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.' 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 the road. 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 themselves in. 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'. 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 as 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 until now. 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.