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'The Paths of Lovers Cross in the Line of Duty.'
WHAT IS A FRACTAL?
Copyright (C) 1990 by Homer Wilson Smith
What is a fractal? A fractal is a picture. A fractal is a picture
demonstrating in color what the output of an equation does for any given
input. The color picture represents the space of all possible inputs,
or some zoomed-in blow up of such a space, and each point in the picture
is colored in a way that represents what the output of the equation does
for that particular input.
Often equations are used to represent the population of living
systems such as a colony of Moths in a Forest. The output of the
equation is the population of Moths at any moment of interest. The
input to the equation is the number of Moths just prior to the moment of
interest and another number representing the total environment the Moths
live in. The Forest in other words.
Thus the equation takes in two numbers, one being the number of
Moths that exist right now, and the other a number representing the
living conditions of the Forest, and the equation puts out just one
number representing the number of Moths at the next moment of time down
Clearly by taking this new number of Moths and plugging it right
back into the equation (along with the second number representing the
Forest), you will get yet another number of Moths even further down the
road. The unit of time being cycled through here can be seconds, days,
weeks, months, years, or anything at all. The purpose of this is to
describe what will happen to the Moth population tomorrow according to
what the population is now and the living conditions they find
The most interesting kind of picture that can be made from this
little game is called a Mandelbrot Map. A Mandelbrot Map is a fractal
and it is a colored picture just like we said, and the colors describe
how long it takes, how many cycles it takes, until the Moth population
dies. Where the Mandelbrot Map is black, the Moths live forever in
happy harmony. Please see the upper left color image of the Mandelbrot
Set on the sheet 'Mandelbrot Sets and Julia Sets'. Black and white
images can be made many different ways. Sometimes they try to emphasize
the contouring on the outside leaving the black inside area white. The
images supplied with this compendium are of this nature. You might
compare the black and white image of the Mandelbrot Set with the color
image of the same.
The picture itself is the space of all possible different FORESTS
the Moths could be in, so it represents the input number that represents
the living conditions the Moths find themselves in during this time.
Forests come in all kinds of shapes and sizes and states of well being.
Thus the Mandelbrot Map shows at a glance which Forests are
conducive to life and which are not. Those for example that are soaked
in Acid Rain, presumably would have a harmful effect on the
survivability of the Moth population. Those Forests therefore that were
deadly to a Moth population would show up as vibrant colors in the
Mandelbrot Map of all possible Forest types. If we were using the
rainbow as our color scheme, then red would be the most deadly with the
Moths dying off in the shortest number of cycles, and violet would be
the least deadly bordering on a good healthy environment. Those Forests
that were truely good for the little beasts would show up as black.
It is a large leap of imagination to go from real Forests to
numbers representing Forests, especially when those numbers are numbers
in the complex number plane. However rather than break your mind with
the details of such things let me assure you that scientists have been
modeling physical things with numbers and equations for a very long
time. In fact they have done quite well in this field for the simpler
phenomena of nature. With this new concept of the Mandelbrot Map the
door has been opened to applying strict scientific scrutiny to things
like weather, and chaotic turbulence which have stumped the best minds
Quickly, things like weather systems always exist in a larger
system from which the smaller system in question takes its life. By
understanding how things survive in their environments, and by having a
tool to describe, compute and predict such relationships, it becomes
possible to study these things at a formal level.
It is pretty obvious that if the whole rest of the atmosphere
around a hurricane were removed, the hurricane would dissipate
forthwith. Thus any hurricane depends for its survival on the calm and
sunny afternoons that day on the other side of the planet.
The fractalness of these pictures comes about because of certain
characteristics that are common to such equations. The primary
characteristic is the tendency of the output of the equation to change
drastically with the slightest change to the input. This is called
INSTABILITY or SENSITIVITY to INITIAL CONDITIONS. Thus the slightest
change in the acidity of a Forest could dramatically alter it from a
living forest to a dead one. And the slightest change in the atmosphere
could precipitate a global ice age.
Fortunately there are large areas in these pictures where things
are relatively stable and NOT sensitive to initial conditions. The
broad areas of common color show this. But where the colors intermingle
in a chaotic frenzy, you know that the slightest change to the input
conditions means a great change in the output result. By making such a
picture we can see easily for the first time where such a system is VERY
unstable and sensitive to initial conditions. Presumably we could check
our present environments to see if they approach these unstable areas
and take heed if caution were indicated.
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