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. '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!).