Fractal Art off-topic but interesting
Mike Dubbeld
Check it out
http://www.skydancers.com/fractalmania/_xenodream_main.html
http://www.skydancers.com/1mainpages/_journeys_of_spirit.html
Do a search on Fractal or xenodream
Mike Dubbeld
Sir Frederick
It's enough to take your fractal breath away!
--
Best,
Martin
sir freddie the unique near sd,ca
*************************
Phrase of the week :
" Tomorrow, and tomorrow, and tomorrow
Creeps in this petty pace from day to day
To the last syllable of recorded time;
And all our yesterdays have lighted fools
The way to dusty death. Out, out, brief candle!
Life's but a walking shadow, a poor player
That struts and frets his hour upon the stage
And then is heard no more. It is a tale
Told by an idiot, full of sound and fury,
Signifying nothing."
-- Shakespeare(? - ?)
:-))))Snort!)
*************************
Dev Singh
How would you define fractal art?
pEACe
Dev
"Mike Dubbeld" <mi...@erols.com> wrote in message
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The Immortalist
"Dev Singh" <de...@ihug.com.au> wrote in message
news:a9gn6v$r12$1...@lust.ihug.co.nz...
> I really don't understand the term.
>
> How would you define fractal art?
>
Fractal Geometry - A Gallery of Monsters by Chris Lucas
Introduction
What do the following have in common ?
A galaxy, a lung, a coastline, a tree
A figure that has more than two and less than three dimensions
A figure with an infinite perimeter and zero area
A solid that contains only two dimensions
A figure which changes its shape the closer you look at it
A figure that looks the same at any scale ?
All of these are related to the same thing, the magic of fractals !
Paradoxical Coastlines Impossible ? In this strange world nothing is quite
what it seems...
Take the coastline of Britain for example. How long is it ?
Nobody knows.
Of course they do you say ! Ah, but they know roughly the area of the
country so they must, by Euclid, know the minimum boundary surely, an
equivalent circle ? Yes, but the actual boundary is infinite ! To see this,
go in your mind to the seaside with a metre rule and measure a section of
rock. You will skip over a few crevices will you not ? Now use a kilometre
ruler instead - this skips over a lot more resulting in a different, lower
reading. Take now a 1 cm measurement, this will go around most
irregularities and give a much bigger total. So, the length is variable
isn't it ? But not infinite surely ?
Move up the coast and you find a river. What do you measure now ? Well, just
continue up the left bank until you re-emerge on the right, the coastline
must be continuous ! But now you are following all the tributaries and
streams and rivulets and....you will never appear on the other side I think
! If you do, take an micron sized measuring stick and try again, by the time
you have measured every soil particle you will have effectively have used
infinite time. Britain thus has an coastline approaching infinity but a
finite area. A 2 dimensional paradox.
More Paradoxes
Now take a tree and measure its volume (easy, just dunk it in a big bath and
see how much water overflows, like Archimedes would). But what is its
surface area ? Yes, that's right it's nearly infinite again (don't forget to
measure down every pore or stomata in every leaf...). A 3 dimensional
paradox now !
How many dimensions did I say ? Two ? Three ? Neither are right, both items
have what is called 'Fractal Dimension' - a fractional number. Ignoring
technicalities, this is a measure of how irregular an object is. A Koch
snowflake (a triangle, with other third size triangles stuck midway on each
side, and so ad infinitum - illustrated) has for example a fractal dimension
of 1.26. In the extreme cases we have seen the dimension become one more
than we would expect, a single dimensional line becomes two dimensional, a
two dimensional surface becomes three. I'll leave to your imagination a 3
dimensional fractal in 4 dimensions !
Self-Similarity
Can we go the other way, reducing a three dimensional object to two ? Yes we
can. Take a pyramid and drill a hole in it. We increase the surface area and
reduce the volume. Keep drilling smaller holes in what is left until you run
out of material, you now have a solid without any volume, made up only of
edges. The formal version of this is called the Sierpinski gasket in three
dimensions, but has a fractal dimension of only two !
While measuring the coast with the different ruler sizes something may have
struck you. The shape of the coast at 1km scale is the same as at 1m scales,
and again the same at 1cm scale. Think of Africa from space, an island on a
map, a rock pool beneath your feet. The shape of the coastline always
appears the same, equally jagged. This is a feature of fractal systems which
we call 'Self-Similarity'. We cannot tell just by looking at a system what
scale it really is. Yet every view is slightly different, the objects are
identical in form but also not identical in detail ! Why is this ?
Mathematical Roots
To see let us look at something quite different. Many people know that you
can find the roots of an equation involving squares of x by using a simple
formula, and it gives two values. Generalising, we can expect n roots for an
equation of x to the power of n. How do we solve these bigger equations ?
Often Newton's method is used, this is an iterative solution and gives a
better approximation by putting the last answer back into the equation,
repeating until the answer doesn't change - that being a solution. For other
solutions we try different starting values and, if lucky, the formula will
converge to an alternative solution. Luck ? This is precise mathematics
isn't it ? Not quite, it is another fractal !
The boundary between the range of values converging on one root and those
converging on another is a fractal curve. If we guess a value close to this
boundary we cannot be sure which root it will converge towards. It make no
difference if we magnify a graph of value versus root, the irregularity and
unpredictability of the boundary is still there, big as ever (as illustrated
in this plot of x6). It is this boundary between part of a system moving in
one direction and part moving in another which is at the heart of fractals.
The Mandelbrot Set
The sensitivity to initial conditions is a feature of the phenomena we call
'Chaos', and a fractal boundary is usually a manifestation of a chaotic
state of the system. Let us now take the most famous fractal image, that of
the Mandelbrot set, what does this show us ?
Well, any two dimensional system can be related to x and y co-ordinates
(just two numbers specifying the position in vertical and horizontal
directions). If we pick two such numbers at random and iterate them in the
same way we did for the Newton solutions, but using the Mandel formula, we
can see what solutions arise. The result may go to zero, or grow continually
(like repeatedly squaring 2) towards infinity. If it goes to zero let's
colour the x/y point on our picture blue. If it grows let us colour it
according to how quickly it exceeds, say 2. If we do this for every x/y
point then we have the normal picture for a Mandelbrot set (right). The
fractal boundary is the edge between the central blue 'sea' and the rest of
the picture. If we zoom in to this edge we obtain all the wondrously
detailed fractal images that you may have already seen.
The formula for the Mandelbrot set is only one of a vast number of possible
formulae that can generate fractal boundaries, in fact there are more
formulae that are fractal at some values than those that are well behaved !
A strange world indeed...
Gea Jones
--
Best Wishes
Gea*
"The Immortalist" <Reanima...@yahoo.com> wrote in message
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Having been using fractals to make pictures am curious to know who Julia is?
Gea
The Immortalist
"Gea Jones" <Geaj...@btinternet.com> wrote in message
news:a9hipa$l2$1...@knossos.btinternet.com...
When only 25 when Gaston Julia published his 199 page masterpiece Mémoire
sur l'iteration des fonctions rationelles which made him famous in the
mathematics centres of his days.
As a soldier in the First World War, Julia had been severely wounded in an
attack on the French front designed to celebrate the Kaiser's birthday. Many
on both sides were wounded including Julia who lost his nose and had to wear
a leather strap across his face for the rest of his life. Between several
painful operations he carried on his mathematical researches in hospital.
Later he became a distinguished professor at the Ecole Polytechnique in
Paris.
In 1918 Julia published a beautiful paper Mémoire sur l'itération des
fonctions rationnelles, Journal de Math. Pure et Appl. 8 (1918), 47-245,
concerning the iteration of a rational function f. Julia gave a precise
description of the set J(f) of those z in C for which the nth iterate fn(z)
stays bounded as n tends to infinity. It received the Grand Prix de
l'Académie des Sciences.
Seminars were organised in Berlin in 1925 to study his work and participants
included Brauer, Hopf and Reidemeister. H Cremer produced an essay on his
work which included the first visualisation of a Julia set.
Although he was famous in the 1920's, his work was essentially forgotten
until B Mandelbrot brought it back to prominence in the 1970's through his
fundamental computer experiments.
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,
Sometime during the late 1970's, following up on the pioneering work of one
of his professors, Gaston Julia, an IBM scientist named Benoit Mandelbrot
made an intriguing leap of reason. Up to then all of the studies of Verhulst
processes and bifurcation theory dealt with values within the real number
set. He proposed that analogs to these ideas existed within the complex
number set. Within a short time, on what is now archaic computer equipment,
Mandelbrot first produced a picture of the famous set that bears his name.
When he first printed the lobular Mandelbrot set, there were very small
"dirt" marks on the printout that were initially dismissed as printer
glitches caused by an unclean roller and ribbon. Imagine his surprise to
find that these "dirt" marks were actually miniature versions of the same
set strewn symmetrically around the large central lobes. His landmark book,
The Fractal Geometry of Nature, was the first book published specifically
with fractals in mind.With the help of Adrien Douady and John Hubbard,
Mandelbrot made a careful and thorough inspection of this set and the
related Julia sets, Cantor sets, Fatou dusts, and Siegel disks. An
incredible wealth of beauty and knowledge spilled forth as a result of their
efforts. The work in our book and this website's gallery is just the latest
child to be born of these forefathers' efforts.
The most notable feature of these fractal objects is one of self-similarity.
That is, if one were to enlarge certain portions of a Julia set, for
example, one would find objects that are identical to the original Julia set
from which they sprang. This repeats over and over ad infinitum. In fact,
because of this endless self-imaging and depth, the Mandelbrot set has been
called the most complicated object ever discovered. For one particular
client, I produced a Julia set fractal at a magnification of one trillion
(10^12) and the magnified picture was essentially no different than the
original. Later I was able to continue this exploration of depth to a
magnification of 169 quintillion (10^18) and still the fractal picture
remained very much unchanged. See our magnification series in this website.
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,
The creator of the Julia set, Gaston Julia is considered one of the
forefathers of modern dynamical systems theory. It was his paper that was
shown to Mandlebrot by his uncle to introduce him to the world of chaos.
Gaston Maurice Julia was born on 3rd February 1893 in Sidi Bel Abbes,
Algeria. In world war one Julia fought for his country and it was during
this that he was wounded. He lost his nose in battle during an attack on a
French front that was designed to celebrate the Kaiser's birthday, due to
this he had to wear a leather strap across his face for the rest of his
life. Despite several painful operations Julia continued his mathematical
studies. I was at the rather young age of 25 that he published the paper
"Memoire sur l'iterations des fonctions" which dealt with the iterations of
rational functions. His paper received the Grand Prix de'l Academie des
Sciences. Seminars were organised in Berlin in 1925 to study his work and it
was during the 1920's that an essay was produced which included the first
visualisation of a Julia set. Julia became very famous in the 1920's, but
his work was soon forgotten and it wasn't until Mandlebrot came back to it
in the 1970's through his computer experiments that it gained it's fame
again.
Both the Mandlebrot set and the Julia sets are based on the recurrence
relation:
Zn+1 = Zn2+C
Thus the Julia set is very closely related to the Mandlebrot set.
There are an infinite number of Julia sets but only one Mandlebrot set. A
particular Julia set is defined by a particular value for C. Then each point
in the Argand plane is taken as a starting point, Z0 of the iteration. That
point either tends to infinity or not. The Julia set is the set of all those
points whose orbits do not tend to infinity. The main difference from the
Mandlebrot set is that the value of C is fixed at an arbitrary value and the
value of Z is the current point you are plotting. When producing a Julia set
if you want it to be connected then the value of C should come from within
the Mandlebrot set. The most interesting patterns occur in the values on the
edge of the Mandlebrot set.
>
>
Gea Jones
--
Best Wishes
Gea*
"The Immortalist" <Reanima...@yahoo.com> wrote in message
news:uboli3n...@corp.supernews.com...
* * * *
sorry, which website,
thankyou for al of the information,the Julia images are the most captivating
to me, at the moment, after four days,
best wishes
Gea
PS how much is a quintillion?
The Immortalist
((**WHAT IS**)) 1,000,000,000,000,000,000 or 10**17
To get an idea of these many thousinds go to:
http://www.kokogiak.com/megapenny/eighteen.asp
http://www.planet-pets.com/plntinsc.htm
>
>
Mike Dubbeld
interesting choice of words. I stumbled on it looking for Quantum
Mechanics stuff. Another good one is on Spherical Harmonics!
How do the orbits of electrons look?
Mike Dubbeld
"Dev Singh" <de...@ihug.com.au> wrote in message
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Dev Singh
answer.
What is fractal art?
pEACe
Dev
"Dev Singh" <de...@ihug.com.au> wrote in message
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ta
"Mike Dubbeld" <mi...@erols.com> wrote in message
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Mike Dubbeld
> I understand everyone's philosophy, but still don't have an understood
> answer.
>
> What is fractal art?
I belive from the mathematics used in the science of Chaos they were
plotting the
results of input data like dripping water from a tap that follows a chaos
function
mathematically and the resulting plot showed up as these weird designs. So
some people just scrapped the Chaos part and used the functions to create
the art. Fractal as in optics.
From Encarta:
Fractal, in mathematics, a geometric shape that is complex and detailed in
structure at any level of magnification. Often fractals are
self-similar-that is, they have the property that each small portion of the
fractal can be viewed as a reduced-scale replica of the whole. One example
of a fractal is the "snowflake" curve constructed by taking an equilateral
triangle and repeatedly erecting smaller equilateral triangles on the middle
third of the progressively smaller sides. Theoretically, the result would be
a figure of finite area but with a perimeter of infinite length, consisting
of an infinite number of vertices. In mathematical terms, such a curve
cannot be differentiated (see Calculus). Many such self-repeating figures
can be constructed, and since they first appeared in the 19th century they
have been considered as merely bizarre.
A turning point in the study of fractals came with the discovery of fractal
geometry by the Polish-born French mathematician Benoit B. Mandelbrot in the
1970s. Mandelbrot adopted a much more abstract definition of dimension than
that used in Euclidean geometry, stating that the dimension of a fractal
must be used as an exponent when measuring its size. The result is that a
fractal cannot be treated as existing strictly in one, two, or any other
whole-number dimensions. Instead, it must be handled mathematically as
though it has some fractional dimension. The "snowflake" curve of fractals
has a dimension of 1.2618.
Fractal geometry is not simply an abstract development. A coastline, if
measured down to its least irregularity, would tend toward infinite length
just as does the "snowflake" curve. Mandelbrot has suggested that mountains,
clouds, aggregates, galaxy clusters, and other natural phenomena are
similarly fractal in nature, and fractal geometry's application in the
sciences has become a rapidly expanding field. In addition, the beauty of
fractals has made them a key element in computer graphics.
Fractals have also been used to compress still and video images on
computers. In 1987, English-born mathematician Dr. Michael F. Barnsley
discovered the Fractal TransformTM which automatically detects fractal codes
in real-world images (digitized photographs). The discovery spawned fractal
image compression, used in a variety of multimedia and other image-based
computer applications.
Contributed By:
Benoit B. Mandelbrot
frac.tal (frak,tl) n. Math., Physics.a geometrical or physical
structure having an irregular or fragmented shape at all scales of
measurement between a greatest and smallest scale such that certain
mathematical or physical properties of the structure, as the perimeter of a
curve or the flow rate in a porous medium, behave as if the dimensions of
the structure (frac,tal dimen,sions) are greater than the spatial
dimensions.