(x,y): x=x^2+c, y=y^2+c This is Kalis idea which he expressed in terms of f1-f2. That got me thinking what about (r,ø)?
The direct idea was (r,ø):r=r^2+c,ø=ø^2+c, which was naive but a start. Now i have a set of vector tools and kombinationlehre to approach the idea .
The idea seems random, but cuts to the heart of understanding the mandelbrot and julia formulations, the transformation of coordinate systems, and parametric and vector functions they entail.
Transforming from extending and rotating to translating and translating at π/2 radians(ie a "hidden" rotation)
z(x,y)= (r2-ø2)cos(2rø)+i*(r2-ø2)sin(2rø) + c(x0,y0}= x + iy + c
Note how the i is now used. This was a great mystical wonder in Gauss's day, requiring Wessel, Argand, Cauchy and Gauss to throw light on it. It was Euler who added some notational simplification, But only Grassmann developed the robust system to demystify it. Hamilton was inclined to be theosophical about it, Grassmann was Platonic, evincing a dialectical exposition. Both however were guided by Gauss who had given some combinatorial consideration to the whole matter, after Bombelli who had initially denoteded them as adjugate magnitudes.Gauss clearly had cone on to consider the quaternions and beyond, but only Grassmann had made it explicit in a linear Algebra, Gauss being too fearful and conniving.He was also very busy on government business and was aware of the pragmatic Zeitgeist that was developing around the industrial revolution. Imaginary magnitudes did not bode well for any one who avowed them. Hamilton di avow them and suffered the consequences in his later life, academically. But for his staunch defense, we may never have retained the imaginary mathesis which seemed so apt to describe the mysteries of the universe and its gods.
Note, The combinatorics can be the same, as the combinatorics are the meta patterns in mathematics, too often not explained or exposited.
The formulae, being implicit is clearly an algorithm or a rule for depicting a relationship in a set of givens, in this case the vectors z(r,ø)
The function form is common to combinations ,functions and vectors and algorithms because it is a combination form by several names! The different names do not add to but rather obscure the combinatorial form. Also the full form makes explicit the givens and the sequences o be used as data sequences.
The relationships are relationships within a whole given form.If the form is not explicitly stated the it is assumed to be some form for which the relationship holds true. It is not a disembodied ntity, but a relationship found in a specific form and seen to be in many other forms, thus the notion of an intrinsic relationship means that the relationship is defining of a form or a class of forms, but not definitive as to shape. Extrinsic means that the specific shape of the form(a class of similar forms) is crucial to the definitions of combination.
INvariance in the symbolism to describe a combination is usually taken to indicate some inherent attribute that has been captured and enshrined within the symbols. It may be the only way to describe thee attribute as the apprehension of it is subjective. These subjective experiences are ofren thought of a evidence of "spiritual truths".
The Philosophical, Theosophical, metaphysical, and Theurgical aspects of any subject are often the subjective and spiritual and religious implication of the ideas expressed symbolically, and the story of how the symbolism was arrived at. Within these deliberations Aesthetics are more than evident as well as personal values and convictions. Very often a clash of philosophies would lead to a symbolic form of combination being declared wrong or false or the work of the devil, when in fact the combination is not only perfectly feasible, but also accurate and utilitarian, and an analogous or alternative form. Such clashes have pock marked the history of mathematics so called.
The set of principles called mathematics betrays this grinding together of controversial and opposing philosophical and religious outlooks, with a mediatorial third party or a board of directors constructing the accepted elements from a utilitarian viewpoint. Mathematics, as a field of study reflects the internecine wars in Academia, the careers and reputations of proud men, more than the natural order of the world. As a consequence mathematics is very human, fallible and fragile, and should not be hyped up beyond what it is and what it has been constructed to do for men,
Now i believe, the board of directors should become subservient to the board of directors of the computing sciences. No matter what, all will be subservient to the boards of directors of Philosophy and Theology.