Definition of Fractal Sequence Set:
For a function f(z,c) and a start value s assign to each point c in the complex plane
the sequence z0, z1, z2,...The sequence z0, z1, z2,.. is generated by calculation of zn+1 = f(zn,c) with start value z0 = s.
A Fractal Sequence Set is defined as the points in the complex plane where each point has assigned this sequence.
Each Fractal Sequence Set is assigned a Fractal Oscillation Set:
Definition of Fractal Sequence Function:
The mapping from the function f(z,c) with a start value s to its Fractal Sequence Set is defined as
Fractal Sequence Function FS:
FS: fs(z,c) → Fractal Sequence Set
In other words: fs(z,c) is the argument of the Fractal Sequence Function, which gets mapped to the corresponding Fractal Sequence Set.
(This has no consequences for defining Perfect Apple Manikin Sets, but is used in the section Parametrized C- and Z-Transformations)
Definition of Fractal Oscillation Set
For each point c in the complex plane assign to each point the number of distinct oscillation values of the sequence z0, z1, z2,...If the sequence does not oscillate (divergent or fixpoint), assign the value 0 to the point c.
The Fractal Oscillation Set is defined as the set of points in the complex plane where each point has assigned the number of distinct oscillation values or 0.
Definition of Fractal Oscillation Areas and Fractal Oscillation Circles
Definition of Fractal Oscillation Area:The points in the Fractal Oscillation Set with the same oscillation value (except the value 0) which are connected are defined as Fractal Oscillations Areas in the complex plane.
Definition of Fractal Oscillation Circle:
A Fractal Oscillation Area which is a circle is defined as Fractal Oscillation Circle.
Definition of a Perfect Apple Manikin Set
A Fractal Sequence Set having an infinite number of Fractal Oscillation Circles is defined as Perfect Apple Manikin Set.Fractal Oscillation Circles of the classical Mandelbrot Set - z^2
