One can probably give no really useful hints, how beautiful fractal images are made. It reminds me of the Woodcarver, who said: The forms are in the wood, you must only recognize it and help a little with the knife. Even if fractals are abstract mathematics, no knowledge of mathematics is necessary in order to produce Fractal images. Only a program, little sense for colors and forms - and very, very much patience.
The mathematics is´nt seen in the images. You see colors. To deal with colors is surely simpler, in any case more intuitive than to cope with mathematics.
Of course it´s better the more you understand of mathematics and fractal theory. But there I have also strong deficits. You can figure out the important things by trying out, questioning and reading.
Concerning color handling, so far the program Fractal Explorer pleased me best.
It is also passably operable, what I unfortunately cannot say of all fractal programs.
Other fractal programs have definitvely more capabilities, particularly the not cost free Ultrafractal
Alas, more program- capabilities require additional abilities from the user or they are useless.
At the moment I try to use some of about 20000 Fractint images which I collected in the Web, as basis for new creations with Fractal Explorer. Parameter files and formulas from the Web are a very good basis for own attempts.
Zooming is always an excellent source for new fraktal images. At least for a fast overview what zooming into fractals can achieve, XAOS is the first choice http://wmi.math.u-szeged.hu/xaos/doku.php
Even with the floating point unit of a PC amazing deep zooms are possible. To express it in figures: With a "medium" zoom of 6*10^12 (6.000.000.000,000) the whole Mandelbrot set would be approximately as large as the distance from Pluto to the sun. The floating point unit can calculate easily zooms of 10^17, about 16000 times as much.
With Fractint arbitrary zooms are possible, more exactly as deep as you abide to wait for the result - when you do deep zzoms, computing times can increase fast to several years for an image.
In any case there is a "Major Rule" for making fractal images. To quote the computer game "Leisure Suit Larry" : Save early, save often!
Probably nobody ever found a distinct fractal image again unless he saved the parameters.
A few words about Apophysis
This program makes only "Flame" frakcals, a kind of IFS www.ultrafractal.com/index.html http://en.wikipedia.org/wiki/Iterated_function_system
Apophysis appeares to be a simple to use program because it produces random fractals without explicit user request. I contradict the "simple to use". I don´t understand IFS fractals and won´t probably learn to understand then.
In my experience randomness very rarely creates nice fractal images and never excellent ones.
I gave up attempts with the "evolver" of Fractint quite soon ( which too changes fractal parameters randomly ). Chaos has nothing to do do with randomness.
Even if I don´t recommend it to beginners, Fractint one is my dearest fractal programs. Not least, because there are innumerable example parameters for Fractint on the Web.
It´s a DOS program that can be run in the Dos Box under Windows. The handling is probably not reasonable for beginners. But with my attempts with Fractal Explorer I often missed painfully some of Fractints capabilities.
By the way I advise strongly against the Windows port WINFRACT - it´s extremely badly operable old program version.
With the Fractint port for Linux I could not make friends too - Fractint seems not suitable for a GUI.
As an outstanding example of a true fractal artist I recommend the homepage of Sylvie Gallet
With the urgent advice not to regard her works as standard for own attempts (that would be depressing).
There are many other Fraktal of programs, most of them cost free. Most have special characteristics, which are worth a try.
A quite extensive list of fractal programs:
I found 2 fractal programs for Mac. I would really like to try out one of them, but I don´t possess a Mac.
http://www.fractaldomains.com/index.html ( 20$ )
http://www.berkhan.com/easyfrac.htm ( 20 Euro, Website german or english )