Do you get bored with the bug-like Mandelbrot fractal image? Here is something
that will startle you. The fractal structure is generated from the basins of attraction in the Newton's iterative method for solving the single-complex-variable equation (refer to your college math textbook):
(1) z <- z - f(z) / f'(z)
z, z0 are complex numbers, f' is the first derivative of f. By variating starting point z0, the iteration will converge into a particular complex value which is one root of the equation
f(z) = 0.
The "Newton Map" is generated by color coding the "basins of attraction" (i.e., which root the iteration eventually drops into from a starting point) for every starting point in a rectangle on the complex plane.
In comparison, Mandelbrot set is created by the iteration:
(2) z <- z^2 + c
But with z all started from (0,0) (or vicinity) and variating c, color coding a rectangle of complex c by the number of iterations before the iteration converges or diverges or drop into a Julie set.
From the simple iteration (1) on the complex plane, it reveals a rich and diverse form of stunning beautiful picture.
Besides the difference in the fractal image from the commonly seen Mandelbrot fractal, there are several important features in this AfterDark module (named "NewtonMap") from other AfterDark modules that display fractal images:
1. Since a fractal image take a long time to produce a complete picture at the best resolution of the screen, you keep feel that a beautiful piece of art is being distroyed when you touch the mouse or keyboard. In this module, the map is saved when it exits AfterDark, then it resumes from where it left over when it enters AfterDark again. In this way, a fractal map which may takes hours or days to complete the calculation can still be accomplished by on and off use of screen saver.
2. When the map is completed at the desired resolution (default set at the
screen resolution), the screen is automatically output to a PICT file, so that you can have a collection of beautiful fractal images.
3. The order of algebraic equation is set upto 21, and it can be randomized; the coefficients of the algebraic equation, the initial starting point, and the size of the map are also randomized. With such a wide variaty of parameters, nobody can predict what the image will be. So just let it run and enjoys all the surpises.
Feel free to use it even when you do not pay the shareware fee ($10). But if you do like the outcome of this module or keep the pictures created by the module, please pay the suggested fee.
Tips:
1. The saved picture file (when the map is completed) is in PICT format, though I set the file creator as SimpleText, you can open it with any other graphics tool. The file is saved onto the top level of your boot disk, named as:
NM-Order**.date;random_number
2. If you want to force it to have a fresh start (rather than from the previous saved results), delete the file "NewtonMap.resume" on the top of the hard disk.
3. You can use the AfterDark randomizer to alternate between the two different "NewtonMap" modules so that the screen will be changed once a while.
4. If you have MPW, you can run NewtonMap.poke to dump out the current status of the iteration (coefficients, roots, etc.)
Please sent $10 shareware fee (for the whole set of "NewtonMap" variations, hopefully there will be more coming in the future) to:
