Fractals: Happiness Within the Infinite

Fractals: Happiness Within the Infinite

For many people life represents a pilgrimage: a rat race to financial success, a scramble to stake a claim on our own little place in the sun, a yearning to engrave a happy ending on our tombstones. The great “pursuit of happiness.” This yearning to “make something of ourselves” requires us to form an order out of the chaos that surrounds us—chaos that seems stupid, random, and infinite.

But what if there already was an order underneath it all? What if all the matter around us, our very bodies and minds, and even the very fabric of space and time was all just a self-replicating pattern on repeat? What if all that we understood was simply a matter of scale? Would we still yearn to run the race?

In 1950, Lewis Fry Richardson, a mathematician, was attempting to draw a correlation between the propensity of two countries to go to war and the length of their shared border. Richardson noticed a strange disparity when trying to ascertain exact measurements of land borders—the reported measurements varied greatly depending on which source he cited. It turns out that measuring the length of any stretch of land is not an exact science but rather a relative one. For example, if you look at a coastline on a map you might calculate a length based on an inches-to-miles ratio. This would provide a rudimentary measure of a country’s coastline. If you then underwent the grueling task of walking the coast with a tape measure, taking into account all the little bays and alcoves that make up a true coastline, your measurement would then be significantly larger than your initial map calculation. A more precise measure would take into account slight variations and indents in the rocks and cliffs and imperfections in each and every pebble and grain of sand. Taken across many miles these minor alterations can have drastic effects. This example of scale relativity is called the Coastline Paradox.

fractal triangle image

FIG. 1: A Koch Snowflake – a triangle intersected with itself ad infinitum is an example of a fractal

A coastline is an example of a fractal, an object or pattern that is self-similar—that is, the shape of the sum is replicated in its parts (See FIG. 1). These types of patterns are deeply embedded into the makeup of nature. Examples include river networks, fault lines, mountain ranges, snow flakes, crystals, lightning, broccoli, blood vessels, DNA, and the branches and leaves of a tree. Fractals demonstrate how simple geometric shapes can form elaborate, complex, and seemingly random designs in nature.

This idea of “self-similarity” is so thoroughly engrained into the DNA of our world that fractal geometry has recently been proposed as the underlying fabric of the cosmos. Nondifferential space—space that is not universally smooth and uniform—implies the very same relativity to scale as a coastline. If this were true then mass itself, the very matter that surrounds us and makes us up, arises as a direct function of the form. In other words, we are the fruits of the fractal rhythm of life.

This means that when we attempt to restrain and to organize a seemingly random world we are actually constructing a prison. The limitations become our limitations.

Fractals represent a type of rhythm of life. They serve as a reminder that incredibly complex patterns do not depend upon a supreme intelligence or order, rather they can result spontaneously from simplicity.

Mandelbrot Set gif

This animation illustrates the self-similarity of the Mandelbrot Set

Going even deeper, fractals also provide insight into the infinite. A fractal pattern such as the Mandelbrot set has a boundary of infinitesimal growth. The length of its boundary, like that of a coastline, continues to expand as you increase your scale of focus, and yet, paradoxically, it is contained within a finite space—infinity within the finite. Our conceptualization of space follows the same pattern: an infinitely expanding universe within a finite sphere of observable space.

This paradoxical relationship illustrates a stunning truth about man’s attempt to organize the world around him into compartments of information: the pursuit of order begets disorder. Infinity and the finite are entwined as concepts that arise mutually out of the mind’s attempt to quantify the natural world.

This also means that infinity is not a specter to be feared, nor is it “stupid.” One can always imagine a smaller number just as one can always imagine a larger number. Infinity, by analogy, signifies that there is no “goal” that we’re expected to reach. There is no particular end to the journey, there can never be a 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 new parts. The only universal truth that we have uncovered is that there are no universal truths.

So by observing and thus accepting that LIFE IS INFINITY and that THERE IS NO GOAL TO REACH, we may finally release ourselves from the bondage of “doing something” with our lives. We might realize that we are, in fact, already doing it.

And we are doing it. We are all doing this great thing called LIFE. To “pursue” life as if it were something to hold onto is meaningless. Whatever life we choose to live is simply a game we elect to play.

By seeing the world this way, laughing comes as naturally as the wind. We’re able to be in the moment, to smile more, to let go, to love, and ultimately to be internally and spiritually content with things precisely as they are—the good, the bad, the ugly, and the beautiful.

And that to me is all the happiness we could ever pursue.

Posted by Zimmerman


For more examples of fractals and chaos theory see our earlier post.

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