Monday, August 30, 2010

Fractals, Fractals Everywhere

I'm not much of a doodler. But ever since childhood, during my rare bouts of doodling, I invariably draw branching patterns. Irregular, but bound by basic rules, tree branches running off the page. There is a simple appeal to this pattern. Of course it reeks of nature, but also math, and (as it turns out) art and psychology as well.

Branching is a common pattern underlying the much more complex shapes called fractals. A fractal is a shape that looks roughly the same at any magnification. For this to be true, there must be an underlying geometric pattern, replicated thousands of times at different scales. Another way to define a fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is... a reduced-size copy of the whole"(Wikipedia).

Nature abounds with fractals or close approximations, such as clouds, snow flakes, crystals, mountain ranges, coastlines, lightning, and river networks. Even food, like this cross between a broccoli and cauliflower. In the human body, neurons and systems of blood vessels and pulmonary vessels are essentially fractal as well.

People also generate fractals as art. The branching image to the right is called a Lichtenberg figure. This sculpture was made by sending high voltage current through a block of plexiglass. The branching discharges, which closely resemble ferns, are thought to extend down to the molecular level.

We are hard-wired to appreciate fractal patterns. Research by physicist Richard Taylor and others has shown that people clearly prefer patterns with fractal dimensions similar to those found in nature. This helps explain the lasting appeal of artists such as Jackson Pollock. Taylor found that the fractal dimensions of Pollock's earlier drip paintings correspond closely to those found in nature. A 1948 painting entitled Number 14 (pictured here) has a fractal dimension of 1.45, similar to that of many coastlines. Fractal patterns may be behind our appreciation for the subtle variations of classical music, and even in movies!

Humans have recognized and found meaning in fractals for centuries.
"According to Vedic science, the universe is a giant yantra. A yantra is a concentration of fractal geometric designs... The implication of a fractal universe is that the micro represents the macro and the macro represents the micro... Even the human individual can be looked at as a microcosm of the universe." (Axiomatica)
(Image to the right: a Sri Yantra, a Vedic mandala)

Over the last 10 to 20 years, researchers in psychology have been finding examples of fractal patters embedded in our behaviors and mental health. Dr. David Pincus summarizes this research in a post to the Psychology Today blog "The Chaotic Life":
"...interpersonal relationships are organized as fractals and most recently that the self-concept is a fractal, with complexity being associated with health in both the psychological and social domains... Fractal personality structure helps us to grow and connect, as do fractal relationships, and each likely has direct influences on physical health by encouraging integration and flexibility among circulatory, respiratory, and immune systems." (Pincus, 2009)
This leads Dr. Pincus to some gorgeous and even bigger speculations:
"...perhaps human consciousness is both simply and profoundly a portal through which such fractal connectivity flows. Perhaps the linkages that so effect our growth and integration at the biopsychosocial scales extend much deeper into the roots of matter, and much farther into the cosmos than modernist science has ever imagined. Science appears to be nearing a period of neo-vitalism, with scientifically grounded ways of exploring the attractive worldview of our root-civilizations - that everything in life is connected and that all of the universe is alive within these connections." (Pincus, 2009)
So we are formed and connected to our environment by complex patterns, the nature and extent of which we are only beginning to understand. One thing seems clear: fractal connectivity suggests that we, like the wide universe around us, have vast and untapped potential for growth and evolution. This, like the patterns themselves, is beautiful.

References and further reading:
Lichtenberg figures: Bert Hickman, Stoneridge Engineering
"Pollock's Fractals," Jennifer Oullette, Discover Magazine, November 2001.
Pincus, 2009. "Fractal Brains, Fractal Thoughts," Dr. David Pincus, blog post in "The Chaotic Life," Psychology Today, September 4, 2009.
Pincus, 2010. "And the Oscar Goes to...Our Brains?" , Dr. David Pincus, blog post in "The Chaotic Life," Psychology Today, March 3, 2010.

Images courtesy of Wikipedia, Tony Smith and Yale University Art Gallery

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