(Editor's Note: This article was originally published on December 4, 2008)

From the Abstract to the Chaotic?

"Designing a planet and associated environment is relatively simple once you have the right parametric equation set and activate the set with a dynamically changing iteration formula. The energy to drive the transformations is built right into the format used to generate the host universe so one merely needs to start up the process and it will unfold mysteriously, wondrously, and unexpectedly in beauty never before brought into manifestation . . ." (see thumbnail at right)

Well, oops, I think I got a bit ahead of myself there! I should have started with a simple insect or bird, then . . but wait, first I need to show you how this all starts. The beginnings of this almost magical branch of mathematics, called fractal geometry, happened with the discovery of what we now call the Mandelbrot Set. The Set was named after Benoit Mandelbrot, who produced the first computer-generated image of it in 1979. The equation and iteration thereof is quite simple: z=z*z+c, where c is a complex number and z starts at 0 (zero). This equation is calculated and the result returned to the equation and recalculated (reiterated) repeatedly to determine the mathematical parameters of the Set and also to graph the resulting image. All results that are equal to or less than 2 after a very large number of iterations are considered to be part of the Set (represented by the aqua colored area in the image below).

Mandelbrot Set imageWhen computed and the results are displayed on your computer monitor, the Set yields the familar "spiny pear" image that is devilishly complex in spite of apparent simplicity (see image at left). With today's computers, anyone can iterate the Mandelbrot Set into the familiar fractal image. With the right program, you can use it to sample concepts such as infinity in full color graphic detail. The study of this Set, along with other fractals, has given rise to what is known as "chaos theory".

So what does all this have to do with nature and your garden? Well, an exploration of the graphics that result from the computation of the Mandelbrot Set yields an infinity of wonderfully beautiful images, some of which are startling in their similarity to natural patterns and objects such as insects and plants. In the insect fractal image below, right, you can see the head, thorax and abdomen quite clearly, and the resemblance to a wasp, bee or ant body is uncanny.

Fractal insectConsider that natural phenomena are not often like geometric shapes with 3 dimensions. If an object has more than two dimensions but less than three, it is said to have fractional dimension, or is fractalic. The Mandelbrot Set is said to have more than one dimension but less than two, making it a fractal. The color visualization of the Set depends upon how you choose to illustrate each point that is iterated, and to what degree a point is iterated before a color is assigned to it. In this way, an amazing variety of images can result from the same equation! And then, of course, you can go on to explore the fractal environment by zooming in on a particular point of detail.

My interest in this subject took off at warp speed when I learned of a computer program that used fractal computations to generate entire planetary landscapes (you read right!), including sky with moons and stars, clouds, oceans, mountains and valleys. Turns out the results can be explored just as though you were flying or floating over the planet, slowly scanning as you go. Oh, and a lot of other manifestations not seen on Earth are possible with this program as well. Knowing that human knowledge in this field is basically just beginning, the concept that planetary simulation is possible at this early stage is, frankly, mind-boggling. But what it also means is that at the heart of what we experience as our natural world could be a complex, dynamically interacting set of mathematical equations kept in constant motion/iteration via a continuous energy flow from the quantum realm to physical reality and back again.

Fractal birdIn future articles I will delve deeper into what can result from this knowledge and show more of how fractal geometry manifests in the natural world, including our gardens. Along the way I'll provide you with a peek at what might be out there in the vastness of space as well! I promise that you will experience a memorable and unique voyage!

Meanwhile, if you want to try your hand at iterating your own fractal images, you can obtain a free fractal generation program by clicking Tiera-Zon. You, too, might be able to iterate a fractal Bird of Paradise like the one at left.

On to Part 2!

Image Credit: LariAnn Garner