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Dec 3, 2022, 12:06:03 PM12/3/22

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ART MATRIX PO 880 Ithaca, NY 14851-0880 USA

(607) 277-0959, Fax (607) 277-8913

'The Paths of Lovers Cross in the Line of Duty.'

INTRODUCTION

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

understood something.

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

mathematical overtones.

PAGE 2

'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

years.

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.

================ http://www.clearing.org ====================

Sat Dec 3 12:06:01 EST 2022

FTP://ftp.lightlink.com/pub/archive/fractals/xfl4.memo

WWW://www.clearing.org

BLOG://adoretheproof.blogspot.com

Send mail to arc...@lightlink.com saying help

================== http://www.lightlink.com/theproof ===================

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.

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