval,

L_{3}\, is a pear curve[3][4].

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

If ER<2 they are inside Mandelbrot set[5].

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[edit]

 /* 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[6]), 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[edit]

  • What is mathemathical description of these curves ?

Rerferences[edit]

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

Licensing[edit]

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File history

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Date/TimeThumbnailDimensionsUserComment
current19:42, 11 January 2009Thumbnail for version as of 19:42, 11 January 20091,000 × 1,000 (73 KB)Geek3smooth and precise plotcurve
10:22, 18 March 2008Thumbnail for version as of 10:22, 18 March 20081,000 × 1,000 (17 KB)Adam majewskiadded 6 lemniscate
08:15, 16 March 2008Thumbnail for version as of 08:15, 16 March 20081,000 × 1,000 (15 KB)Adam majewski{{Information |Description= |Source=self-made |Date= |Author= Adam majewski |Permission= |other_versions= }}
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