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July 2009 Issue 29
The Dollmaker has always loved the spiral
as a motif in her own Crazy Quilting.
Spiral Obsession can be a lifelong preoccupation...
so...
It follows that...
When the Dollmaker first saw
Fractal Imagery
her spiral antennae extended all the way out... what was this?
wow!
Egad! Double WOW!
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The Dollmaker Explores
Fractals
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Yes we are now about to tackle the question...
What the heck is a Fractal?
ok, this is the whole idea behind fractals....
observe what is happening here:
We take a simple straight line and add a triangular bump to it
If we keep doing this over and over,
adding a triangle to every straight line,
we get a most intricate design.
Check it out. We get a star shape...
And then, a bunch of smaller and smaller stars
which all resemble the larger star.
This may end up as ART, but it starts out as MATH!
Wikipedea states:
Geometry is a branch of mathematics concerned with questions of
size, shape, relative position of figures, and properties of space.
Geometry is one of the oldest sciences.
It is a body of practical knowledge
concerning lengths, areas, and volumes.
(image from dreamstime.com)
so...
Fractals
are shapes made by lines...
just like you find in your basic geometry
where every line is represented by a number
when you change the numbers you change the shapes made by the lines
When you start doing more and more crazy things with the numbers,
the lines go crazy too, and when you feed a whole blackboard full of numbers
into an equation which produces countless repetitions
you can have a very simple design which goes berzerk, repeating itself to infinity
or at least beyond the reach of ordinary human perception.
When the Dollmaker was a kid, geometry didn't look like this!
Yet, this snakey yum-yum thing is just a mathematical formula
repeating itself with wild abandon.
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An Example using the +
(this ceaseless repetition is called "iteration")
Here we have a straight line, crossed once to make a "plus".
Keep crossing every straight line... (that is the "Production Rule" in this case)
and you get a much more decorative plus made up of bunches of little plusses...
so this would be called
the "PLUS FRACTAL"
no matter how we ZOOM, inwards or outwards,
the details and shapes, whether large or small,
resemble the shape of the original image.
This is what is meant by self-similarity.
A Fractal can be anything that contains
self-similar images within itself.
The smaller branches look like the larger tree.
Coastlines, clouds, rivers, and leaves
all contain self-similar patterns within themselves.
The human circulatory and bronchial systems are fractal in nature.
Blood vessels and bronchial passages
keep dividing into smaller and smaller self-similar branches.
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty."
-- Bertrand Russell, from The Study of Mathematics: Philosophical Essays
Now of course, with a very rigidly self-similar image,
there's not really much point in zooming in...
other than to hypnotize yourself....
as everything is the same;
small detail looks like large detail,
continuing on into infinity
as the Fractal keeps repeating itself.
like they say... As Above, so Below...
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+ -> t
Not Quite So Similar
Fortunately for fractal enthusiasts, that isn't all there is to it...
Perhaps it is just in our natures to love deviance,
to slide away from rigid systems which repeat themselves endlessly.....
but it is a great joy to just go in and
change the parameters....
Just a teeny deviation in the formula
+
in the location, length or angle of our friendly little PLUS...
( like, if instead of a + the original geometric shape changed to a t )
KA-POWWY!
Any tiny change anywhere in the formula- multiplied thousands of times-
can and will cause major mathematical
(and therefore visual) tsunamis.....
Slight Changes in Production Rules
let loose Giant Twists of COLOR!
The shapes and forms tease us with a little bit of familiarity,
they tantalize us with new and unexpected twists.
We can explore the passageways of a single fractal design for hours.
We could end up seeing something that nobody has ever seen before.
This is what makes fractal exploration so intriguing.
Of course we have to shout a mightly HAIL!
for the computers who do all the iterating (pattern repetitions)
and a big cheer for Programs which translate the endless mathematical configuring
into such glorious fantastical imagery, that we might happily zoom around,
changing the parameters of images and losing our way in the dense twisted forests
of color and form.
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Fractals
Demonstrate Chaos Theory
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Remember in the film "Jurassic Park"
when Jeff Goldblum (playing Ian Malcolm, a Mathematician)
starts talking about
Chaos Theory...
"Chaos" implying the existence of unpredictable or random behavior
with a Sensitivity to the idea that small differences
introduced into initial conditions
have dramatically significant effects on the system in the long run.
In Jurassic Park, with reference to Chaos Theory (and breeding dinosaurs),
the Mathematician warns that complex systems,
even nice neat linear systems which ought to be predictable,
will often exhibit wildly unpredictable behavior.
And it's true that Scientists in many fields,
stumped by mysterious deviations from expected results and feeling uncomfortably human,
have had to grudgingly admit that it would be a good idea
to find a way to start factoring in Chaos.
The Dollmaker only brings it up because
Chaos Theory definitely describes Fractals.
(It also describes weather, art, and raising children...)
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"Prediction is difficult, especially the future."
-Niels Bohr
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...who knows....
where the definition of a Fractal may lead...
how the Thinkers of our Time will apply Fractal Theory globally
and also zoom in to a personal level...
After all, Fractal images do tend to whisper to us
to ponder the ceaseless self-similar repetitions in our own worlds...
a Fractal State of Mind
http://www.rochester.edu/news/show.php?id=967
til next time.....
Most of the fractal images used here are from Enjoey Designs
(delightful and free... check them out)
http://www.sgeier.net/fractals/indexe.php
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Want to know more....?
For a wonderful simple tutorial on
http://arcytech.org/java/fractals/
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Animated Fractals... awesome site:
http://abm-enterprises.net/animations/celticfractal.gif
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curious about...
Chaos Theory
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also, google:
Free
Fractal-Making Programs
for a free passage into other worlds....
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If you feel powerless, learn something. Knowledge is
power.
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