File:Lemniscates5.png
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Contents
Summary
[edit]DescriptionLemniscates5.png | 6 lemniscates of Mandelbrot set. Computed using implicit equations. |
Source | self-made with help of many people, using free CAS Maxima, Gnuplot and implicit_plot package (by Andrej Vodopivec) |
Author | Adam majewski |
Other versions | lemniscates for Julia set |
Compare with
[edit]-
DEM and sobel
-
LCM/J; ER=1000
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LCM/J but better algorithm, ER = 2
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LSM/J B&W; ER = 1000
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LSM/J colour ( probably made with Fractint ); ER = 2
See also:
Long description
[edit]" 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 :
where
is Escape Radius, bailout value, radius of circle which is used to measure if orbit of is bounded; it is integer number
are complex numbers (points of 2-D planes )
is point of dynamical plane ( z-plane)
is point of parameter plane ( c-plane)
One can compute first few iterations :
and so on .
Then :
...
is a circle,
is an Cassini oval,
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
[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[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
[edit]- What is mathemathical description of these curves ?
Rerferences
[edit]- ↑ You tube video
- ↑ lemniscates at Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo
- ↑ LSM/M
- ↑ Weisstein, Eric W. "Pear Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PearCurve.html
- ↑ Mandelbrot lemniscate at 2DCurves by Jan Wassenaar
- ↑ Polynomial_lemniscate
- ↑ | Weisstein, Eric W. "Mandelbrot Set Lemniscate." From MathWorld--A Wolfram Web Resource.
Licensing
[edit]Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue |
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File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 19:42, 11 January 2009 | 1,000 × 1,000 (73 KB) | Geek3 (talk | contribs) | smooth and precise plotcurve | |
10:22, 18 March 2008 | 1,000 × 1,000 (17 KB) | Soul windsurfer (talk | contribs) | added 6 lemniscate | ||
08:15, 16 March 2008 | 1,000 × 1,000 (15 KB) | Soul windsurfer (talk | contribs) | {{Information |Description= |Source=self-made |Date= |Author= Adam majewski |Permission= |other_versions= }} |
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