Category: Escape-time fractal

When I tried to program the Mandelbrot set myself the first time, I made the following mistake: instead of returning the ('original') point (x0, y0) that does not escape the radius of 2 about the orgin, the procedure returned the values of xn and yn which contain the values of the point (x0, y0) after its n-th iteration (here 25). (There is also the term orbit for the trajectory of a point being evaluated by the iteration formula z_n+1 := z_n² + c).

> restart: with(plots): 

> alexerror:=proc(x, iter)
> # written by Alexander F. Walz
> # May 29, 1996
> # March 02, 1997
>    local n, pts, xn, xnold, yn, y;
>    pts:=[0, 0];  
>    for y from 0 to 1.1 by 0.01 do
>       xn := x;
>       yn := y;
>       for n to iter while evalhf(xn^2+yn^2) < 4 do
>          xnold:=xn;
>          xn := evalhf(xn^2-yn^2+x); 
>          yn := evalhf(2*xnold*yn+y)
>       od;
>       if n >= iter then pts := pts, [xn, yn], [xn, -yn] fi;
>       # as a variant try: 
>       # "if n >= iter then pts := pts, [x, y] fi;" 
>       # for the Mandelbrot set  
>    od; 
>    pts 
> end:

> i:='i': 

> PLOT(seq(POINTS(alexerror(i/100)), i=-200 .. 70), SYMBOL(POINT));

The Mandelbrot set.

Merging of the Mandelbrot set and alexerror
after 100 iterations, both at the same scale.

MAPLE V FRACTALS AlexError #1.01 current as of May 23, 1999
Author: Alexander F. Walz, alexander.f.walz@t-online.de
Original file location: http://www.math.utsa.edu/mirrors/maple/mfralex.htm