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)

 

Enlightenment

Enlightenment

On a recent post, some people were critical of the term “enlightenment” and its application toward the human race. Now, this term was being used in conjunction with the “Age of Enlightenment,” something altogether different from our modern American use of the term. Students of History understand, know, or at least have heard of the Western European “Age of Enlightenment,” so called because of the explosion of knowledge, science, and access to those tools that brought forward many of our modern inventions and way of thinking.

According to Websters, enlightenment is explained thus:

inˈlītnmənt/enˈlītnmənt/

noun

1. the action of enlightening or the state of being enlightened. “Robbie looked to me for enlightenment”; synonyms: insight, understanding, awareness, education, learning, knowledge.

2. a European intellectual movement of the late 17th and 18th centuries emphasizing reason and individualism rather than tradition. It was heavily influenced by 17th-century philosophers such as Descartes, Locke, and Newton, and its prominent exponents include Kant, Goethe, Voltaire, Rousseau, and Adam Smith.

Someone very wise once told me that Freemasons fall into two general “modes,” if you will, with regards to their approach to Freemasonry: intellectual and devotional. This is a wide spectrum; and, we all have elements of both within our personalities while some people will fall more towards one side than the other. It’s difficult for someone who leans more toward an intellectual bent to understand a devotional way of being, and vise versa. An intellectually-bent person might look at Freemasonry as a tool to intellectual discovery, a place for concrete fraternal relationships, and a more inward view of life. Analysis. A devotional-bent person may want to explore the esoteric and occult side of Freemasonry, feel more reverential toward their deity through their Masonic work, and perhaps be more inclined toward personal, service-oriented relationships. Feeling. Each person has to some degree these modes of operation. Yet, as a Freemason, they are perhaps brought visible.

braincogs

Why does this matter when discussing enlightenment? It seems that each of these people view enlightenment in very different ways. Is knowledge derived from a pure scientific approach? Analysis? Is knowledge derived from a pure empirical approach? Feeling? The interesting thing is the judgement that goes along with how each other views the opposite approach. There’s an intellectual snide comment here or there when the devotional Freemason approaches enlightenment with an emotional response. There’s harsh condemnation of science when the intellectual produces a theory based on their analytical approach and disregards the “human” element.  What is interesting is how each immediately judges the other’s approach to enlightenment, as if there is only one way. Even the non-religious discussion can evoke a dogmatic high-horse.

Is it so difficult to imagine that you can have both approaches, and both are valid?There’s also this “great quest” toward enlightenment, as if it’s something that can be achieved through one method, one voice, or one frame of mind. Some think that we can achieve enlightenment in a lifetime, like a Buddha or Christ. Some think that scientists could never achieve enlightenment, no matter how intelligent, because they have no “devotion.” Some think that only scientists could achieve enlightenment because they have “purer” processes. Some think that humans can achieve enlightenment one being at a time, and still others insist that it must be an all or nothing endeavor. I think enlightenment is far greater than the individual, and enlightenment isn’t something sparkly, pretty, easy, or fun. There’s no flash of sudden godhood nor individual ascension into the realms of all-knowing, having-no-use-of-bodies beings that will provide us some unknown fascinating wisdom. I don’t think that we get out of this corporeal manifestation anytime soon.

The idea of enlightenment, as in The Age of Enlightenment of the 17th and 18th C., is really about letting go of the shackles of tradition. It’s about embracing change and using knowledge to propel us, individuals and humanity, forward. Enlightenment isn’t everyone achieving godhood. It’s about all of us realizing that we are already in control, and have the tools inside of us to solve those problems. Deepak Chopra said, “I was an atheist until I realized that god was inside of me.” When asked about his religious views, Einstein replied:

“Your question is the most difficult in the world. It is not a question I can answer simply with yes or no. I am not an Atheist. I do not know if I can define myself as a Pantheist. The problem involved is too vast for our limited minds. May I not reply with a parable? The human mind, no matter how highly trained, cannot grasp the universe. We are in the position of a little child, entering a huge library whose walls are covered to the ceiling with books in many different tongues. The child knows that someone must have written those books. It does not know who or how. It does not understand the languages in which they are written. The child notes a definite plan in the arrangement of the books, a mysterious order, which it does not comprehend, but only dimly suspects. That, it seems to me, is the attitude of the human mind, even the greatest and most cultured, toward God. We see a universe marvelously arranged, obeying certain laws, but we understand the laws only dimly. Our limited minds cannot grasp the mysterious force that sways the constellations. I am fascinated by Spinoza’s Pantheism. I admire even more his contributions to modern thought. Spinoza is the greatest of modern philosophers, because he is the first philosopher who deals with the soul and the body as one, not as two separate things.”

Is finding “God” enlightenment? Born just prior to the Age of Enlightenment, Baruch aristotleSpinoza laid the groundwork for radical thought (in 17th C Europe) regarding the existence and definition of God. Much like Mozart as the pinnacle of Baroque music, Spinoza was the pinnacle of Latin academic writings in rationalist philosophy. According to Spinoza, God is Nature, and Nature is God.

The fascinating thing about Spinoza is that he worked, day to day, as a lens grinder. His passion was philosophy, ethics, religion, and the question of the divine. He did not content himself with or define himself as his day-to-day paying job. He did not accept honors or rewards based on his writings and thought. He died young, at the age of 44, but seems to have accomplished a great deal for the human race in that short of a time. One can read Ethics and Spinoza’s other works at Project Gutenberg.

One would like to think this is true enlightened human being. Spinoza was an everyday man who engaged in deep thought, the search for Truth, and produced that Truth in service to Humanity. He propelled the next generation, and several after, to continue to explore and discover knowledge. He was an individual who kept the greater species in mind, literally. He was not concerned with some idea of heavenly admittance, some monetary gain, or some brilliance that only he could attain. This is someone who is on the path to enlightenment and bringing others along with him by virtue of sharing what he thought. It’s not purely the result of his work that causes him to be enlightened; it is the fact that he is bringing the entire species up to a level of awareness not previously found. He’s enlightened because of his humility and selflessness. Perfecting the human to perfect humanity.

The Age of Enlightenment did that as well; it brought different cultures to new heights of thought, awareness, and knowledge. As a species, it was a leap forward. Each leap of knowledge is usually obvious but not always grand. One cannot leap from the valley floor to the top of a mountain in one go. It also is visible in hindsight, rarely in the present. Enlightenment seems, to me, to be gently incremental. There are no five easy steps to enlightenment, no matter what anyone says. There is no golden knowledge at the end of all the degrees. Enlightenment is work. Hard work by many, many people. And…we can only bring humanity up if we work toward its good, bringing it all up with the talents and gifts that we have, be it a lens grinder or a philosopher.

lightbulb

And why not both? What is stopping us from pushing away from the TVs and video games and doing what Spinoza did? Nothing, as far as I can tell, except our own laziness. We are tempted by many things which bring down our humanity, or at the very least, stagnate and stall our progress. We need to be self-discovering, exploring ourselves, our environment, nature, our own natures, the universe, looking at things we know and don’t know, with both our natures – intellectual and devotional. Science and nature. Analytical and feeling. We might not find “enlightenment” at the bottom of a test tube but we may find wonder, delight, and wisdom on the journey. The results, of the destination and the journey, are the seeds of Enlightenment.

Know Thyself: The Ship of Thieves

Know Thyself: The Ship of Thieves

“I am not the person I was.” We hear that a lot, especially when it comes to growing older and, one hopes, wiser. Indeed, we’re not the same person we were. Over the course of time, our cells die, regenerate, add, delete, change, morph, and eventually we have all new cells. But we retain our name, our memories, our lives. Are we not the same person?

One would argue that of course we are. Or are we? Really?

We cling to our identities like dryer sheets to hot cotton shirts. In our minds, we are who we always have been. We are that twelve-year-old child who swam in the lake as well as that adult who had their first job in fast food. We remember events, creations, or possessions and claim them to be ours.

Conversely, we claim our “self” to exist because of those things. We do not change, or if we do, it is at a glacial pace. We affix our identity in time and space, and like an astronaut, place a flag on it and proclaim it to be ours, to be “true” identity: knowing who we are.Theseus_Helene_Staatliche_Antikensammlungen_2309_n2

In a recent conversation with a fellow Mason, I was discussing the Ship of Theseus. The paradox is quickly explained in this video: The Paradox of the Ship of Theseus. In essence, the question is this: at what point does the ship cease to become Theseus’ ship and become something else?

If we take one plank from the ship and replace it, we generally can agree that the ship is still Theseus’ ship. At what point, however, do you fix enough broken pieces that the ship becomes something else? My colleague was convinced that the ship remained and always remained Theseus’ ship. For him, the idea of identity stays with the generally recognized “thing” even if the sum of its parts is not original.

Conversely, the argument is this: if I am a thief, and I slowly steal the pieces of Theseus’ ship, replace them with identical parts,  take the original parts, and put them together in my backyard, who has the ship of Theseus? The original owner, or me?

My friend said that the original owner did. I disagree. If I take a painting from the Louvre, and replace it with an identical painting, and everyone recognizes it as the “painting,” who has the “real” painting? In my colleague’s eyes, then, have I really stolen anything?

identityI contend that I have, if nothing else, I have stolen the certainty of the Ship of Theseus. I have stolen, or potentially stolen, the idea of the ship. But these painful musings do have a purpose: they help us work out our identity – the answers to the question of: Who am I?

A brilliant article on this is found on Brainpickings. I would encourage you to watch the other short videos on this site: not only is the one on Who Am I thought-provoking, but there are links to life’s other huge questions. How do I know I exist? What is the Nature of Reality? But, I digress.

The question is, at what point is our self no longer “us?” Is it when all the cells in our body have replaced themselves? What about new neural pathways or brain cells? If we replace a leg or arm or heart, are we the same person? 

Freemasons live by an adage of “Know Thyself,” which also adorned the Oracle of Delphi  at the Temple of Apollo. We must first understand what it is that makes up our “self” and when does that “self” become something else. I think this is a life long exploration and, since the self is constantly undergoing change, are we always who we were? Perhaps not.

But then, where did “we” go? Does our identity persist? If it does so, how? What makes us, us?fingerprint

I asked my fellow Mason about clones, which sent us down an entirely different path, discussing identical twins, and the like. Does time make a difference? If a plank is rotten on Theseus’ ship, and it is replaced, does that make identity linger, as opposed to replacing a “new” plank? If I change my mind about how I feel about something, am I still the same person? What if I create new habits? What then?

We are ever seeking to understand our true natures; yet, our true nature is ever-changing. Freemasonry teaches us about the cycles of life, death, rebirth, nature. and science. It teaches us all of Life’s Mysteries. If stagnation is death and change is life, how can we ever be the same person moment to moment? Perhaps that is the mystery that we must ever follow: a constant, persistent discovery of who we are, and what we are doing.