The Infinite Spiral
If you've ever seen a cut-open Nautilus shell or a fern leaf fiddlehead unfurling, you've seen examples of fractalic structures in nature. The Romanescu cauliflower (see thumbnail at right) is as "classic" an example of spiral fractal manifestation as any I've seen. This inflorescence is extremely similar to structures I've seen in images generated from the Mandelbrot Set equation. The spiral can also be seen in certain flowers, such as roses, and succulent plants that grow as rosettes, such as Sempervivum.
Revisiting the Mandelbrot Set image (introduced in my article here), the area of most interest is the perimeter that encloses the Set, not necessarily the Set itself. Surprisingly, the images that can be generated are all dependent upon how you define, or dimensionalize, the calculated points. For example, starting with the rule that the final value for each point must be equal to or less than 2 in order to be a part of the Set, you ask, what about all the rest of the points? Well, they do not all calculate alike, as some will increase towards infinity much faster than others. Even numbers within the Set will vary as to how quickly or slowly they advance towards 2. By assigning color or intensity values to the points based upon their performance, you can visualize different structural characteristics of the Set or of the outlying area.
This, in my view, is similar to some of the manifestations we see in the natural world. Of course, all the natural structures are not governed by the same equations! Mountain ranges, coastlines and clouds are showing the results of different fractal calculations than the fractal fern leaf (image at left) or rose flower, yet they can all be shown to have fractal geometry in their manifestation. Additionally, the natural world is dynamic, which means that it is undergoing constant changes. Conceivably, this effect can also be modeled in a fractal graphic, most likely by using separate re-iterating equations and a much more powerful computer. This dynamic equation set would act to force an automatic and periodic re-iteration of the entire scene or object to reflect the small changes specified by the equation at each reiteration interval. The dynamic equations would be equivalent to various driving forces in planetary changes, such as temperature, pressure, and incoming radiation changes as happen in day vs. night. These, then, could also be set to interact with each other as well, yielding unpredictable and interesting outcomes.
The dynamic aspect is one crucial difference between a static fractal, such as the computer-generated Mandelbrot Set, and a real-life mountain range or thunderhead.
A Rose, by any other name would . . . be a Fractal?
In the image at right, you can see one of several "roses" that I've generated with the Tiera-Zon fractal generating program. Unlike a rose in your garden, these don't appear at the end of the kind of branches you'd expect, although I'm sure a creative artist/fractal master could compose a set of equations that would do something similar for you. What this image shows, though is how these kinds of organic or natural shapes can be generated with mathematical equations governed by dimensional parameters. By making small changes to the equations that produced this rose, different flower shapes can be generated and discovered in the image output.
Something Completely Different . . .
I won't close here without giving you a look at something from "way out there". The picture at left is a strange alien moon generated with the MojoWorld program, using fractalic equations. As you can see, the mountains are very sharp and high, and this moon has an atmosphere. I imagine this is possible because of the extra gravity provided by some neutron star material at the core of this moon. When generating these kinds of images, the imagination goes wild with the possibilities. Watch for more fractal wonders to come!
On to Part 3!
Image Credit: LariAnn Garner and GNU Free Documentation License