The Ornate World: Fractals and Complexity #photochallenge

Romanesco cauliflower

Sunflower

Fern

Julia set

Mandelbrot set

This week’s WordPress photo challenge is to celebrate the ornate. Where do we find complexity and complication, and find it beautiful? What make complexity beautiful, at least for me, is when there is a pattern behind the complexity. Something that makes each shape inevitable, even as the whole is intricate and surprising. This is one of the hallmarks of fractals, and perhaps of good writing as well.

Fractals are forms created mathematically when a simple formula is executed over and over, replicating the same pattern or nearly the same pattern at every level, adding more and more detail  in such a way that no matter how far in you zoom it looks the same at every scale. This kind of self-similar pattern is often found in nature, as is shown in the first few images in the gallery. Romanesco cauliflower has a flavor somewhere between cauliflower and broccoli, and is such a wonderful example of a fractal, self-similar form that it’s what you find when you google “fractal vegetable.” The overall cone shape is repeated on all sides in a swirling spiral of smaller cones, each of which sprouts a new spiral of even smaller cones, and so forth down the levels. The leaves of a fern show the same idea. Each green front has the same overall shape, which is repeated in smaller and smaller scales down the leaf. The center of a sunflower is another example, where the spirals of seeds trace smaller and smaller spirals in toward the center. Of course, being actual physical objects there is a limit to how many levels of self-similarity there can be.

The last two images in the gallery were not photos that I took, but are examples of mathematical fractals created by feeding the result of a formula back into the formula to iterate again and again. A Julia set arises when the result of feeding the function back on itself over and over produces an explosion of points that trace an intricate and beautiful pattern in two or more dimensions. Even more famous is the Mandelbrot set, a kind of set of all Julia sets, which is breathtaking (and mind-blowing) in its chaotic complexity. Zooming in deeper and deeper in either of these sets brings an ever-repeating but ever-changing kaleidoscope of astonishing shapes. I recommend spending a few minutes on YouTube.com watching a short journey into a Julia set and a longer journey deep into the Mandelbrot set.

This stuff is fascinating and gorgeous in its bottomless complexity, and it doesn’t need me to hang any extra weight of meaning on it, but I will anyway. The self-similarity of these fractals is a useful metaphor for most creative endeavors. What makes a piece of music great? We humans mostly don’t care for the careful randomness of 12-tone or atonal music. We like something with a key, a pattern we can follow. On the other hand, music that is too simple and repetitive quickly becomes boring. We need to hit the perfect balance of predictability and surprise. The same is true for those of us creating fiction. It is important that the reader be surprised, but that the surprises be fitting, so that looking back they seem inevitable. Here’s another application: Randy Ingermanson explicitly applies the Koch snowflake (another fractal structure) to his snowflake method for writing a novel. You also need a certain level of self-similarity, or consistency between all the levels of a work. If the grand sweep of the story is a battle between good and evil, it helps to have smaller, more personal good-and-evil struggles play out in the chapters and the individual scenes. Not exactly the same way, but as self-similar, fractal echoes of the bigger struggle. Like a Julia set.

Let’s bring the ornate beauty of the fractal into our writing.

In response to The Daily Post’s weekly photo challenge: “Ornate.”