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'The Paths of Lovers Cross in the Line of Duty.'
The enclosed material is a short compendium of material on fractals
for the layman. It ranges from the truely simple to the slightly more
complex. I have made an effort not to lessen the subject matter by
writing watered down analogies that would not make any sense to a true
scientist and would give the beginner a false feeling that they had
Thus each of the following pieces invites thought and understanding
by presenting fractals as they really are. The first few pieces do not
assume that the reader has any mathematical back ground at all, and in
fact approach the subject in such a way that a mathematical background
is not needed.
However fractals are eventually mathematical, thus to understand
them one has to foray into the subject of mathematics at least a little
bit. The remaining pieces go into the mathematics of fractals in
greater detail. You should follow this material only as long as you can
continue to understand it and take from it what you need.
Everyone starts off learning about mathematics at the beginning, I
was no exception, therefore you should not feel uncomfortable if you are
at the beginning of this subject. Many people give up mathematics
because it became boring to them, it became useless. Our intention with
fractals is to get people interested in mathematics again even if they
are 90 years old. With fractals it happens every day. People who are
retired and living off their well earned pension are suddenly calling me
up telling me about the crazy new pictures they have just made on their
PC. Just like kids again. And yes their mathematics is rusty.
The Theory behind 'The Cell and the Womb'.
This is by far the easiest to read and requires no mathematics at
all. It was written long after 'The Cell and the Womb' in order to give
a basic understanding of the principles. The idea that mathematics
could be applied to the growth of the human body has met with some
resistance and cat calls. My intention with this piece was to point out
a use of the principles involved that was so simple and so obvious that
no one could refute it. It was hoped that this might then open the door
to the more speculative ideas contained in 'The Cell and the Womb'.
'What is a fractal?'
This provides a quick overview of the nature of fractals and what
they might be used for in the biological sciences. It goes over again
in a simpler light the material covered in the first piece and in fact
could be used as a sort of press release about what fractals are and
what they are useful for. This is placed second only because of its
'Do Fractals Explain Everything?'
Besides trying to answer this question directly, this piece is
directed to a thorough definition of the terms evaluation and iteration,
input and output, periodic and chaotic, and stable and unstable.
Although little to no mathematics is required here, this is meant for
serious study and contemplation.
'The Cell and the Womb'
This is an effort to apply the simplest fractal concepts to a
living system in an environment. It is a semi detailed description of
how the fertilized cell growing in the womb might be described by a set
of equations that give rise to very fractal behavior.
Mandelbrot Sets and Julia Sets.
This was written for teachers and serious students of the subject
who wanted their first introduction to the mathematical foundation of
fractals. It defines in detail the mathematical terms and concepts
surrounding the computation of fractals such as iteration, basins of
attraction, periodic cycles and what a Julia Set is.
There is a definite drum being beaten here for fractals. Namely
that they are important, easy to understand and beautiful. They are
important in that many phenomena of nature not yet susceptible to formal
scientific analysis will soon be. They are easy to understand in that
your child in the 10th grade should be able to program them on his or
her PC. Their beauty is obvious and should need no further explanation.
Even if you yourself feel that fractals are beyond you, you should
consider that fractals are for children for they are the scientific
pioneers of our future. Fractal mathematics will be common curriculum
in 10th grade high schools around the country in a few years if only as
a motivating factor to keep kids interested in math and computers.
Every person that is touched for the first time by a fractal recognizes
in their heart that they have found something new and wonderful,
something that THEY want to be a part of and they will be endlessly
thankful to the person who first turned them onto the subject (you!).
The products that we offer at ART MATRIX are the result of years of
research and millions of dollars worth of computer time and resources.
These funds have been given gladly by the various institutions involved
in the name of research. From this research have come a few token
products that we bring you in an effort to help the rest of the world
see what we have been so fortunate to be buried in for the past few
We are not paid by Cornell or anyone else to do this research. Art
Matrix is our only source of income, thus by helping us get this new
found wonder out to the population at large you are also taking a direct
hand in keeping the research going. At the same time you are inspiring
the little people of this planet to hold onto their hope and give them a
new and wonderful way to contribute to the problems that need to be
solved in the years ahead.
Sat Dec 3 12:06:01 EST 2022
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Learning implies Learning with Certainty or Learning without Certainty.
Learning across a Distance implies Learning by Being an Effect.
Learning by Being an Effect implies Learning without Certainty.
Therefore, Learning with Certainty implies Learning, but
not by Being an Effect, and not across a Distance.