Fractals: A Roadmap Through Infinity

Western culture has often analogized life to a pilgrimage—a challenge to overcome, a race to the end, a dare to succeed in the face of failure. We stand before an infinitely large universe that is utterly out of control. Death is coming at us from all directions and the cards feel as if they are stacked against us. Something is always standing in our way. This adversity seems to give humanity a purpose—an adversary to fight for control of our fate. After all, we’ve got to do something with our lives!

And yet we seldom stop to wonder whether this paradigm might be creating the very obstacles we seek to surmount. In fact, nature is quite consistent in demonstrating how order and direction come not from attrition, but from unimpeded chaos. It is not when life is dictated to but when left to fumble about on its own that it seems to work out with wondrous results for all things, including our not-so-humble selves. This is a clear pattern in nature and man has been struggling to account for it for some time.

In 1950, Lewis Fry Richardson was attempting to draw a correlation between the propensity for two countries to go to war and the length of the border shared between them. He quickly discovered, however, that measuring land borders was anything but an exact science. Countries are not perfect geometric shapes. Territorial borders follow the geological contours of the Earth. They are jagged, like coastlines and mountain ranges. Richardson discovered that the length of any land mass entirely is subjective to one’s scale of measurement.

image_coastline_paradoxAs you look at a map you might calculate a length in inches, but that wouldn’t take into account all the little bays and alcoves that make up a true coastline. Maybe instead of using a map, you undergo the grueling task of walking the coast with a tape measure and recording your findings. This time your measurement would be significantly larger than your initial map calculation, but it still wouldn’t account for the slight variations and indents in the rocks and cliffs. These slight indents, while small, drastically elongate the length of a coastline and this carries into such a micro level as the imprecations of every grain of sand. This is known as the Coastline Paradox and it says the length of a coast is ultimately immeasurable since the possible variations are infinite.

Koch_SnowflakeA coastline is an example of a fractal, an object or pattern that is self-similar—that is a pattern derived by a single shape replicated over and over again (See FIG. 1). These types of patterns are deeply embedded into the makeup of nature: rivers, fault lines, mountain ranges, snow flakes, crystals, lightning, broccoli, blood vessels, DNA, and the veins of tree leaves are all fractal patterns.

Fractals, in this way, represent a type of rhythm to life. They demonstrate how simplicity can form incredibly complex, elaborate, and seemingly chaotic patterns. They serve as a reminder that incredible beauty does not depend upon a sculpting hand, it can result spontaneously from simplicity and time.

image_mandelbrot_set

Animation of self-similarity in a Mandelbrot Set

Going even deeper, fractals give us insight into infinity. A fractal pattern such as the Mandelbrot set has a boundary of infinitesimal growth. The length of its boundary, like that of a coastline, will grow infinitely larger as you increase your scale yet it is contained within a finite space. Infinite growth within a finite space.

So, coastlines, fractals, geometry, infinity… how does it all tie together? What can we learn from the paradox of chaotic systems?

Everything revolves around perspective. The paradox of an infinite measurement residing within a finite space is resolved when we define infinity as a product of observation. The complexity of a shape or pattern is nothing more than a view from one particular perspective. Change the perspective and you change the complexity. Yet the pattern itself remains the same. Complexity and simplicity are not traits of nature, they are traits human of human observation. They are our invention.

In metaphysical terms, man’s attempt to dig into the makeup of molecules, atoms, particles, quarks, strings, and so on might just be a wild goose chase. One can always imagine a smaller number just as one can always imagine a larger number. There is no particular end to the journey, no reachable summit at which one might finally decree “this is it.” Try to touch the furthest boundary of the universe and you’ll never get there. Try to break down matter into its smallest components and you will always discover something smaller. There is no higher purpose, no one thing humanity is to overcome. The terms of our struggles in life are set by ourselves. We are, in fact, always in control and always have been. There is no struggle after all. Life is merely a game we’re playing with ourselves.

Perhaps, then, we can release ourselves from the bondage of “doing something” with our lives. We are, in fact, already doing it.

And we are doing it. We are all doing this great thing called LIFE. So what’s there to stress? Let life happen and it will work out. Create adversity and you will struggle against it. The choice is yours at every moment.

Choose wisely.

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