The Nature of Fractals – Part III: Our Chaotic Reality

The Nature of Fractals – Part III: Our Chaotic Reality

This is Part III of a three part series on the Nature of Fractals. Readers can view the first two installments here: Part I and Part II.


In the previous part of this series, I introduced the thought that quantum mechanics are related to fractals. That combination further implies the idea that quantum objects represent the combination of spirit and matter, which themselves exhibit fractal properties.

The Hebrew Letter Kaf

A concept alluding to a transition akin to that seen in quantum mechanics can be found in mystic  interpretations of the Hebrew alphabet. The word “kaf” is made up of two symbols from the first letters of two other words – koach, meaning potential, and poel: “…suggesting that Kaf enables the latent power of the spiritual (the potential) to be made actual in the physical…”1  The symbol kaf represents a palm with a Yod in its middle where potential becomes reality and hearkens back to the concept of a quantum object collapsing into its matter form.

The combination of spirit and matter (see featured image) further suggests a direct link between Humanity and God. Our very thoughts may be thought of as quantum objects and underlie our reality, bringing into existence that which we interact with on a daily basis.

Imagine a being, a being so advanced and evolved, that he is One with the Word – he is indistinguishable from the Word. His thoughts are very special – they appear in his mind as quantum mechanical potential energy and take on a life of their own. Within the thought, there is Life – spirit and matter combine to form consciousness, one facet of the being as described in the dual-aspect theory.

In this manner, Sungsang and Hyungsang interact and form a human being. The thoughts or beings exist in the Universe of his mind, evolving themselves until they have each “had their due” and dissolve into nothingness. In this manner, each thought works to merge with the Word – thus further purifying the being. Thoughts beget thoughts of their own. which become fractal tessellations perpetuated throughout eternity. Fractals are left behind throughout the infinite hierarchy as evidence of the being’s work. At the End of Time, the being himself dissolves into nothing, a passing thought in a greater being’s mind.

Along similar lines, both the Bible and Mormonism support beliefs that are consistent with the previous paragraph. In the Bible we read in Romans (8:16-17):

“The Spirit itself beareth witness with our spirit, that we are the children of God and if children, then heirs; heirs of God, and joint-heirs with Christ; if so be that we suffer with him, that we may be also glorified together.”

In the Mormon faith, it is the belief that each individual can progress to the point to inheriting a universe of their own. 

“…Each one of you has it within the realm of his possibility to develop a kingdom over which you will preside as its king and god. You will  need to develop yourself and grow in ability and power and worthiness, to govern such a world with all of its people.”2

Lorentz Attractor

From a different perspective, fractal mathematics are a representation of chaos. “Fractals are related to chaos because they are complex systems that have definite properties.”3 It was discovered that there is indeed a pattern to chaotic systems. Using a number of different initial points for a given chaotic system and running the system for quite some time (a chore perfectly suited to automation), they all eventually resolve into a two-dimensional projection of a butterfly shape called the Lorentz Attractor. 4

When one thinks about the Earth and the moon revolving around each other due to gravitational attraction, the Earth is an attractor for the moon. In chaos theory, the “orbital” patterns seen in the butterfly shape seem to form orbits around what are called “strange attractors.” The butterfly shape is described in terms of fractal dimensions, which means that the shape’s dimensionality is not an integer (2-D, 3-D, etc.), but a fraction between two integers.

“…So, a fractal image is a visual representation of a strange attractor (or fractal space) that defines the orbit of a deterministic system that behaves chaotically…”5

So when we consider that fractals represent a form of order ubiquitous throughout Nature, it can be seen that they are indeed the embodiment of Ordo ab Chao – Order out of Chaos.


1 The Letter Kaf/Khaf, n.d.

2 “. . . the Matter of Marriage” [address delivered at University of Utah Institute of Religion, 22 Oct. 1976])  (Will Exalted Mormons Get Their Own Spirit Children and Worlds?, n.d.

3 Blumenthal, n.d.

4 Chaos VII : Strange Attractors, n.d.

5 Wrigley, 2017.

The Nature of Fractals – Part II: Of Spirit and Matter

The Nature of Fractals – Part II: Of Spirit and Matter

In the previous part of this series, I introduced the concept of a fractal. It is a construct which appears frequently throughout our daily lives and Nature herself.

So what could be driving this self-similarity at multiple scales? It is my belief that quantum mechanics is related to the phenomenon. Recently, scientists demonstrated that quantum mechanics has fractal properties.  (Dacey, 2010) Recalling some basics of quantum theory, sub-atomic quantum objects are thought to have both wave-like and particle-like (mass) properties. When a quantum object is in the wave state, it exists in a number of probabilities – a number of locations. When the wave function collapses, it becomes a definite particle of matter discernible by our instruments at a specific location. At this point, the particle not only takes on position and mass, it also takes on fractal properties.

As alluded to by Dr. Pincus in his quote in the previous article, forming a fractal mass particle is very efficient and effective in terms of energy conservation. I believe that matter retains both wave-like and matter-like quantum properties no matter its form, just to varying degrees between the two. It is the wave aspect of matter that allows it to interact with other object waves. When a given particle replicates to either a smaller or larger scale, the primary form is determined by the original object at the smaller or larger scale (thus the similarity), but that form is affected by wave interactions with adjacent objects.

Thus, the variability we find in Nature for naturally occurring fractals. In other words, if those interactions did not occur, then smaller and larger replications would be a perfect copy of the original. “Adjacent” particles do not necessarily correspond to normal concepts of space and time given the quantum mechanical property of entanglement, which states that a given particle can affect another particle at some physical distance. It is also my belief that this quantum object is the embodiment of Spirit and Matter – that the wave function seen in quantum particles is an indirect indication of the existence of Spirit. Our current instruments cannot detect or measure Spirit directly.

Sungsang and Hyungsang

Fractal mechanics in nature as described above suggest a dynamic relationship between the spiritual and the concrete – matter. A number of belief systems support this line of thought. An adjunct to Chinese philosophy’s Yin and Yang is the concept of Sungsang and Hyungsang. Where Yin and Yang describe attributes of an object, Sungsang and Hyungsang address the composition of the object – what it is made of. Sungsang refers to the non-visible aspects of an object while Hyungsang refers to the visible aspects of the same object such as mass, shape and the like. As an example, the Sungsang for a plant is Life. Both concepts are thought to stem from the Original Image, their origin.

Sungsang and Hyungsang exists in every physical object, from minerals to plants to animals to human beings in that order to form a kind of hierarchy. As one moves “up” through the hierarchy, the higher form introduces new attributes while taking on those lower down in the hierarchy. So plants, for example, introduce “Life” as it Sungsang, but also inherits physico-chemical aspects from minerals. The diagram to the right illustrates the concept. Sungsang and Hyungsang is one additional way to view and explain our inter-connectivity with Nature.

Note that the human level of the hierarchy adds the spirit mind and spirit body to the mix. Adherents to Unification Thought, those that believe in Sungsang and Hyungsang, believe that at the end of our lives, the material body – the portion of us that we inherit from the animal kingdom – falls away and our spirit, mind, and body continue to live on indefinitely.  (I. The Universal Image of the Individual Truth Body, n.d.)

Mosaic Pavement – Spirit and Matter

Another concept coming from the psychology of mind domain seems to exhibit aspects of  spirit and matter as well. Dual-aspect theory or dual monism addresses the mind-body problem – how the mind and body interact and exist alongside one another. As can be imagined, because mental processes are involved, some of the theories can get kind of exotic – and dual-aspect theory is no exception.

Under this Theory, the object under consideration, normally a person, is neither physical nor mental, but an inseparable combination of the both. A term used to refer to this combination in the readings I explored is “phental.”

The combination is not the simple combination of the mental and physical, nor is it reducible beyond its normal form. The mental and physical are then aspects of the phental.

A further refinement of the theory postulates that the phental is unknowable and that the mental and physical aspects are but a portion of the true self being exposed. This last part has some interesting implications and is a potential reference to what is stated above regarding the wave-like portion of a quantum object. As stated previously, I believe the wave-like portion to be an outward expression of the imbuing Spirit. This line of thought then makes one wonder what could be the underlying truth regarding the physical aspect of the phental.

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)

 

Triangles Everywhere

Triangles Everywhere

I was recently exploring the idea of the triangle – its form, function, stability, and meanings. In Freemasonry, as in many traditions, the triangle holds significant influence in symbolic meanings.

A triangle is a polygon with three sides and three vertices. There are many forms of triangles – right, equilateral, obtuse, acute, isosceles, and scalene. There are also oblique and degenerate triangles. Triangles may be multiple types. Triangles are generally believed to be two-dimensional objects whose interior angles, at least in Euclidean space, equal 180 degrees. They can be various shapes but the ones most often seen are right triangles and equilateral triangles.

Of the triangle knowledge from history, the famous philosophers Pythagoras, Plato, and Euclid are known best for theorems, ideas, and esoteric supposition surrounding the form. The form is so basic that it’s most likely older than written history. Ancient petroglyphs, such as those from Columbia, the Sierras in North America, and Mexico, show humans with bodies and heads in the form of triangles. This is a basic shape that mimicked the human form, with wide shoulders and narrow waist, or a wide head crown and narrow chin. There isn’t anything to indicate, in-depth, the symbolic meaning of the triangle other than it being incorporated into the human form.

The Egyptians used the triangle quite often, generally in the realignment of land plots after the Nile floods but also in architecture. In a 2000 thesis article regarding the “sacred triangle,” the author asserts that Egyptians knew and used, even in the Old Kingdom, the “sacred triangle” of 3:4:5. Indeed, the author goes on to state that using straight vertices, or a “simple, straight vertical pole,” to find location or identify specific time of day or days of the year. While this is a heavy-mathematics article, the reader might find some deeper, symbolic meanings in the geometry.

During the 6th C. B.C.E., the School of Pythagoras became known for its theorem regarding the formation of the ‘sacred triangle.’ Pythagoras left no mathematical writings of his own, while Euclid and Plato did. Thales of Miletus is really the creator of basic mathematics and geometry, and probably the first to give us theorems about the triangle. Pythagoras, who created the words philosophy and mathematics, is more well-known and did much to bring the form of the triangle into deeper meaning.

To Pythagoras, the number 10 was the holiest of numbers; the tetractys is a triangle form of 10 dots, created by interlinking the dots into nine triangles forming the 10th, larger triangle. It is used to symbolize the creative forces of the universe. From ancient-symbol.com, “In the figure, the first row has a single point that is representative of the Creator, the active principle, the divine power behind all creation and is associated with wisdom. The second row contains two points that represent the passive principle and are associated with friction, movement, impulse, strength, and courage. The third row with three points signifies the world coming out of the union of the above two, a union of physical and mental balance and is associated with harmony. The fourth row has four points that represent the four liberal arts & sciences that complete the world. These four points symbolize the four elements of earth, fire, air, and water.” This was, generally speaking, the first time that the philosophical meaning of a number, its holiness and perfection, being derived from pure mathematical reasoning rather than from inductive reasoning. It was more than the total of our fingers on our hands. Another interesting article on the triangle and tetractys, among other things, can be found here: http://www.projectawe.org/blog/2015/12/21/up-and-down-the-monochord-part-iii-triangle-trinity-unity. The author of this blog does a very good and thorough job of digging into these ideas, and I would highly encourage everyone interested in these subjects to read it.

In the alchemical writings of the Middle Ages, the classical elements of hermeticism were based off the form of the triangle, turned upward or down, with a line to denote the opposite or without to indicate the base elements. The conjoining of fire and water is indicative of balance and achieving perfection. The triangle is also seen in the “triangle of art” also known as Solomon’s Triangle. The circle in that triangle represents the space where spirits are called, with the triangle representative of the safe space from which the magician worked.

Triangles in astrology are seen as very positive, and a grand trine, or golden triangle, is seen as a creative, harmonious flow of energy in a person’s life; they generally are composed of the objects being in the same elements, in the form of an equilateral triangle.

Triangles are a form of stability, where two extremes are balanced by a third point. Triangles are everywhere in Freemasonry, overt and subtle, and have different stories surrounding each. These different stories speak to individuals differently even if the core remains the same; depending on the degree being worked and studied, the aspirant may find different aspects of the same truth. These truths are not much different than the ancient Egyptians and Greeks found and used in their daily lives. There are always extremes and balance is achieved by that third, divine point. One might also see that all emanates from the Divine, the single point, which may also turn into that point within a circle which is perfect balance. The perfect man may be the one who finds equilibrium during whatever storm shakes him. Taking this symbolism into our daily lives and applying it to our relationships with people is really the value of the study of symbol. We can work toward being the middle point between extremes, able to see both sides in equal measure. A more holistic view of those things that permeate our lives creates a better person.

Hidden Mysteries of Science

Hidden Mysteries of Science

Science is “the intellectual and practical activity encompassing the systematic study of the structure and behavior of the physical and natural world through observation and experiment.” We all perform scientific acts each and every day. Being aware and present in our actual work, home life, educational pursuits, and leisure all encompass some aspect of “science,” as described above. Do we not learn relationship interaction through observation and experimentation? Of course we do! Do we study others and then experiment with things like cooking, clothing ourselves, cleaning the house, and raising children? Absolutely. Life is science.

And yet… there are the science doubters. The Washington Post did an article, in 2015, on science doubters. Entitled, “Why is Science so Hard to Believe?” the article goes on to discuss confirmation bias, the discipline of the scientific method, and why so many people would rather believe media hype or misinformation from friends rather than actual science. Media is not science and it is not gospel. We consume the media that’s easy to consume rather than do the work for ourselves. It’s easier to doubt than to verify.

Neil deGrasse Tyson has an interesting quote: “The good thing about science is that it’s true whether or not you believe in it.” He also said that “the universe is under no obligation to make sense to you.” Both of these quotes speak to the hubris of humans – we think we know much more about the word than we really do.

In a quote from an article on National Public Radio, the author quoted his friend, a professor of Jewish philosopher, as saying “science tries to make magic real.” The author goes on to specifically outline activities, now commonplace human activities, as ones that we once thought of as magical, for example, flying. We fly without a second thought; yet, 500 years ago, to say one flew was heresy, possibly leading to death. Other examples are the knowledge of “invisible” animals capable of making humans ill, or being able to see great distances into space (the past) through a telescope. The ability for our phones to “think” and talk with us would have been quite astounding to the medieval mind.

The author continues his journey with the main difference between science and magic: his belief is that the power of magic originates within us, where as science’s power originates outside of humans. Science is a set of immutable laws of the universe. Right?

Well, no. Science updates theories based on knowledge gained from further expressions of the scientific method, and then new theories are postulated. Science is evolving, a never-stagnant set of data that we are constantly testing and proving or disproving. Magic is generally seen as not obeying the laws of nature, being outside of those “rules” or “metaphysical,” as it were. Yet, we’ve all said it: couldn’t what we see as magic just be unexplained scientific laws that we do not understand quite yet?

Why are Freemasons charged to examine and study nature and science? Nature AND science? It seems that it might be because the world is made up of both the understood and the mystery. We have many questions to answer about nature and we use science to get there. Perhaps we could say we have many questions to answer about magic and science is the method. There’s no reason we can’t have wonder and reason hanging out together in our minds. We can appreciate the brilliant stars and the awe of an eclipse and still want to know how it happens. Knowledge does not take away wonder.

I want to believe that perhaps science and magic are part of the same evolutionary cycle – what starts out as magic becomes understood by science, which breeds questions within our curious minds, wonder at something unknown, triggering us to embrace the tools of science to explore. Freemasons get to play in both realms, being co-creators on the path of humanity.

How Do You Know?

How Do You Know?

Our modern times have brought us many great advancements. We find ourselves living longer, becoming more globally connected, and enjoying medical ingenuities, such as antibiotics, blood transfusions, and artificial organs. There are many amazing necessities and niceties that are enjoyed by the human race in varying degrees because of Science. Science has given us a lot to be thankful for. Or has it?

In recent years, there have been debates, and at times heated arguments, over the likes of genetically modified foods, vaccinations, and global warming. Even the effectivenessFlat Earth of Western medicine has cropped up in many personal conversations over the years. Ideas such as Flat Earth have come back to the scene in modern discussions and often with contention.

Once thought for certain by the general populace, many scientific concepts are met with skepticism. But before you believe this blog is about winning you over to one side or the other, I ask you read on, because it not. There is something greater underneath these debates, and it has everything to do with you.

When researching the Philosophy of Science the other day, I came upon a very intriguing
question: How do you know your knowledge is authentic?

What a wonderful question, and it has given me more than a pause. Now before we reduce this question to reducto adsurdum, and say how can we really ever know anything, let’s try to accept the question for what it is: an invitation to know ourselves a little bit better.

Knowledge. It is a formidable due to its ubiquitous nature. It is an invading species that finds life in the uninhabitable regions of our brain. It plants its roots and digs deep so it cannot be easily removed, often without our realizing it.

Thus, when we allow “knowledge” to pass our acceptance filters and impregnate itself in our world view, it becomes almost impossible to remove. Especially if it comes from an authority – like science, religion, or a person of a particular importance. But are these sources enough to make an idea become an organism of knowledge?

One of the greatest lessons science has taught me is that it is only at its best when it is being challenged, and I find that this true of human knowledge in general. Authenticity cannot exist if challenge is not present. Growth is a product of conflict, not peace. Knowledge that is real will survive and become stronger; the ideas that do not deserve to be uprooted and replaced with a more genuine concept.

How do you know your knowledge is real? We listen and we give the other side their due. This is a very Masonic and scientific principle. In doing so, the only danger we will face is the danger of becoming more authentic in what we know. That doesn’t sound so bad, does it?

 

When Did We Stop?

When Did We Stop?

It is easy for life to sweep us away on the current of self-importance. I don’t think we mean to; it just happens to be the way our culture works. Fast and busy and “me” centered. This way of life isn’t just an adult thing. We have shown our children how to do it. They, too, are pounded with the every day commitments we give them and allow. This way of living is like a fierce version of the Tango but at a pace it was never intended to be danced at.

This is my life as well; I made the same choice you did, to be a part of this me-speed machine.

Two events recently occurred that has made me slow my dance steps down and see those around me better: the launch of Falcon Heavy and a philosophical discussion on whether we should migrate to Mars.

The only word that I can give to the launching of Falcon Heavy is wonder. Watching the launch left my mouth open but with no words. There was something eerie when the sideFalcon Heavy boosters landed on Earth again. This shouldn’t be happening, I told myself. Side boosters don’t come back, they just don’t. Again, the wonderment had me re-watching several times over until the busy day I had, had dragged my eye lids closed.

Two weeks later the philosophical debate on whether humans should migrate to Mars coincidently dove-tailed with the SpaceX’s launch. The discussion was an interesting juxtaposition to my earlier experience of watching the Falcon Heavy launch. I entered the discussion, as I do monthly, with great enthusiasm about the topic. How could I not with this particular idea? We were going to talk about the possible expansion of our kind. To me, the feeling I had could be analogous to what people must have felt when travel to the New World seemed impossibly possible. The feeling was akin to infectious hope sprinkled with reservation. The New World, that is Mars, seems so alien, so inhospitable, could we ever truly make a life there?

It was after this debate that I have felt my mouth go dry with disappointment and my inner Tango stumble with the memory of a statement made earlier in the discussion, “What did schlepping to the Moon ever get us?” I shouldn’t judge I know… but I did. This question has forced me to understand the alternative purpose that Elon Musk had when he sent his Tesla roadster into space. He didn’t use his car solely as payload… he used it to get our attention.

I have to ask; I have to know. When did we stop looking up? When did we stop finding continual inspiration in the stars and unimaginable possibilities in worlds that seem saturns_shadowunreachable? I cannot help but to understand Elon Musk’s strategy. He needed to pull our eyes off the ground by wowing us with his fancy car whizzing around Earth’s orbit because a rocket that brings us one step literally closer to Mars, wasn’t and isn’t enough.

My hope has been temporarily dampened, but it still remains because it is possible to change the rhythm by which we live to include the stars. Space exploration isn’t about man schlepping through the cosmos; it is about us making a bigger place for ourselves in that inky black sky. And the possibility that we are closer than ever to doing just this gives oxygen to that small flicker of hope.

 

Astronomy and the Quadrivium

Astronomy and the Quadrivium

Perhaps your first thought, as was mine, is: “How can Astronomy be an ‘art?'” Furthermore, how can Astronomy be called a ‘liberal’ art? From a very interesting (and worth exploring) website called “Arts of Liberty,” we have a snippet for explanation:

“To call astronomy an ‘art’ can come as a shock to a modern reader… Perhaps without thinking much about it, we think of “science” as being a genuine and exact knowledge, whereas ‘art’ is more expressive, or touchy-feely.  But, that is not quite adequate, since medicine is also an ‘art,’ and it is anything but touchy-feely… And while ‘science’ and ‘art’ do not appear to be synonyms, it could very well be that the same discipline can be called both a ‘science’ and an ‘art,’ although for different reasons.

To understand this properly requires us to consider a sense of the word ‘science’ not in common use today.  The word ‘science’ comes from the Latin word scientia, which meant a very exact knowledge, a rigorous and sure knowledge of things deduced from self-evident truths.  The ancient Greeks would have called such knowledge epistémé...  

In the vocabulary of the ancients, an ‘art,’ like a science, meant a carefully reasoned-out knowledge, but more than that, it meant a knowledge of how to produce something.  Where there is no ‘product,’ there is no ‘art.’  So it is possible for a form of knowledge to be a ‘science’ but not an ‘art.’  For example, Aristotle considered the study of god to be a ‘science,’ a body of knowledge rigorously reasoned out from self-evident principles, but not an ‘art,’ because it did not teach us how to make gods, or how to do anything about god.”

Ptolemaicsystem-smallClaudius Ptolemy (100-170 CE) was a Greek mathematician living in Alexandria. His work The Amalgest was one of the most influential astronomical works until Galileo’s discoveries in the 17th C. The Amalgest documents many mathematical and astronomical treatises, including works by other mathematicians – works thought to be lost. The most significant piece of this Amalgest (total of 13 books), is the documentation of the geocentric model of the universe. Ptolemy’s work became the accepted theory of the structure of the planets and stars, with the Earth central to all.

This influenced not only astronomy and mathematics but also theology, philosophy, and fine art. Three centuries after it was written, Hypatia and her father Theon, genius mathematicians, added to the work with their own commentary, throwing in their thoughts of elliptical orbits, the procession of the equinoxes, revising Ptolemy’s Handy Tables, and introducing the sexigesimal calculation systems. It’s believed that this rendering of the Amalgest, with the Hypatia/Theon commentary, is the one that was used for the majority of the Middle Ages although no direct reference to Theon survives and Hypatia is mentioned only in a passing 10th C. reference.


And thus, the Quadrivium ends. I hope you’ve enjoyed my answer to the Bro.’s challenge of finding a significant event, work, or person who influenced each specific liberal art. The question was posed, should Freemason’s really learn the liberal arts? The answer, to me, should be obvious. Not only should we learn the liberal arts individually, but understand their context in the whole of being educated about the natural world. Human beings can be taught easily to survive; we cannot just “pick up” how to thrive, generate ideas, and create a better world.

An example of this “Freemasonic mindset” is James Madison, even though he was not a Freemason. In his early twenties, when the United States was in its infancy, he gave up much of his career and life to studying the histories and government of world cultures. He was relentless in his pursuit of the histories and knowledge of government administration, what worked, what didn’t; he studied philosophy, history, theology, art, classical literature, geography – the liberal arts and more. By the time he finished, and began his work in the new nation’s government, he was arguably the single biggest influencer in shaping the United States Constitution and the framework of our Democracy. By learning the past deeply, he was able to innovate and create a new world. To me, that is a main goal of the Service of Freemasons.

Geometry and the Quadrivium

Geometry and the Quadrivium

Whist sitting in school, slaving away with compasses and a ruler, one hardly remembers that geometry is the study of the measurement of the earth. Earth. The thing we sit on, utilize, and finally rest in when this is all over. The geometry in schools today looks nothing like the geometry of 3000 years ago. It is difficult to divorce geometry from the other liberal arts when we take into consideration the scale to while discoveries are interconnected. Geometry arose from the needs of agriculture, civilization, and war. For so much of this, we can thank Archimedes of Syracuse. A student of Euclid in the 3rd c. BCE, his advances in the field of geometry furthered irrigation (Archimedes’ Screw), astronomy (the first planetarium), and weights & measures (Archimedes’ Principle). The most interesting, to me, is The Method of Exhaustion (remember Dialectica) also known as “The Method” or “Archimedes’ Method.”

“…, to estimate the area of a circle, he constructed a larger polygon outside the circle and a smaller one inside it. He first enclosed the circle in a triangle, then in a square, pentagon, hexagon, etc, etc, each time approximating the area of the circle more closely. By this archimedes_circleso-called ‘method of exhaustion’ (or simply ‘Archimedes’ Method’), he effectively homed in on a value for one of the most important numbers in all of mathematics, π.” 1

Linked together with this Method is the “Method of Mechanical Theorems.” Proofs are everything to the mathematician, and in his Method of Mechanical Theorems, Archimedes had none that would be accepted. He set out using Eudoxus’ The Method of Exhaustion to prove what he knew to be true. In a letter to Eratosthenes, in manuscripts discovered in 1906, Archimedes outlines his thought processes. This document is known as the Archimedes Palimpsest.

Certain theorems first became clear to me by means of a mechanical method. Then, however, they had to be proved geometrically since the method provided no real proof. It is obviously easier to find a proof when we have already learned something about the question by means of the method than it is to find one without such advance knowledge.

The importance of these discoveries and the methods by which Archimedes came to them may be obvious – who doesn’t need π? However, it is also fascinating to peer inside the mathematician’s mind and view it with a Freemason’s perspective. Here was a man who could see the Plan, understand the Plan, and only needed to bring it to life: a divine spark of wisdom, the will to discover, and beauty in its presentation.

For an interesting and short expose on The Method and the “Archimedes Palmipsest,” whence this Method is documented, review  “The Illustrated Method of Archimedes” by  Andre Koch Torres Assis and Ceno Pietro Magnaghi. The PDF can be found here.

Additionally, the originally translated letter from Archimedes to Eratosthenes can be downloaded here. (Thank you, JSTOR.)


Just a note (1): The Story of Mathematics, Luke Mastin – http://www.storyofmathematics.com/hellenistic_archimedes.html – I’ve done my best to verify statements here, and so should you.

Mathematics and the Quadrivium

Mathematics and the Quadrivium

Personally, I struggled with Math in school. Faced with a math test, any math test, I froze, cried, banged my head against the desk, and ultimately gave up. I saw mathematics as an isolated “thing” to be conquered. You were either good at math, or you were not.

How little I knew, and how little I was taught, about true mathematics. More than numbers, factorials, and fractions, Mathematics is about relationships – of numbers: how they work with each other, work for us, against us, and can talk about any situation. There are mathematics of money, elections, government, science, music, agriculture, capitalism, socialism, any -ism. Math is language and structure: it is a bridge between all aspects of liberal art. Which leads us to the Bridges of Koenigsberg.

bridgesLeonhard Euler, a Swiss mathematician of the 18th Century solved, sort of, the problem of the Seven Bridges of Koenigsberg (Russia, at the time). Koenigsberg had two islands connected by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point (touching every edge only once). Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree, and stated without proof that connected graphs with all vertices of even degree have an Eulerian circuit. The bridges did not meet this condition and therefore, no solution could be found to the problem.

Yet, what this Eulerian circuit eventually did provide is the basis for modern topology , which has expanded into areas of quantum physics, cosmology, biology, computer eulernetworking, and computer programming. For example, the Eulerian cycle or path is used in CMOS circuit design to find an optimal logic gate layouts. For anyone wanting to read the paper outlining these paths in the original Latin, it can be found here.  English translations do exist. A good page on the history of topology is here.

Leonhard Euler was a fascinating individual in that he saw mathematics as something that infused all of life. Though his writings, he made applied mathematics accessible to the layman and his scholastic peers alike. An excellent and thorough biography, written by Walter Gautschi, can be downloaded in PDF form here. With a varied interest in all aspects of mathematics  (arithmetic, geometry, algebra, physics), music, anatomy, physiology, and astronomy, he truly was a man of the “Enlightenment.”  While he was not a Freemason from what I can tell, he seemed to hold much regard for the idea of true science, and creating a better world for his fellow man: a Freemason’s true ideals, to be sure.