Fractal Geometry

Last winter in my travels to Tucson and the Grand Canyon, I had a conversation with a guy named David about fractal geometry.  The subject came up when we were discussing agate formation.  A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole -- a property called self-similarity. The use of fractals enables the prediction of patterns. Roots of the idea of fractals go back to the 17th century.

There are several examples of fractals, which are defined as portraying exact self-similarity, quasi self-similarity, or statistical self-similarity. While fractals are a mathematical construct, they are found in nature, too. Natural objects that are approximated by fractals to a degree include clouds, mountain ranges, lightning bolts, river water sheds, snow flakes, various vegetables (cauliflower and broccoli), animal coloration patterns, and in some cases -- agates.

Many cacti have fractal patterns.

So do tree limbs...

And ice crystals.....

And leaves....

And flowers.....

And river drainage patterns.....

Of course, a kaleidoscope is sort of a way to cheat the fractal creation process -- using mirrors.

One of the most intricate and beautiful images in all of mathematics is the Mandelbrot set, discovered by Benoit Mandelbrot in 1980. Most people within the mathematics community, and many people outside of the discipline, have seen this image and have marveled at its geometric intricacy. 

The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape. The Mandelbrot set is generated by iteration. Iteration means to repeat a process over and over again. In mathematics this process is most often the application of a mathematical function. For the Mandelbrot set, the function involved is the simplest nonlinear function imaginable, namely x2 + c, where c is a constant. To iterate x2 + c, we begin with a seed for the iteration. This is a (real or complex) number.

Many agates also demonstrate fractal patterns.