From: John Armstrong <hagbard@glue.umd.edu>
Newsgroups: sci.math
Subject: Re: Mandelbroth Set
Date: Fri, 13 Mar 1998 08:41:26 -0500

Omer Ahmad wrote:

> Does anyone out there know about the mandelbroth set?  I'm sure you do.
> I'm just starting to program, and I was wondering if anyone could help
> me with a personal project (I swear it isn't an assignment).  How can I
> program the mandelbroth set in C++.  You don't have to give me the code,
> but maybe some hints as to what I should be trying to do.
>
> Thanks...
>
> Omer.

  Firstly, it's the Mandelbrot set, but we won't hold that against you.  The
Mandelbrot set is the boundary of the set of points c where the set of z in
the complex plane with bounded orbits under the map z_n+1 = z_n^2 + c is
connected.  To cut through the drek, for a fixed c, plot the points of C
which don't diverge to infinity under mapping z_n+1 = z_n^2 + c.  This
actually means that they don't go outside the circle |z| = 2.  Now this set
can be shown to be connected iff z_0 = 0 is in it.  Thus to plot points in
Mandelbrot's picture, for each c in the complex plane, start with z = 0.
Loop up to a certain number of times, seeing if you're outside |z| <= 2, and
stop if you are.  If you aren't at the end, plot a black point.

c = C
z = 0;
for(int i = 0; i < 2500; i++)
{
  z = z*z + c;            // you'll have to use a complex multiplication
}                             // either use the prepackaged one (yuck)
                               // or write your own (yay)
if (mag(z) > 2)
  plot(c,white);        // plot you'll have to write too
else
  plot(c,black);


For coloured plots, say 256 colours:

c = C
z = 0;
for(int i = 0; (i < 2550)&&(mag(z) <= 2); i++)
{
  z = z*z + c;            // you'll have to use a complex multiplication
}                             // either use the prepackaged one (yuck)
                               // or write your own (yay)
if (mag(z) > 2)
  plot(c,i / 10);         // make sure you code it to use integer division
else
  plot(c,0);                // 0 is usually black

    JRJA
==============================================================================

From: israel@math.ubc.ca (Robert Israel)
Subject: Re: Scanning the plane with Maple
Date: 9 Aug 1999 08:03:56 GMT
Newsgroups: sci.math
Keywords: code to generate the Mandelbrot set

In article <37ACB7DC.73D0@ath.forthnet.gr>,
Ioannis Galidakis  <jgal@ath.forthnet.gr> wrote:

|>I want to explore some fractal images, and I have two choices:
|>One is to code the program in Pascal or C, but I have no explicit
|>algorithm for the LambertW function there, so I cannot calculate my
|>fixed points in my orbits which are determined by this function.

|>If I try to do it in Maple, it will be SLOOOOW. I have heard that it can
|>be done though. In any case, in Maple I have the following problem:
|>How to scan the plane, say [-1,1]x[-1,1], since Maple plots only
|>finitely many functions at a time, and in my case I need a bit-continuum
|>of scan lines. If I do something like:
|>line:=(x,y)->y; with variable y, I can do a plot(line(x),x=-1..1); is
|>there any way to plot the entire range of those scan lines
|>bit-simultaneously with y ranging continuously from -1 to 1, (i.e. in
|>one plot) with increments dy=dpixel?
|>Better yet, how can I approximate the dpixel parsing of a plane square
|>or rect if the above is not possible?

Write a function F of two variables: the colour of the point [x,y]
will be determined by F(x,y).  E.g. for the Mandelbrot set, 
you could use

F:= proc(x, y)
   local zr,zi,tmp, m;
   zr:= x; zi:= y;
   m:=0;
   to 200 while zr^2+zi^2<4 do
     tmp:= 2*zr*zi+y;
     zr:= zr^2-zi^2+x;
     zi:= tmp;      
     m:=m+1;
   od;
   m;
end:


 Then try

> plot3d(F, -1 .. 1, -1 .. 1, grid=[100,100], shading=ZHUE, 
style=PATCHNOGRID, orientation=[0,0], axes=BOX); 

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2