Fract & Friction

by Danny Malvert

[Disclaimer : This article is written from a pro-mystical perspective and as such contains anti-scientific bias,because of this I personally distance myself from comments made within the text,and include what I consider to be corrections or rebuttals to the views of the author.
Although the main body of the text is factual the philosophical conclusion drawn by the author is misleading, since it attempts to put a pro- nature worshipping slant upon the facts. When reading this it should be remembered that it was "blinkered scientists" that created fractals with supercomputers and not druids chanting over crystals.The contempt shown by the author and the ignorance of the subject matter seem to me to derive from sour grapes that science has once again "gloriously urinated" on mysticism -LB]

WAV 242K

For centuries mankind has sought to discover the secrets of nature. An insight into the superficially simple yet beautifully complex components of the perceived universe can be seen in the observations in prehistoric cave-paintings to the use of natural knowledge in stone circles and participation in the very systems of nature.
We can trace the interaction of  'orthodox' science with natural systems, the transition from 'tentative' observer to 'sleeves rolled up' participant, back to the Middle Ages. The mystics of this time strove, through the use of mathematical and sympathetic magical principles, for the ultimate answer.
They sought to induce spiritual changes upon their material world through complex geometric forms which were often upheld as talismans and attributed with great power. The unblinkered scientists also sought meaning which ascended the sum of its composition and continued the search for a symbolic vehicle to express their doctrine.
In 1202 the Italian mathematician Leonardo Fibonacci published his influential treatise Liber Abaci (The Book of the Abacus). This was the first European work that dealt with Indo/ Arabian mathematics and consequently Fibonacci is best remembered for giving us our numerals and numbering system. His recreational concerns, however, are not so revered yet are of no less importance. Parts of Liber Abaci mused on light-hearted matters, such as the following problem.

'How many pairs of rabbits can be produced from a single pair in one year if it is assumed that every month each pair begets a new pair which, from the second month , becomes productive?' By adding the two previous numbers to gain the next, the following sequence is generated.

Jan	Feb	Mar	April	May	June	July	Aug	Sept	Oct	Nov	Dec
1	1	2	3	5	8	13	21	34	55	89	144

... ad infinitum Subsequently, in the 19th century Fibonacci numbers were discovered in many natural forms such as in the botanical phenomenon known as phyllotaxy (the study of the arrangement of leaves on a stem). Starting from any leaf on a plant, a spiral is traced in an anti- clockwise direction from leaf to leaf. Upon reaching a leaf directly above the first, the number of leaves and turns around the stem within the spiral are constant, irrespective of the starting point.
Although the number of intervening leaves and turns around the stem varies for differing species, certain figures re-occur. It transpired that the whorls on a pine-cone, the petals on a sunflower, the pattern of snail shells and the genealogy of the male bee all follow a sequence of Fibonacci numbers. These have been extensively studied and the relevance of their properties seems inexhaustible.

The Golden mean
In 15th century science had difficulty containing itself over the implications of the Divine or Golden Proportion devised by the Italian mathematician Luca Paciola. The attributes of this 'golden mean' (1:1.6180) [Calculated as (1+ Ö5)/2 -LB] were held by medieval artists and scholars alike to be the most aesthetically pleasing of any ratio.
The inclusion by many of the word 'divine' in this context reinforced the fact that it was seen as some kind of miraculous, esoteric breakthrough. Consequently the golden mean can be seen in the dimensions of many medieval buildings, the work of countless great painters of the period and, even now, in the humble playing card.
Mathematician Robert Simson noted in 1753 that as the Fibonacci numbers increased in magnitude, the ratio between succeeding numbers approached the golden number - 1.6180. Although the theories of Fibonacci and Paciola were separated by 200 years, there seems to be a significant link between them.
King Oscar II of Sweden once set a problem concerning the stability of celestial bodies in general, known as the 'three body problem.' Then, in 1889 a French mathematician , theoretical astronomer and philosopher Henri Poincaré, was awarded the winning prize for his solution.
One hypothetical scenario he propounded consisted of a Moon's orbit being affected by the gravity of two stationary planets. In his initial study and submission, Poincaré surmised that the orbital pattern of the Moon in these circumstances was stable and predictable.
After publication, however, Poincaré discovered a pivotal error in his calculations. What he found was a much underrated scientific milestone : apparently logical well ordered systems could produce effectively random results.
For the pattern of the Moon' s orbit seemed to be beyond accurate prediction. This was nothing less than a revelation at the time, one that has since become known as Chaos theory.

Chaos and fractals
Poincaré is widely considered as the father of Fractal and Chaos mathematical theory, both having their origins in his work on the 'three body problem.' The term fractal, from the Latin fractus (broken or fragmented) , was coined by the Polish mathematician Benoit B. Mandelbrot, in 1975.
For many years fractals were considered to have no relevance to reality. Their true wonder was not realised until the dawn of the computer age in the 1960s and 70s. Only then were mathematicians able to perform the thousands of calculations necessary to build up a fractal image.

[Note that the author admits that modern "blinkered" scientists were the ones who discovered fractals by use of computers.Poincaré only had manual methods and he plotted the first few points,but could not really get a handle on the myriad complexity of the system he had uncovered.In essence computers are absolutely necessary to truly see fractals -LB]

There can be few of us who have not seen a fractal, perhaps unknowingly. They are all around us, in snowflakes and tree formations, in the way that clouds form and streams and rivers weave their paths.
Fractals have enabled science to create models of many irregular natural phenomena such as the behavioural patterns of animal groups, the erosion of coastlines and the distribution of galaxy clusters throughout the universe. Every time you scrunch up a piece of paper you are creating one, even though you may be unaware of this fact.
Although fractal images appear complex, the rules controlling them are relatively simple. The result of each cycle of mathematical calculation is the input value for the next; this is known as reiterative calculation.
The Fibonacci sequence demonstrates a simplistic form of this. Fractals are said to exhibit the property of self-similarity, an object's component parts resemble the whole and vice versa. One particular fractal shape, Barnsley's Fern, is a useful example of this because it relates directly to the characteristics of the actual fern. Its overall shape is duplicated in the fronds, the protuberances from the side of the stem. Along the frond' s edge are smaller prominences which represent the overall shape, a scale smaller, and along those protuberances are smaller prominences, and so on.
Barnsley' s Fern is created by the repetitive application of four relatively simple mathematical rules. It is too perfect to be a product of nature as each component part is an exact copy of the whole. Natural fractals are more irregular and the component parts often bear only a fleeting resemblance to the overall structure. Yet the smallest part of a fern, as well as many other natural patterns, may look remarkably like the original shape.

Principles
Fibonacci' s principles also hold good here. For example, when the spiralling of phyllotaxy reaches the apex of a stem, leaf formation becomes progressively more condensed towards the shape of a flat helix - the microscopic shape of one of a plant' s smallest components, RNA. This is the genetic library of the plant and, as with DNA, the equivalent substance in animals, it is held to be the stuff of organic life itself.
It could be argued that the known universe shows a tendency towards self-similarity, between the macrocosm and the microcosm. The resemblance between photons and stars, the likeness between moons orbiting planets and electrons orbiting atoms.
Natural fractals conceal a wealth of information concerning their formation and subsequent evolution. Science acknowledges that, as yet, it simply isn't clever enough to extract it.

[Whilst this is actually true,the carping,sniping nature of the comment as "blinkered scientists will never understand the esoteric aspects of nature which is ultimately beyond their grasp" belies the fact that science has understood a great deal of nature,and the comment suggests that mystics are in touch with something which science can't grasp which is "beyond understanding". Science will become clever enough to understand it,that's how it has progressed,there is not a special thing that only mystics know about such that they can look down their noses at scientists as their inferiors as if somehow they had a special knowledge.
If it were left to mystics,fractals would never have been discovered,and it is because of science that the author has a chance to "embrace" them. One wonders why such an ungrateful and condescending tone is taken to refer to the people who gave birth to something which the author thinks worthy of assimilating into his belief system.What we are witnessing is science being plagiarised by pseudo- science,and being exploited to make a point -LB]

In Fibonacci's time scientists were little more than well informed observers. Breakthroughs made in understanding nature therefore had no significant pro-active purpose.
The arrogance of modern science acknowledges that nature finally appears to fit into its principles. Small fragments of knowledge allow science to meddle ["Meddling" assumes that we don't know what we are doing.The author shoots himself in the foot,if we didn't then we wouldn't be able to exploit chaos for weather prediction -LB] with things it could once only observe.
Thankfully and, let's face it, not surprisingly, nature still has a few tricks up her sleeve. With the gift of fractal understanding came its sibling: chaos theory. To scientists, this is an unwelcome, confounding concept but, to those of us who value nature more than quantum mechanics, it is to be embraced with arms flung wide.

[This is a total lie,Chaos is not confounding or unwelcome.All it did was throw a spanner into the Newtonian clockwork universe.Again there is the idea that somehow mystics already knew that the universe had these attributes or could be described by them.They DIDN'T.
In "Does God Play Dice?" (p133) Ian Stewart says:-

We will see later that the butterfly effect is as much a blessing as a curse,but we will see that only by considering it's nature in some depth...."

If Danny Malvert actually bothered to study nature in some depth instead of worshipping it,he might not draw false conclusions.
The author also makes the faus pas of loving nature more than quantum mechanics.First nature IS quantum mechanics (at least latterly described in terms of) and quantum mechanics describes a free universe,which one would have thought the author would have "embraced" since it has the same effect as Chaos theory.Both systems are deterministic in principle,in that they are subject to rule systems,and presumably mystics hate determinism and things being reduced to rules. But both systems render a freedom to the universe and make it fundamentally unpredictable,so there is no reason to embrace one and reject the other.More absurdly still both systems become tied together in Quantum Chaos,and so to embrace one and not the other makes no sense whatsoever.This is what happens when you don't know what you're talking about,and have a poor understanding of your material.
 It's even more bizarre if you consider that both systems free the universe which is presumably to mystics liking,but their own beliefs seek to straitjacket nature into doing their bidding or tie it up into a fatal future that is already predefined. The idea of a "destiny" is very much tied up in this view and is a complete anathema to a free universe.One wonders then why a fatalist would embrace a basically unpredictable system which says that you cannot say what will happen next.This is a direct affront to the belief system,but in the mixed up world of pseudo science it doesn't matter if it's consistent as long as you like the bit of science you've found,and can find a way to warp it into your belief system,and after all Fractals are pretty pictures and resemble Mandala so they must be mystical.More fundamentally fractals hark back to the hermetic geometries done by the natural philosophers to which the author referred to as "unblinkered scientists".When these people were working they drew analogies between the orbits of planets and geometric solids and created 2D plane drawing symbols and imbued them with mystical power,because of some ill-understood (or well-understood) mathematical principle.In fractals New Agers have found another bit of mathematics to plagiarise and assimilate into their hotch potch patchwork quilt philosophy. The early scholars were not cleverer than now,they laid foundations.Some of their work is fallacious and some brilliant.Hexagons don't have mystical powers. The length of a line joining adjacent vertices of a pentagon is the golden mean,this is not magic, it's mathematics-LB]

Chaos is not randomness or chance. Small uncertainties in any natural system grow so large that they render any definite long-range prediction of outcomes impossible. Calculations are so complex and of such quantity that even computers cannot calculate and memorise chaotic sequences with in finite accuracy. Various computers treat mathematics slightly differently. If numerous computers were given the same mathematical model they would, in time, produce effectively unrelated results.

[This is stretching a point.The calculations do depend on very small perturbations and in principle various computer systems ways of holding data would render variations in the final outcome,but modern computers can hold numbers to accuracies in the region of 1x10^-32 or 0.00000....(31 zeros).........1,the essential flavour of the fractal would be maintained on all computers up until they diverged in accuracy at a point around this figure.
In "Does God Play Dice? (p143) Ian Stewart says:-

 However,computers do not store numbers to infinitely many decimal places;they use "finite precision" arithmetic which introduces round-off errors, so the fine detail of the Lorenz attractor looks different on different types of computer.So is the Lorenz attractor just an artefact of the finite precision approximation, or is it present in the exact solution to the original equations? The answer is not at all obvious,and a few diehards have even stated that the irregular behaviour found by Lorenz is entirely due to computer error.
 We know now that it is not an error,thanks to Konstantin Mischaikow and Marian Mrozek at the Georgia Institute of Technology.


Ian goes on to explain how this proof is obtained.The fact is it's largely irrelevant whether finite precision of computers cannot handle the infinite complexity of a fractal,the fact is it can be proven that the chaos is real and not due to computer error.
All the author is able to say is that finite computers aren't able to calculate with infinite accuracy.That's hardly a revelation,neither is it an incapacity of computers.There are mathematical techniques that can deal with infinity.
In essence all modern computers would produce essentially the same image for all intents and purposes, and the variation in accuracy is a limitation of all computers,it is not an indication of the futility or inability of science to deal with nature.
Euclid for instance discerned that there are an infinite number of prime numbers, without actually counting them all.He deduced an in principle argument that showed that one could construct a new prime from a previous one in never ending succession,showing that there was no biggest prime number. Techniques such as this mean that just because something is infinite or infinitesimal doesn't make it out of bounds from scientific analysis.David Hilbert has also done work showing how mathematics can defeat infinity.So if the author draws on infinity as a means to showing science's fallibility,he is grossly mistaken -LB]

The commonly quoted notion that the butterfly flapping its wings in Tahiti can cause a hurricane in Britain is not wholly accurate. The hurricane would almost certainly have happened at some time anyway. The flapping of the butterfly's wings do not cause the hurricane but simply make it impossible for science to predict when it will occur.

[This is not entirely accurate.Butterflies wings do cause hurricanes,that is why it is impossible to predict the weather.The hurricane wouldn't have happened anyway. What the author is trying to articulate is the misunderstanding of the literal nature of the analogy as being that hurricanes requiring butterflies wings flapping in order to start. The butterfly analogy is meant to emphasise the unpredictable nature of weather prediction based on the smallest fluctuation.In that weather systems would create hurricanes anyway, regardless of any butterflies wings flapping,the author is correct. But the slant given to the sentence is to show that "once again scientists are defeated by nature" and that it is lack of precision and inability to cope with the infinite that "urinates" on them.
In "Does God play Dice?" (p131) Ian Stewart explains:-

What does this mean for our proverbial butterfly? Can it really cause a hurricane?
What the butterfly does is disturb the motion of the point in phase space that represents the Earth's weather. Assuming that this point lies on an attractor,albeit a highly complex multidimensional one,then the tiny flapping of the butterfly can divert the point off the attractor only very briefly,after which it rapidly returns to the same attractor. However,instead of returning to the point A that it would have reached if undisturbed,it returns to some nearby point B. The trajectories of A and B then diverge exponentially,but because they lie on the same attractor, they generate time -series with the same texture. In particular,a hurricane -which is a characteristic weather motif - cannot occur in the perturbed time-series unless it was (eventually) going to occur in the original one. So what the butterfly does is to alter the timing of a hurricane that - in a sense- was going to happen anyway.Don't take that too literally: the butterfly may trigger the conditions needed to make a hurricane form,or prevent one that would have formed.But most of the time it will just have a minor effect on where and when a hurricane that has been building up for global reasons - the right kind of warm,humid air in the right place - will occur.

and on p132,he says:-

Given all this,it is an exaggeration to claim the butterfly as the cause of the big changes that its flapping wings sets in train.The true cause is the butterfly in conjunction with everything else.

I would suppose that a mathematician who actually works with Chaos Theory for a living is much more likely to know what he is talking about than a columnist for a mystic magazine.

In fact many modern methods can make quite good predictions based on models of what has gone before,this is in stark contrast to mystics ignorantly hoping for the best,or casting runes,portents,and looking for conjunctions or chanting over crystals in order to bend nature to their will,which is a total fallacy -LB]

Over the centuries nature has given away some of her deeper secrets and then, with a splendour which reflects the very systems concerned, has gloriously urinated on the of all scientific law. Consequently, at least for some , the discovery of science 'fract' has merely led to science friction.

[Finally the author ends on a crescendo saying that fractals are a exemplar of nature defeating science.What baloney.Fractals are an exemplar of science comprehending nature,if indeed they truly represent aspects of nature which they appear superficially to resemble.Fractals have not confounded science,they have aided it.Weather prediction has progressed because of chaos theory,not in spite of it. Virtual reality simulators can run at real time speeds primarily because fractal simulations of environments make rendering of images quicker. Your average pilot's simulator would be up a gum tree without them.CD-ROMS contain myriad images and many of which may use fractal image compression to store them.So fractals aren't a thorn in the side of science,this is a gross lie.It was science that discovered them,not mystics,and yet like the other animals after Henny Lenney's cake they want a piece but weren't prepared to bake it.They are just jealous that science is much better at understanding nature,than their pathetic attempts to get it to do their bidding,and thus try to make it sound like they know better,and somehow have an a priori claim on things like fractals, which should they take the time out to actually understand the equations (note the author quotes no equations at all) they would find that in fact,the ideas are at odds with the fatalism present in mystical belief.In short,as usual,they make a farcical mess of trying to comprehend things they don't understand -LB]

Virtual Mode by Piers Anthony

Colene is a young woman with everything going for her.She's attractive, intelligent, popular. Colene is a young woman with a problem.She's screwed up,depressed,suicidal.On the outside she sparkles,inside her soul is rotting.Her heavily scarred wrists are testament to this.For her,the idea of death represents freedom.But she can't bring herself to take the plunge,to escape. Then,one day,she finds a man unconscious by the roadside,with no identification on him. She helps him,for he is injured,hides him,for he seems very strange,and learns to communicate with him,for he speaks a strange language unlike any she has ever heard before. Slowly,a story more fantastic than any she could have imagined emerges.The stranger's name is Darius and he is a ruler of part of his world in a universe where magic works.But the source of his magic is depleting and,desperate to find a partner who can help him replenish his power,he has embarked on a venture fraught with risks - to explore the infinite network of worlds - or Modes - in search of such a person......

Back in the study hall she brought out her compass and wiped the point on the tissue,just to be sure.Then she brought out her geometry homework,so that no one would wonder about the compass.Geometry was a snap;in fact it was boring, because it was two dimensional.It would have been more of a challenge in three dimensions,or four.If only they had a class in cubic geometry,or multi-dimensional constructions.Or fractals:now there would be one she could truly sink her teeth into. Class,today we shall take our little pencil and graph paper and define the complete Mandelbrot Set. Colene stifled a smile.The Mandelbrot Set was said to be the most complicated object in mathematics.Even mainframe computers could not fathom the whole of it.Yet it was simply an exercise in algebra,plotted on paper.How she would love to explore that beautiful picture! To lose herself in its phenomenal and diminishing convolutions,for ever and ever Amen. But this was a mundane school,where brains were routinely pickled in trivia.No hope here.

Science and Mathematics are the real magic!

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