fractal geometry musings

I watched this program on fractals last night. Benoit Mandelbrot "discovered" them in 1975 when he worked for IBM. He's a brilliant mathematician who irritated the European establishment; so he said, "Screw you guys. I'm going to America." As I understand it, he came up with an equation that explains repeating patterns in nature and some 19th century math conundrums. Thus, creating fractals. Mandelbrot came up with "fractals" from the Latin for "broken". Geometry and most of math is looking at smoothness; whereas, fractals investigates roughness. It's a way of looking at parts instead of the whole or in relation to the whole. It's a way of applying math to the visual world. Until Mandelbrot's breakthrough.

The two main components my brain latched onto were 1)self-similarity and 2) inter dimensionality because I have a natural fascination with these concepts. (Our mom should have let us play with Barbies so we would have a fighting chance at normalcy.)

Self-similarity is how the whole and the part reflect each other's pattern. For instance, think about an oak tree and a twig from that oak tree. Stand the twig up to mirror the tree. Its branching system is the same just to a different scale. This is true with clouds and blood vessels. And, the larger the system the more efficient it is. A elephant is 200,000 times the size of a mouse but only requires 10,000 times the amount of calories-- which means it's twenty times more efficient. There's a built in economy of scale.

Does self-similarity exist in history? Is there a natural lapping over of micro and macro history? Does my personal history reflect the larger history of the US or according to the Scriptures? Is Yahweh and Israel's relationship the pattern for my relationship with him? Is my identity crisis endemic of the US as a whole?

Then there is inter-dimensionality in respect to fractals. In Euclidean geometry, there are clearly defined dimensions. A line is one-dimensional. A quadrilateral is two-dimensional. A cube is three-dimensional. But, is it possible to for something to exist between two separate dimensions? Fractal geometry says YES! Something can be 2.376-dimensional. The higher the fraction the rougher the edges and closer to the higher dimension. It's a really interesting version of liminality. Liminal comes from the greek for "threshold"; its the space in between two known spaces.

This concept intrigues me because it seems to bridge a link between broken-ness and growth, which I think is true. We know it's true with building muscle and that is why rest is so integral to athletic training. This concept reminded me of my favorite Puritan quotation, "God breaks every heart differently" and the Buddhist concept of a storm breaking before beauty is born. Craziness and broken-ness can be interpreted as progress in some circumstances.

I like how Mandelbrot bent math to fit reality. He didn't adapt nature to a principle but played with the principle until it fit the reality of nature. This is the same reason I'm so much more comfortable with biblical theology than systematic theology. Biblical theology is 3.8976 dimensional to systematic's 3.2374 dimensional. It's rougher but more realistic.