On the drive into work this morning I was musing over my past experiences in the rainforests in the Pacific Northwest (I was once a resident of Seattle). Thinking of each of the different parts of the forest, the moss, the rock, the water, the tree, the flowers, the ants, the animals, etc., I began to contemplate the multiplicity of interactions in this many-faceted jewel of the forest, and all it’s apparent complexity, and yet how finely tuned, how perfectly orchestrated. I imagined then talking with my own internal skeptic (we all have one whirring away inside!) about how this complexity was the same, metaphorically, as the complexity of the Mandelbrot set, in that it has a perfect order and simplicity and a hidden law that keeps it all flowing in balance. The skeptic seemed to reply, “Why does the Mandelbrot set show its order at every scale and every view, while nature in the wild seems chaotic at every scale and every view?”
Here is how they are the same.
The complexity of the Mandelbrot set is not fully manifested in the imagery created from it. It fools one this way, really, because the full complexity of it comes from the fact that the entire thing is created from a very simple process. What lies hidden by that process is a far greater mystery not at all explained by the seductively regular and tangible geometric patterns that emerge from it. What is hidden is, how does that simple process produce such lucid, ordered geometry? What is the actual mechanism? In what space, in what place does that Geometry exist? The beautiful created forms of the Mandelbrot set are nowhere discernible in the process that creates the imagery with which we conceptualize it. And yet, they emerge in all their glory directly from that simple process. Much as the most startlingly beautiful flowers, of intricate form and color, emerge from a tiny, simple seed.
As the seed-to-flower simile implies, the actual “thing” behind the Mandelbrot imagery (the equation seed) contains much more complexity than we can see in that simple seed. Nature is the same, but much more complex. The integrated complexity of a rain forest is mostly hidden. If you could see with the eye the rules and laws that order the rain forest, then in addition to the actual visible aspects of it, you would indeed see a flowing geometry of order and pattern. It could be extra-dimensional (i.e. more than 3 dimensions in space), maybe something the eye couldn’t even grasp, assuming it could be represented in this way. Nevertheless in theory, the result would show a perfect flow of order and structure, probably not made of regular polygons, but showing a perceptible geometry that lay hidden previously.
It’s the great beauty of the Mandelbrot that, while precisely (if not completely) defined as a mathematical entity, it yet affords so much room for imagination and metaphor. It’s the great joy of life that, while diverse and wondrous in the extreme, there is nevertheless a perfect underlying harmony of reliable principles operative at every level.
Below is an image I mined from the Mandelbrot set, using a freely available program (there are several on the internet).
Mandelbrot Raw Image 1
I thought this looked like a snow storm, so I took it into Photoshop® and altered it to emphasize that aspect of the image.
A Mandelbrot Christmas Card
I enjoy taking photos of trees silhouetted against the sky. I find that, apart from being fascinating and beautiful, they spectacularly illustrate the existence of hidden, rigorous principle in nature.
An Oak in the Sky
Of course, the more obvious evidence (apart from clear-minded observation of one’s own life — ref. Heraclitus’s Logos) of hidden principles, are the products of scientific research. One such lovely product is this image of an ice crystal.
An Ice Crystal (click image to learn about it's source)
And this beautiful image of a cloud chamber event (for some reason it’s very difficult to find such images online!).
Cloud Chamber Image
Although it’s rather low quality, you might enjoy this realtime video of cloud chamber events. The paths of energetic subatomic particles ebbing and surging across the screen remind me of water splashing in a puddle.
A fascinating exploration of the Mandelbrot Set producing mystical imagery can be found here. This “Buddhabrot” is discussed more thoroughly, with “deeper” imagery, here. And even more beautiful renderings may be found here. “Grow” the Buddhabrot in real time here. At the same sight, you can “grow” the regular Mandelbrot set in real time a well.
Buddhabrot (home grown)
