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MathGroup Archive 2004

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Re: Mandelbrot Set & Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48029] Re: Mandelbrot Set & Mathematica
  • From: AGUIRRE ESTIBALEZ Julian <mtpagesj at lg.ehu.es>
  • Date: Sat, 8 May 2004 01:23:34 -0400 (EDT)
  • References: <c7fhp4$oar$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

On Fri, 7 May 2004, fake wrote:

> I'm looking for a program using Mathematica commands to obtain the
> Mandelbrot set representation without using the .m file "Fractal"
> downloadable from Mathworld. Please report the Timing parameter if you have
> done some tests.
> TIA

This is what I did for a Dynamical Systems course. It is based on code
from the help files. It includes knowledge about points that are in the
Mandelbrot set.

Clear[c, test, niter, BlackWhite, mandelbrot];
BlackWhite = If[# == 1, GrayLevel[0], GrayLevel[1]]&;
niter = 100;
test = (Abs[#] =BE 2) &;
mandelbrot[c_] := 0 /; Abs[c] > 2;
mandelbrot[c_] := 1 /; Abs[c + 1] < 1/4;
mandelbrot[c_] := 1 /; 16 Abs[c]^2 < 5 - 4 Cos[Arg[c]];
mandelbrot[c_] := (Length@NestWhileList[(#^2+c)&,c,test,1,niter]-1)/niter;
DensityPlot[mandelbrot[x + y I], {x, -2, .5}, {y, 0, 1},
    PlotPoints -> {600, 300},
    Mesh -> False,
    ImageSize -> 600,
    AspectRatio -> Automatic,
    ColorFunction -> BlackWhite];

Color can be added defining new color functions. I like

rainbow = Hue[.8(1 - #)]&

Julian Aguirre
UPV/EHU


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