Freemasonry and Geometry: What are the Symbolic Meanings of the Most Basic Geometric Forms?

Freemasonry and Geometry: What are the Symbolic Meanings of the Most Basic Geometric Forms?

The more abstract a concept is, the more meaning that we can find in it. Is this because the meaning is all projected from within our minds, or because there is something objectively real there to be found? Plato or Pythagoras certainly believed the latter, as did Jung with his archetypes, and certainly the abstraction of mathematics has enabled much of our modern scientific grasp of the workings of the world around us.

In Freemasonry, we deal in symbols constantly. The Temple itself is a symbol, as are all the rituals which take place within, and there are layers of meaning which are only revealed as one progresses through the degrees. Geometry, in particular, is very significant to Freemasonry. This stems not only from the practical geometrical knowledge that was required to build grand cathedrals and other structures by operative masonry, but also from the Western wisdom teachings stemming from Pythagoras, Plato, and others, often referred to as Sacred Geometry, which sees a greater symbolic significance in geometrical forms.

While its common to find interpretations of more complex forms like the Flower of Life, or Metatron’s Cube, some of the geometric forms I find most fascinating are the simplest. This is because the simpler a form is, the more universal it is. In the simplest forms, I see the very foundations of all creation. So, what is the symbolic significance of a few of these simple geometric forms?

What follows are not in any sense “official” masonic interpretations of these forms, but merely one mason’s reflections on them. Books can and have been written to interpret these and other geometric forms, so here we will just be sampling a few.

The Point

pointThe point is the beginning of all forms, and all finite things. In that sense, it is the most abstract representation of finity, of a finite form. It does not even have definitions, specific boundaries, size, or dimensions, and therefore is the most abstract finite thing that can emerge from the formless infinity. It is also the beginning of duality, because in the point, you have one thing which is separated from everything else, the thing and its surrounding context, self and world. Yet, without a clear boundary, it still contains an element of infinity within it. The point can be viewed as infinitely small, or infinitely large. Without boundaries and other objects to be compared to, it has no reference.

In the point, we can see the fundamental essence of all finite entities, including each person: the Self, consciousness, existence.

The Line

HorizThe line is the first movement of the point, the first dimension of form. As such, it represents the most basic possible expression of time and motion. In the line, we can see the progression from moment to moment of what was first seen in the point. While the point is motionless and still, the line implies a trajectory, and a continuity. It is also the first impression of a space in which motion and change can take place. Yet, the line is only the simplest, most primal of forms with a boundary, and does not yet demonstrate a space in which objects may exist, although it begins to imply them.

In the line, we can see the fundamental essence of all motion and change, linearity, past progressing into future, and memory.

The Cross

sacred geometryThe cross is where two lines meet at a perfect right angle, and the simplest indication of the second dimension. In the cross, we can see the implied 2-dimensional plane stretching out in four directions. Also within the cross, we can see both the singular point, and the line doubled. Like every new form, it is the expansion of what came before it. The cross is where two intersect to become one; two planes, two people, two forces, and the point at which they meet is also their origin.

In the cross, we can see the fundamental essence of all duality, multiplicity, and complexity. It is the simplest possible mandala, and also represents the multiple dimensions of self, whether masculine/feminine, thinking/feeling, or ordered/chaotic, intersecting at the central point of consciousness or spirit.

The Circle

infinity circleThe circle is the perfect return of the trajectory of the point traveling in a line back to its origin. Simultaneously, it is an enlarged and expanded representation of the point, being equidistant in all directions from the center point. In the circle, the completeness and perfection of the process of the formless point expanding into form can be seen. Within the circle, we can fit any number of mandalic or sacred forms, such as the pantagram, the hexagram, the cross, or the equilateral triangle. It contains all of these in essence, and yet is beyond all of them.

In the circle, we can see the totality of form, the Ouroboros, the nearest form to infinity.

The Circumpunct

circumpunct freemasonryThe circumpunct is the completion of the journey from finite point to infinite circle. In the circumpunct, all of the dimensions of self which have progressed outward into form are in complete and total balance and alignment with their formless origin. In the mind, it is the personality in complete harmony with its essence and origin. It also represents the essential boundary of the self. While the point had a finite existence, it still had a certain infinite quality, in it’s lack of defined boundaries. In the circumpunct, we have both the boundary-less point of origin, and the boundary, with self on the inside, and world on the outside.

In the circumpunct, we can see the completion of all existence, the simultaneity of infinity and finite form, the perfection of self, world, and the relationship between the two.

Further Reflections

expanding dimensionsThe truth is that neither you nor I have ever really seen a point or a line, only things that are shaped like them, usually drawings on paper. But the truest sense in which a point and a line exist is in the abstract realm of ideas, yet they are clearly significant. We could not build anything without them, and this is true of mathematical abstractions, in general: they are ephemeral, yet essential. But can they be regarded as the true foundations of all creation, of reality itself?

What I am struck most by these simple forms is how each dimension, and the basic geometric form it represents, implies and transcends the next. The moment you create a point, it imples that that point could transcend the 0 dimension, and be a line, if only it moved beyond that 0 dimension. The moment you create a line, it implies that that line could move in a different direction, transcending that 1st dimensional existence, creating a plane. The moment you create a plane, it implies that that plane could be bent or folded, and that therefore a third dimension must exist. The moment you have a 3-dimensional object, it implies that that object can shift and change across time, thus implying a fourth dimension, transcendent from 3-dimensional space.

This seems to me to imply that the world in which we find ourselves is in fact a manifestation from the infinite 0 dimension, into the four dimensions where we currently find ourselves. Its almost as if a divine infinite mind, in similar manner to an architect, charted out and traced the fundamental principles of reality by which a creation could manifest.

But is that really where it ends? For us, the 4 dimensions are the extent of dimensionality we can see and understand, with even the 4th being somewhat mysterious to us. But what if this is merely because of our position along the spectrum of dimensions, and in actuality, this process of one dimension implying the next goes on infinitely, with the 4th implying and creating the 5th, the 5th the 6th, ad infinitum?

String theorists and various other kinds of physicists currently posit some number of dimensions, often 11 or higher, but what if there is no ceiling to the dimensional hierarchy extending into the unseen, unknown realms of possibility? In that case, we truly could be said to be dwelling in an infinite creation.

infinite dimensions

Eastern Sages, Western Secrets: What Has Vedanta to Do with Freemasonry?

Eastern Sages, Western Secrets: What Has Vedanta to Do with Freemasonry?

Like unto that of a man blindfolded and carried away by robbers from his own country is a man’s condition. The folds of cloth over his eyes being removed by a friend, he recovers the use of his eyes and slowly finds his way home, step by step, inquiring at each stage. So also, the good teacher instructs the seeker of Truth and helps him to unloose his bonds of desire. ~ From the Chandogya Upanishad 6:14:1/3

As a young seeker of Truth, I found the wisdom of the East long before I found the hidden wisdom of the West. After I had turned away from the exoteric religion of my upbringing, and briefly embraced the alternative orthodoxy of nihilistic materialist scientism, I began to seek truths beyond the tiresome antagonism of the “religion vs. science” debate, and it was not long before I discovered the Bhagavad Gita, Taoism, Buddhist teachings, and Western interpreters like Alan Watts and Ram Dass. Of all the Eastern traditions, Vedantic philosophy (rooted in the Vedas, Upanishads, and related texts) has influenced my worldview more than any other system of thought.

Now, as a newcomer to the Western wisdom tradition of Freemasonry, and to some extent also Theosophy, Alchemy, and Hermeticism, I am struck by the similarity in essence, but difference in expression between East and West. Although I still have much to learn, I sense that the one great Truth, described by Aldous Huxley as the Perennial Philosophy, is fundamentally the same in the Western traditions as in the East, but is clothed in secrecy and symbolism, and in some ways emphasizes certain values over others. The contrast between East and West is particularly interesting to me, as they are two halves of humanity’s collective mind, just as they represent two poles on our globe.

Torch Bearers from the East

eastern wisdomTo begin with, we can be fairly certain that the connection between East and West goes back at least as far as all of Western history, as we know it. The history of great Western philosophers is also in part a history of those who journeyed to the East, learned, and came back with new insights which had to be clothed and couched in the prevailing worldview of whatever Western culture they were returning to, in order to be understood and accepted; even then, they were often rejected, sometimes violently. Bearers of the torch carrying light from the East are notoriously persecuted upon their return to the West, and often meet a gruesome end.

Some think that none other than Jesus of Nazareth is one such example, although this is still a highly controversial theory, with inconclusive evidence that he visited the East during his “missing years.” Another is Pythagoras, who is thought to have traveled extensively in his youth, at least as far East as Persia, and who also was killed by the ignorant. One that is more recent, and therefore also more certain, is that of H. P. Blavatsky, one of the founders of Theosophy and originators of Victorian spiritualism, which also preceded and influenced much of what people consider to be “new age” or “new thought” ideas, today.

In general, I think that we underestimate the degree to which people traveled, and teachings were shared or spread via the Silk Road and other trade routes between East and West, throughout our history. It doesn’t help that our mainstream historians are hesitant to acknowledge Eastern influence on Western thought. So, with all this cross-pollination, why are the West and East not identical?

Wisdom in Contrast

east and westThe most striking difference to me between East and West, in terms of the mysteries, is that in the East they simply aren’t mysteries. There isn’t much secrecy in Vedantic, Buddhist, or Taoist traditions, teachers are prone to publicly say things like, “So long as God seems to be outside and far away, there is ignorance. But when God is realised within, that is true knowledge [Sri Ramakrishna].” The volumes of Vedantic and other Eastern teachings are filled with things like this, which in the West might be merely scoffed at today, but in the past, could have led to a burning at the stake or crucifixion, for speaking so blasphemously.

Here, we arrive at what seems to me to be the chief reason for the secrecy of the Western traditions, which is the millennium of history in which the desert religions of Abraham turned from their mystic origins to the darkness of fanaticism and ignorance, and dogma spread like a plague, reigning over the West with the fiery whip of religious persecution for roughly one thousand years. While our history lessons often breeze over this period as the “Dark Ages,” with some discussion of feudalism and monarchy, the harsh reality is that Western culture underwent an intellectual and religious cleansing, where all ideas that ran contrary to the dogmas of the church (or the mosque) were punished by torture, imprisonment, and gruesome death.

No wonder, then, that those who held the wisdom of the ancient traditions of the West were forced to seek shelter in organizations like operative freemasonry, which provided secrecy, as well as a highly effective organizational structure, and fertile ground for the symbolic coding of wisdom in the tools and principles of masonry. Meanwhile, our Eastern neighbors, safeguarded by distance, geographical features like the himalayas, and their own kingdoms and power structures, held the wisdom passed on from ancient times, and continued teaching it in a relatively open manner. This is an oversimplification, but is generally more accurate than not, I would say.

Aside from being hidden vs. open, what else separates West from East? Perhaps there is a more essential difference, due to differences in temperament and culture of the two peoples, shaped in part by their climates. I could make an argument that the harsh climates of Europe, mostly in the North, bred a spirituality more focused on action, intention, and the overcoming of obstacles, while mostly tropical Eastern environments, particularly in India, bred a spirituality with a more passive focus on meditation and surrender. This theory may have some merit, but ultimately we’ll never know for sure. It does seem to me that the West is more focused on building and actively working to perfect the human, while the East is more about dissolving, letting go, and becoming free from attachment.

At the risk being accused of over-simplifying neuroscience, the general dichotomy of the left and right sides of the brain could also be said to correspond to the same principle. In many ways, West and East are akin to the left and right. The West/Left is all about clearly defined logic, boundaries, and places a greater emphasis on intellect; the East/Right is more about direct perception, dissolving boundaries, and places greater emphasis on intuitive realizations. With all these differences, is there common ground? What is the corpus callosum of East and West?

The Bridge and the Stairway

philosophyI would say absolutely, and as the beginning of this post alludes to, there are striking similarities, at times, between the teachings of the Lodge or Western esotericism in general, and that of the Swamis. Both speak of the human condition as a state of darkness and ignorance striving for light; both design their sacred structures to resemble the human body; both tend towards idealism, or the belief that consciousness rather than matter is fundamental; both ultimately teach that God dwells within.

The commonalities are surely greater than the differences, and the essence, I believe, is one. In many ways, co-masonry in particular may be an excellent bridge between the two systems, with its close ties to theosophy, a much more Eastern tradition than many of the other Western systems, and its focus on adopting the feminine into the masculine lodge. Whatever the bridge we use, it seems clear to me that we must blend East and West, right and left, action and contemplation, intellect and intuition, if we are ever to rise above, and climb the stairway to a greater truth, some transcendent gnosis.

Ultimately, I believe that we are all approximating this same truth, like a middle point of a circle which both East and West circumambulate in their own ways. The classical human mystical experience, which all these traditions are ultimately based on, seems to be more-or-less universal across cultures, and to differ only in it’s interpretation post-experience, and the cultural context which either allows it to flourish, or forces it to hide and conceal itself. Either way, the truth will out, and the light will not be concealed for long, for it is what every human on this planet thirsts for, in the depths of their soul.

The Sacred Tetractys: Why do Freemasons connect the dots?

The Sacred Tetractys: Why do Freemasons connect the dots?

If Pythagoras found himself transported to the modern world, he would have much to learn about technology, science, and human thought. But is there something Pythagoras can still teach us today in his symbol of the tetractys? What do the dots reveal? How is it significant to Freemasonry?

To start with, tetractys refers to a symbol of the Pythagoreans which consists of four rows of dots containing one, two, three, and four dots respectively which form an equilateral triangle. Many have found the tetractys full of sublime meaning.

When did the tetractys first come about? To answer this question may be more difficult. Very little is known about the real Pythagoras, or rather too much is “known” about him, but most of it is surely mistaken. The biographical trail is scattered with contradictions. It combines the sublime, the absurd, the inconceivable, and the just plain weird.

The teachings are elusive because he never wrote anything down. His treatises are only known to us through other Greek researchers. Consequently, it is up to present-day scholars (and there are many) to sift through these works in order to find a common thread that can be genuinely ascribed to Pythagoras. Jean Leblond I - 1605-1666

We do know that Pythagoras was born in Samos in the sixth century B.C.E. Pythagoras was both a mystic and a scientist, although some scholars tend to praise his mathematical prowess while looking away with embarrassment at his perceived “mysticism.” For Pythagoreans, they were one and the same.

The Science of Number was the cornerstone of the Pythagoreans. It describes, if not yet everything, at least something very important about physical reality, namely the sizes and shapes of the objects that inhabit it.

The Pythagoreans influenced the world by the simple expression:

“All is number.” – Pythagoras

What Did Pythagoras mean by this famous motto “All is Number?”

Is it possible to listen to this message today afresh, with Pythagorean ears? What teaching does the tetractys offer a Freemason?

The Tetractys: A Masonic Lecture by William Preston (1772)

FINAL Tetractys PrestonFreemasons in earlier times thought highly of Pythagorean philosophy. Brother Manly Palmer Hall, a 33° Mason dedicated an entire chapter in his work “The Secret Teachings of all Ages” to the both mystical and philosophical qualities of Pythagorean numbers.

Hall wrote:

“The ten dots, or Tetractys of Pythagoras, was a symbol of the greatest importance, for to the discerning mind it revealed the mystery of universal nature.”

Hall states that if one examines the tetractys symbolically a wealth of otherwise hidden wisdom begins to reveal itself.

The Prestonian Lectures (1772) give us further insight into some of the possible masonic thinking on the tetractys in the 1800’s.  It was the subject in one of the series of lectures written by Brother William Preston for instruction and education of the Lodge members.

An excerpt of the Lecture (1772) goes as follows:

“The Pythagorean philosophers and their ancestors considered a Tetractys or No. 4:

  • 1st as containing the decad;
  • 2nd as completing an entire and perfect triangle;
  • 3rd as comprising the 4 great principles of arithmetic and geometry;
  • 4th as representing in its several points the 4 elements of Air, Fire, Water and Earth, and collectively the whole system of the universe;
  • Lastly as separately typifying the 4 external principles of existence, generation, emanation, creation and preservation, thence collectively denoting the Great Architect of the Universe Wherefore to swear by the Tetractys was their most sacred and inviolate oath.”

In other words, it is taught to Freemasons that a four-fold pattern permeates the natural world, examples of which are the point, line, surface and solid and the four elements earth, water, air and fire. Musically they represent the perfect consonants: the unison, the octave, the fifth and the fourth.

The Divine Creator in Freemasonry is sometimes referred to as The Great “Architect” or Grand “Geometrician” always building the universe through the creative tools of the geometer. Tetractys itself can be interpreted as a divine blueprint of creation. Grand Geometrician

Some say that Pythagoras and his successors had two ways of teaching, one for the profane, and one for the initiated. The first was clear and unveiled, the second was symbolic and enigmatic. In order to achieve mastery of this universe, a person has to discover the veiled meaning of numbers hidden in all things.

I have often wondered if we could hypothetically peer into the mind of the Grand Geometrician, and the veil was lifted, what design would we see?

The Grand Design

Perhaps we would see how the Master Builder has ordered all things by measure and number and weight. Throughout the structure of the universe the properties of number are manifested. Geometry is fundamental to the work of the masonic builders. It is engaged with the first configurations of the Plan upon which the form is erected and the idea materialized.

Examining numbers symbolically, they represent more than quantities; they also have qualities. Brother H. P. Blavatsky in the “Secret Doctrine” tells us the numbers are entities. They are mysterious. They are essential to all forms. They are to be found in the realm of essential consciousness. They are clues to our evolution.

Blavatsky emphasizes that the study of numbers is not only a way of understanding nature, but it is also a means of turning the mind away from the physical world which Pythagoras held to be transitory and unreal, leading to the contemplation of the “real.”

Personally, I find that the masonic teachings in all their many symbolic forms a good way to study numbers. The reason I continually come back to Pythagorean philosophy is the tradition of music theory. In music, the Divine patterns of the Grand Geometrician are expressed in musical ratios. Harmony through sound, therefore, can be applied to all phenomena of nature, even going so far as to demonstrate the harmonic relationship of the planets, constellations, elements and everything, really. The reason being that all life vibrates, like the string. Tetractys Portal

Why do Freemasons connect the dots? Like many symbols, the tetractys can lead a craftsman down a rabbit hole of self-discovery. By rabbit hole, I mean a portal into a mysterious and infinite wonderland of formulas filled with beauty, confusion and intrigue – a place to encounter all sorts of adventures with concepts beyond our wildest dreams that keeps us coming back for more.

“The more deeply we study the processes of nature the greater in every direction becomes our admiration for the wonderful work of Him who made it all.”

– C.W. Leadbeader


Note: The full Prestonian Lecture on the Tetractys and Masonic Geometry can be referenced in Ars Quatuor Coronatorum, Volume 83.