Chaos, Order, or Nature?
by
Michael Theroux

It is indeed a rare thing that we should not only witness her beauty, but 
wholly embrace the visage of that divinity we baptize as nature. As the 
naturalists sought out the qualitative in life, so the fundamentalists of 
science tried ever to define nature's character in quantitative analysis.  
Herein lies the elementary problem concerning chaos theory.  For a 
time,  I found myself disregarding chaos theory altogether, owing to the 
fact that I regarded myself as one whose mission was to sustain that 
which could be rendered life-giving, and to slight all that was not.  
Finally, it was time to investigate what all this nonsense was about.  Of 
course, within this investigation, I would adhere to my subjective 
intuitions.
	The idea of chaos theory is that a seemingly random (chaotic) 
function displays periods of order, and that these periods will be self 
similar; that is, in perfect relation to one another (without delving too far 
into the history of this so- called "chaos" I will explain some of the high 
points in its association). A fine example is known as the Bifurcation 
which is an illustration of population growth and has been studied by 
many biologists. Consider a time line, insert the right formula, and 
geometrically you have a representation which exhibits stable and 
unstable regions of growth. The interesting fact is that the stable 
regions appear in a perfectly ordered fashion, contrary to what is 
expected. Again, the problem with the theory 
is that one may think that order has been created out of something 
appearing to be chaotic, but in reality there is nothing but an already 
ordered system which has subtly revealed itself. A contradiction. Alas, 
this is the way with nature - or is it?
	It was Benoit Mandelbrot who came up with the term "fractal" 
and it is his efforts that have brought the whole science of chaos into 
the spotlight. His paper entitled, "How Long is the Coast of Britain" 
opened up a new avenue of thinking in the mathematical and theoretical 
sciences. This idea of just how to measure the coastline would naturally 
depend upon your viewpoint. But, Euclidean measure could do no 
justice to the demarcation of irregular shapes and this is where one 
must turn to the perception of dimension. Many were quite familiar with 
this concept including such early pioneers as Charles Proteus 
Steinmetz who expounded on the nature of space in relation to 
projective geometry and the theory of relativity.  If dimension altogether 
relies on your point of view, from what distance does one measure an 
object? Advancing from the macroscopic to the microscopic levels of 
perspective, one will begin to perceive certain definable details, and 
although reminiscent of the whole, display a quality of irregularity and 
immeasurability. An infinitude in either case, if you will. 
Through his studies of irregular patterns in natural processes, 
Mandelbrot came upon the quality of self-similarity. This new geometry 
was to be hailed as nature's own.
	Anyone who has seen a fractal has seen the now famous 
Mandelbrot set. The formula for its creation is quite simple; take a 
number, square it , and add the original number (z^2 + c). The process is 
then repeated indefinitely with the iteration being either rational or 
irrational. Those that are rational become a part of the "set" and those 
that are not, do not. With the magnification of the set, the self-similarity 
reveals itself by displaying a replica of the  original. It is an extremely 
complex and beautiful form developed out of a very simple formula. 
	But, what about a fractal's relationship to nature (or using the 
new science term "dynamical system")? The difference is that when 
nature forms something it follows a general rule called the Golden 
Section. Some have tried to define this "irrational" number as just that - 
a number. More correctly it is a ratio, not an endless number string. Pi, 
Phi (the Golden Mean), the squaring of a circle, and other mathematical 
conundrums of so called irrationality have kept the quantitative 
theorists methodically labouring for centuries with no answers in sight 
(but probably keeping them employed). Phi, as an irrational number, 
may be familiar to most in the form of 1.6180339...etc. The more 
appropriate geometric representation is: sqrt (5 + 1)/2. This function 
shows itself repeatedly in all of nature and has thus been deemed an 
intrinsic component of what has been referred to as "Sacred Geometry". 
From the phyllotaxis of plants (their spiraling arrangement of leaves) to 
the Nautilus shell, the Golden Section or sacred geometry is found 
throughout all growth in nature, and has been dutifully emulated by both 
ancient and modern artist in sculpture, architecture, and music.
	In order for the fractal idea to model nature's processes I felt it 
necessary to insert the Golden Section into the established formulas for 
experimentation. In the initial stage of the experiment I added the sqrt (5 
+ 1)/2 and sqrt (5 - 1)/2 to the extant z^2 + c formula (where c actually 
became the Golden Section) and was not completely surprised to see 
the patterns of growth exhibited by the Nautilus shell emerge. Most 
showed the characteristic self-similarity described by the modern chaos 
theorists. Continued manipulation of the formula (z^3 and z^4) produced 
the resemblance of the reproductive organs of both female and male 
respectively.  Here, in deep magnification, the issuance of  spiralling 
forms would reveal a striking similarity to the vortexian mechanics of 
flowing water and galactic streaming so important to the theory of 
cosmic superimposition. But, what I found most fascinating was that 
upon extreme magnification, these implosive tendrils of form 
experienced periods of orbital irregularity. At once these spiralling 
shoots would appear to be gravitating toward a common center but then 
in certain instances would sprout new shoots not following the typical 
orbital pattern, as though they had found a new source of entrainment.  
Through successive manipulations of the basic formula using the 
Golden Section, I arrived at a form appearing to have no self-similarity 
up to the highest magnification possible. It is the imperfection of this 
"coastline" form which reveals a part of the genuine beauty of nature. 
This would naturally be true chaos, if indeed it is correct. What is 
interesting in the most qualitative and subjective aspect of nature - is 
that there is a seeming chaos in which her processes work. For 
instance, a typical IFS (Iterated Function System) fractal - one that 
utilises self-similarity to create objects somewhat resembling plant life - 
in its generation, is perfectly formed. However, in nature, the formation 
of plant life most often produces anomalies in character which do not 
typically follow the self-similarity code. The same error proves true for 
what are known as L-Systems fractals. These were developed in 1968 
as a method for modeling the growth of living organisms. The 
mathematical processes involved with these fractals mechanically 
instruct the computer how to generate the image and really have no 
relation to the growth of anything living or dead. Only by the inclusion of 
Golden Section geometry into the formulas are we to gain anything 
resembling the structure of nature, and dimensionally, these images are 
still quite lifeless. Only the motive properties of projective geometry will 
produce the growth factor into Golden Section geometries and this 
brings about the limitations inherant in computer generated 
iconography. This does not mean that something cannot be 
gleaned from researching the fundamentals of form through the use of 
computer generated images. The intricacies of these Golden Section 
images are in themselves a portal through which we may understand 
nature's dynamical systems more closely, but we must remember it is 
certainly not the complete picture. That the orthodoxy of science would 
represent nature with such mechanistic examples like the L-Systems 
and IFS models, is typical of their wanton destruction of the living. 
Nowhere can we define nature with such absurdities, and nowhere are 
we to "order" chaos. Nature has defined for herself what is to be order 
and what is to be chaos. The two are inseparable yet distinctly separate 
entities.
	I feel that the idea of the so-called "chaos" theory is of great 
importance in understanding certain basic functions of mathematical 
concepts such as self-similarity, but more importantly we should be 
concerned with the non-Euclidean geometries associated with spatial 
perspective and growth in the living. The inappropriate expression 
"chaos" would seem to be another avoidance of  "nature" by those 
bound solely to quantitative analysis, but this should not limit those 
with a desire to understand quality as well as quantity in all "dynamical 
systems". That nature works on defined geometric principles is 
undeniable. It is therefore undeniable that nature's geometry be 
included in "chaos".

NEW FORMULAS

;Please note that the file below is set up to run with Fractint 17.2 and 
needs no modification.

;All of these formulas are based on the Golden Section or Phi as it is 
found everywhere in nature. Exhibiting true patterns of growth, these are 
new to the genre and have many interesting characteristics not found in 
the "secular" world of fractals.  All of these fractal formulas work with 
Fractint 17.2. Enjoy.                                     

Michaelbrot(origin) {    ;based on Golden Mean
z = pixel:
z = sqr(z) + ((sqrt 5 + 1)/2), 
|z| <= 4; 
}
Raphaelbrot(xyaxis) {  ;phi
z = pixel:
z = sqr(z) + ((sqrt 5 - 1)/2) 
|z| <= 4;
}

DrChaosbrot1(xaxis) { ;more phi
z = c = pixel:
z = sqr(z) + (((sqrt 5 + 1)/2)+c)
|z| <= 4;
}

DrChaosbrot2(xyaxis)   { ;more phi
z = c = pixel:
z = sqr(z) + (((sqrt 5 + 1)/2)+c)
|z| <= 4;
}

Natura(xyaxis)  {  ;phi yoni
z = pixel:
z = z*z*z + ((sqrt 5 + 1)/2) 
|z| <= 4;
}

Element(xyaxis) { ;phi lingam
z = pixel:
z = z*z*z*z + ((sqrt 5 + 1)/2) 
|z| <= 4;
}

;try inside=maxiter and outside=imag on these

test(xyaxis) { ;+phi 
z = ((sqrt 5 + 1)/2)/pixel:
z =  z*z + pixel*((sqrt 5 + 1)/2)
|z| <= 4;
}

test1(xyaxis) { ;+phi
c = pixel 
z = ((sqrt 5 + 1)/2):
z =  z*z + pixel*((sqrt 5 + 1)/2) + c
|z| <= 4;
}

test2(xyaxis) { ;+phi 
z = ((sqrt 5 + 1)/2)/pixel:
z =  z*z*z + pixel*((sqrt 5 + 1)/2)
|z| <= 4;
}

test3(xyaxis) { ;+phi 
z = ((sqrt 5 + 1)/2)/pixel:
z =  z*z + pixel*((sqrt 5 + 1)/2)/ ((sqrt     5 - 1)/2)
|z| <= 4;
}

Tetratephi(xyaxis) { ;?
z  = c = pixel:
z = c^z + ((sqrt 5 +1)/2) 
|z| <= 4;  }

REFERENCES

1. Chaos - Making A New Science by James Gleick 1987, Penguin 
Books.
2. Fractal Creations by Timothy Wegner and Mark Peterson 1991, The 
Waite Group Press.
3. Fractint 17.2 Freeware program for generating fractals, available from 
the Graphic Developers Forum on Compuserve.
4. The Golden Section by C. Fredrick Rosenblum J. of Orgonomy, 8, No. 
2.
5. Cosmic Superimposition by Wilhelm Reich, o.p.
6. Four Lectures on Relativity and Space by Charles Proteus Steinmetz 
1989 reprint, Borderland Sciences.