"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."

-- Benoit Mandelbrot, 1983

There are some significant changes in the eBook that are not in this draft. You may purchase the eBook here

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Nature is not naturally smooth edged. Smooth surfaces are an exception. Euclidean geometry, the geometry that we learned in high school, describes ideal shapes -- the sphere, the circle, the cube, the square. Euclidean shapes are man-made -- not nature made.

Fractal geometry is the geometry of irregular shapes that we find in nature. Fractal geometry gives us the power to describe natural shapes that are inexpressible using Euclidean geometry. One can easily observe fractal patterns in trees, rivers, mountains, and the structure of mammalian lungs. The term "fractal"was coined by Benoit Mandelbrot in 1975 and was derived from the Latin word "fractus" meaning "broken" or "fractured."

The key characteristics of fractals is that they are irregular and self similar. Self-similarity means that as the magnification of an object changes, the shape (the geometry) of the fractal does not change. The shape of a magnified portion of the object looks approximately the same as does the original portion. A fractal pattern looks the same close up as it does far away. When we look very closely at patterns that are created with Euclidean geometry, the shapes look more and more like simple straight lines, but that when you look at a fractal with greater magnification you see more and more detail.

Visualize a leafless tree in the dead of winter where all we see is a complex series of branches that vary in size. The farther away from the tru**n**k, the smaller the branches. Ultimately, the branches become twigs.

Now pretend you are holding a magnifying glass. Look at a magnified view of the branches. Increase the magnification and look at a smaller section of the tree. What do you see ?

All three views look similar-- approximately the same. All three views portray branching structures. Within limits, all three views look like any other tree. Parts of the tree have the same shape as the whole tree no matter what the magnification. This repetition of the same pattern (approximate or exact) at different magnifications is self similarity.

Self similarity is ubiquitous in nature. We see self similarity in a fern leaf, a snowflake, our lung structure, the path of a forest fire, villi in our intestines, the Internet, temporal processes such as music, social behaviors, and many other patterns in nature. Many of nature's patterns depend on the principle of self similarity to function properly. A good example is the mammalian lung.

Alveoli are the tiny pockets in your lungs that store air for brief periods to allow time for oxygen to absorbed into the blood-stream. In order to permit the absorption of sufficient oxygen into the blood stream, the alveoli must have a large total surface area. In fact, human lungs contain 300 million alveoli with a surface area of 160 square meters -- the size of a singles tennis court. The volume of a human lung contained in the chest cavity is only about 6 liters. So, this huge surface area is contained within this small volume. This can happen only because the geometry of the lung structure is fractal or self similar.

The efficiency of the lungs in diffusing oxygen from the inhaled air is directly proportional to the available surface. So, for a given volume of lung it is highly advantageous to maximize the complexity of the surface area. The internal surfaces of lung tissues are highly complex. Fractal dimension is a measure of the complexity of a structure. Lungs have extremely high fractal dimension (about 2.9 our of a possible 3.0).

Fractal geometry is very suitable for modelling all sorts of natural phenomena because it provides a very good representation of aspects of real life. Fractal theory offers methods for describing the inherent irregularity of natural objects. Since many of nature's patterns are fractal, we will find it useful to build models with fractal features if we wish to gain a deeper understanding.

A fractal pattern often has the following features:

- It has a fine structure at all magnifications.
- It is too irregular to be easily described in traditional Euclidean geometric language.
- It is self-similar (at least approximately or stochastically).
- It can be easily constructed with a simple, recursive algorithm.

The “Useful References” link provided below offers a great deal of detail on fractals.