Unfortunately, most of the concepts of modern science are counter intuitive and are difficult to picture. Newtonian physics makes much better intuitive sense than does Relativity Theory or Quantum physics; or Chaos Theory and Fractals.

The abstract scientific theory or mathematical formulae that describe a physical system are models of the actual processes. As models of real world systems, they are idealized symbolic metaphors through which we relate to events in the outside world with our particular thinking mechanisms as human beings. They are our interfaces to the physical universe. This allows us to 'create' order in the material universe through which we gain a measure of control over things. For instance, once the models for atomic energy, i.e. Relativity and Quantum Physics, were understood, people started building the devices (atomic bombs and lasers, among other things) that the models said they should be able to build, and they worked! It is important to realize that the model is not reality. A particular mathematical scientific model is one possible interface between us and the physical world that gives the impression to many that it is the reality itself. This is not true, and NO model can EVER be the actual reality.

Until the discovery of Fractals and the recursive mathematics which describes them, many natural phenomena were dismissed as random disturbances or noise and not included in the equations for the systems in question. Benoit Mandelbrot (from whom the Mandelbrot Set gets its name) studied the static in data transmission lines for IBM because their engineers could not figure out what the cause was. He described the nature of the static as being related to a fractal called Cantor Dust. Based on Mandelbrot's analysis, IBM's engineers formulated plans to deal with the static which were entirely different than the approach they were considering when he showed them the real cause. Or, perhaps more accurate, he showed them a cause for which he had a model to explain. Their classical training did not contain the model to understand the static. The point is that Fractal Geometry decribes many real phenomena very well and there are already implications of great importance for all of physics and philosophy.

The idealized objects of Euclidean Geometry are lines, triangles, circles and all the other shapes that every school child studies. The idealized objects of Fractal Geometry are Mandelbrot Sets, Sierpinski Gaskets, Peano Curves, and many other exotically named and visually appealing objects. Euclidean Geometry is the underlying mathematical basis of Classical Physics and Fractal Geometry is the basis for Chaos Theory.

It is the nature of Fractal Geometry's idealized objects that they are generated by recursive formulae. Remember how you made a graph in school by plugging in different values of X, calculating Y, and plotting the resulting points on the X-Y axis? You could plug in ANY value of X and quickly calculate the corresponding value of Y. A fractal object does not work that way. Each point on the graph for a fractal is the result of plugging the present value of the function back into itself. You cannot figure out the value of the function for any point on the graph as you can in linear Euclidean graphing. Even though a fractal's shape may be totally determined by the iterative formula which describes it, the only way to determine its state at any point in the future is to iterate your way there a step at a time. In contrast, a non fractal object such as a sine wave may be exactly specified at any point by plugging the value(s) in at that point and doing the calculation. Fractals are not computable downstream because they do not have a formula that describes the relationship between the variables in a linear way. The math of Newton's physics is The Calculus. It is based on the continuity of the formulas (functions) that make up the mathematical model of the Newtonian universe, a deterministic, totally predictable machine; a clockworks. Its fundamental assumption is the truth of Euclidian geometry. The essence of Newton's universe was expressed by the eighteenth century mathematician, Pierre Laplace. Laplace envisioned an intellect sufficiently powerful to be aware of and process all the universe's data. Such an intellect would know the state of the universe and everything in it at any time, past, present, or future; a totally deterministic and computable universe based on Euclidian geometry and Newton's Calculus.

A fractal model of a physical process, on the other hand, does not allow any intellect, regardless of its power, to predict its future states without allowing the process to actually happen. Small changes in initial conditions have unpredictable and sometimes great effects on the future states of the system. The weather unfolds before us in ways which confound any attempt to predict. Small variances in local conditions today snowball into large effects somewhere else tomorrow. One of the early investigators of fractals, Edward Lorenz, modeled the weather with a very simple fractal formula on an early computer around 1960. Other examples of Chaotic (fractal) processes are a pendulum swinging, a heart beating, a brain deciding, a jet's exhaust, and many more.

There are natural fractals such as mountains, coastlines, clouds, and radio static. Then there are mathematically generated fractals such as the Mandelbrot Set or the Sierpinski Gasket. To see a fractal, simply look at the clouds. One difference between ideal fractals and natural fractals is that the ideal fractals possess their fractal qualities at all scales whereas natural fractals have upper and lower scale limits. But, the fact is that within these limits, Fractal Geometry does provide a basis to study and model phenomena that heretofore were viewed as unknowable.

The picture of fractals which begins to emerge contains some very interesting implications. It is hard to see where the discovery of Fractal Geometry and Chaos will lead. It may be that we will only learn how much we can never know.