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]
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.
... 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.
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
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
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
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
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.
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
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)
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
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
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
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
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
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
What does this mean for our proverbial butterfly? Can it really cause
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
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
Science and Mathematics are the real magic!
Predictions July 1997 File Info: Created --/--/-- Updated
15/3/2014 Page Address: