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

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)


Hidden Mysteries of Nature

Hidden Mysteries of Nature

Recently, I was with a group of Freemasons having a passionate discussion about the word “magic.” Some of the members of the discussion group felt that Freemasonry is “magic,” while others disregarded the word as superstition and illusion. Still others were exploring different meanings, trying to find within themselves how the word made them feel, what it made them think, and what was their own relationship to magic. As Freemasons, we regularly discuss religion, or rather, being religious. We sometimes specifically compare religious symbols to one another and generally explore spiritual diversity and messages. Often corrupted by men, we lose site of what being religious truly is. We almost never talk about magic, even in free-thinking circles and in public, you only hear “magic” discussed, generally, with humor, disgust, or fear.

Most humans may lose sight of what being “magical” is. Our current world is corrupted by the thoughts of the fearful in so many ways, it’s often hard to tell that we’ve been conditioned by it, by ourselves, by our family, media, and friends. For example, when we use the word magic, it tend to conjure up thoughts of either something horrific, like ritual sacrifice or Voldemort (Yes, I said his name). It might bring to mind witches, burned at the stake, or witches doing strange things in forests at night. Yet, the word magical also tends to bring us to Disney artifacts (Tinkerbell, anyone?), gigantic film special effects, or even dreamy, personal experiences – think, Christmas at Rockefeller Center. The point is, we have not explored the word magic as much as we’ve explored the word religion. However, both may be important to humanity and the Freemason as well. Our ingrained fears stop us from talking about the word and stick it in a cave, hidden from the rest of the world. It’s time to do a little word spelunking.

img_0249The word magic is presumably derived from Old Persian and possibly from the proto-Indo-European language as meh-gh, which means “to help, power, to be able to.” It’s taken many forms over the years, from everything to indicate the workings of scholars, sages, Zoroastrian priests, rituals, spells, and eventually related to something or someone not of your religion. If you didn’t understand it as part of your personal religious upbringing, it was considered magic, especially by both Judaism and Christianity (13/14c C.E) . In Frazer’s The Golden Bough, he illustrates a very thorough journey from folklore, myth, magic, and religion, to the science of modernity. From what I have so far deduced and experienced, the knowledge and wonder of discovering how the natural world works is what magic has been for thousands of years. It’s learning, understanding, exploring, and working in conjunction with the natural world. Forget the word’s baggage and take it back to its origins: the wonder of the natural world that brings us awe and teaches us reverence and respect.

We’ve all learned that humans put their own connotation on the words we use, and shared and agreed-upon usage are how they become “fact.” We should do our best discard dogma; if something imparts an emotional response, it seems to be time to explore it, not shun it or parrot someone else’s belief. Understanding the words we use, like understanding ourselves, gives us authenticity and gives the words power.

Understanding the truth of what magic is seems to be related to how we are in relationship with our natural world. I understand magic to be the physical laws of nature and the universe that I do not currently comprehend thoroughly, and and magic is the process of continually learning how to “be” and be in harmony with our universe. This is not so far from what we perceive herbalists do when they understand plant lore and heal the sick, or weirdly enough, the gymnast who understands the laws of gravity and motion in his body, and can execute the most incredible flips and jumps. Have you ever had someone throw a ball in your direction and you reached up your hand to grab it at the perfect time, even if you might not have been looking at it coming toward you? How did you do that? Magic? Perhaps you understand the laws of motion and the physics of gravity well enough to make the catch. Others may not. To them, it appears as magical.

img_0250The “magical” feelings evoked are the impetus for the process of discovery. We first see something that entices us, intrigues us, gives us a certain spark of interest and imagination. What did we just see? What happened there? Then, we may try to recreate it, seek its origin, find out how to do what it is we saw. “To be able to” means we’re learning magic. From the learning how to do, we wonder and our interest continues. We start dissecting, breaking apart the machine of nature to figure out its meaning, its purpose, and its origin. We might take a path through religion to get there, or we may jump right to science – either is an option. Once we find the how, we seek the why.

There is a quote from a book by Arthur E. Powell, The Magic of Freemasonry, which takes me toward the part Freemasonry plays. It is this:

“Why do men love Masonry? What lure leads them to it? What spell holds them through the long years? What strand is it that tugs at our hearts, taut when so many threads are broken by the rough ways of the world? And what is it in the wild that calls to the little wild things? What sacred secret things do the mountains whisper to the hillman, so silently yet so surely that they can be heard above the din and clatter of the world? What mystery does the sea tell the sailor; the desert to the Arab; the arctic ice to the explorer; the stars to the astronomer? When we have answered these questions mayhap we may divine the magic of Masonry. Who knows what it is, or how or why, unless it be the long cable tow of God, running from heart to heart.”

So, is Freemasonry magical? Not in the way that Disney or Satanists or even fundamentalists of any religion would have the world think. That is fear and ignorance asserting themselves.

img_0253I believe it’s the discovery of the world around us that is magical. It persuades us to keep seeking and searching for the mysteries of nature and science. It speaks to us of understanding our world – not just the laws of men but also the laws of nature and whatever source it is that keeps us all “together.” Some may call it God, The Force, Allah, Jehovah, Yahweh, Diana, Odin, the Tao, Krishna, and a host of other names. Perhaps they are just human mirrors of the same “thing” that ties us together. Perhaps that is the thing I am truly seeking: smashing the mirrors to understand what lies on the other side.

I would say that Freemasonry encourages magic and magical behavior, magical thought, and a magical mind. Ritual of any sort has a purpose and the structure, words, ritual, and trappings of Freemasonry are not as simple as to call them purely “magic.” Freemasonry requires a curious mind to work on its initiates. If one is not curious about Freemasonry and about the world in general, they will see Freemasonry as an institution, made for charity work, a fraternity in which to socialize, and a series of rituals that just encourage the participant to gain degrees. Maybe, for those masons, that is a first step, and maybe if there are more lives than this, we keep Freemasonry going for theirs, and our, future selves.  I see it as the Freemason’s duty to continue to keep our minds open and test our theories, test the world, be inquisitive; thus, perhaps Freemasons are magical scientists.

I do not think that magic is the antithesis of science. I think it is a step in the process of discovery, of which science is another. Science, which is “such knowledge, general truths, or such a system of knowledge concerned with the physical world and its phenomena” is another charged word, especially in the information and technology age. Is Freemasonry scientific? Take your own voyage and let me know what you think. This is your journey, too.