Basics

Fractal images as shown here is all about generating sequences of complex numbers and rendering these sequences.
First let me introduce the mathematical basics.
More basics you can find in: Banach fixed point theorem and dynamical systems.
An often asked question: What about the real world?
In a second step a mathematical rendering method is shown.
A variant of generating sequences is the so called "Julia style".

Perfect Apple Manikin Sets

Apple manikin which is "Apfelmaennchen" in German is a German word for the Mandelbrot Set.
Generating fractal images this way you soon realize there is a kind of "perfect" images.
These perfect images have an infinite number of apple manikins. (The number is infinite, but also the apple manikins get infinitely small).
These apple manikins look like small versions of the basic apple manikin, the Mandelbrot set.
I have made my own try to define such "Perfect Apple Manikin Sets".
The trick is: as soon as you have a mathematical definition, you can compute, if
a fractal image is a Perfect Apple Manikin Set.
Under which condition is a fractal set a Perfect Apple Manikin Set?
It depends on the start value s of the sequence generation: Convergence to Perfect Apple Manikin Sets.

Classification

Searching for Perfect Apple Manikin Sets I have found repeating patterns,
which results in a classification of Perfect Apple Manikin Sets.
Next is an image table showing examples for each class.

Basic Formulas

Like z2 + c is the basic formula for Mandelbrot-like sets, I am searching for basic formulas for other classes of fractal sets.

Transformations

Other variants of generating sequences are so called
C-Transformations and Z-Transformations.
One can go a step further and do parametrized C- and Z-Transformations.
An interesting property of fractal sets shows when applying the function several times to itself: Idempotence of Z-Transformations.
Another variant doing transformations are I-Transformations.
An interesting property of I-Transformations, invariance of the form, is shown in I-Transformations of the Zatanz and Zloglog1z set.