# The Nature of Fractals – Part I: The World Around Us

Man continually seeks deeper understanding of the world around him. From the deepest reaches of space, to the depths of our oceans, to the smallest particle, Humanity seeks to gain ever more profound insight into this world we all experience together. However, what if the clues to gaining some insight into our existence lie right before our eyes?

As I journey through my life, it continues to amaze me how complex and yet simple our existence really is. Humans have a remarkable ability to discern patterns. Repeating patterns are a phenomenon seen throughout nature, such as the fractal. Could our ability to discern those patterns and their existence be an indication of deeper truths for this reality?

Example of a Fractal

A fractal is defined as a “natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale.” No matter what magnification the observer uses, the same pattern is evident, just at a larger or smaller scale depending on the magnification used. The Mandelbrot Set is one such fractal and is illustrated to the left. Mandelbrot described the fractal as “…a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole…” (New World Encyclopedia, n.d.)

One example of a fractal is seen in a hyperbolic fractal tessellation. A tessellation is a closed, countable set of tiles arranged so that they do not overlap with a repeating pattern. They essentially form a two-dimensional shape within the Euclidian Plane. A hyperbolic fractal tessellation combines the traits of a tessellation and a fractal in a manner similar to the illustration at the right.

Fractals can be seen in our daily lives. The manner in which this article was assembled has fractal patterns – start at the highest level, build a framework (outline), select one of the subsections and write to that, inserting a sub-framework around which the words are assembled, repeat until the depth of detail desired is reached. The antennas used in cell phones are fractal in design as well. This design was selected to solve an early problem with cell phones – the large number of different frequencies each phone had to receive. The length of an antenna must be a whole fraction of the wavelength of the signal for the signal to be received. Dr. Nathan Cohen discovered in 1988 that an antenna designed as a fractal could receive multiple signals because a fractal antenna realized antennas of multiple different lengths, either matching or a whole fraction of the wavelengths of the received signals.

Fractal Pattern in Nature

Fractals are ubiquitous throughout nature as well. From a certain perspective, the fractal antenna above was successful because it replicated the concept seen in nature. Some of the more commonly seen fractals include trees and ferns. For trees, think about how the trunk is the base for multiple large branches, which form the foundation for smaller branches, so and so forth to the leaves at the end of the smallest branches. Certain sea shells also exhibit a fractal pattern. You may wonder why natural systems behave in this manner. As quoted from Dr. David Pincus:

Essentially, fractal systems have many opportunities for growth, change and re-organization. Yet they also are very robust. They maintain their coherence; they hold together well, even under tough circumstances. They are balanced in this respect, between order and chaos. They are simple, yet also very complex. This balance is often referred to as “criticality.”

And the term “self-organized” is often added because systems tend to become fractal on their own, simply by putting a lot of system components together and allowing them to exchange information. Think of a party. All you need to do is come up with enough people at the same place and time and they will start to form complex patterns of connection with one another.”  (Z.McGee, n.d.) I like to think that fractals are so complex that they are simple.

Fractal Pattern in the Brain

It turns out that the brain is fractal, both in the way it is organized physically and functionally. On the physical level, at the smallest scales are the pyramidal neuron, which is the most common neuronal structure in the brain. These form into cortical columns, consisting of numerous pyramidal neurons. Finally, the Columnar Complex consists of a number of cortical columns. All of these structures exhibit branching both into and out of the arrangement.  (The Fractal Brain Theory, n.d.)

Indeed, illustrations of the neuron and its surroundings depict a fractal type of construction. Even the way the brain works is fractal in nature. Psychologists discovered in recent years that behavior patterns and social behavior adhere to those principles. So Humanity exhibits a fractal nature from the smallest to the most gross scale, which may explain our connectivity to Nature itself. One author describes this connectedness as “broadband connectivity” and explains how that may be related to our consciousness.  (Ph.D., 2009)