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A Brief Introduction into Sacred Geometry

Updated: Oct 7, 2021

By: WB Daniel Genchi Gila Valley Lodge 9 (F&AM) Florence, AZ


“Let none enter here who are ignorant of Geometry.” This is said to have been inscribed above the entrance of Plato’s Academy.  A quote which this author found, at the very least, to be interesting and at the very most obsessively infatuating.  I must warn on the onset before we begin on this journey down the rabbit hole, you must bear in mind that this topic is only the beginning.  It is the scratch on the surface. 

What you are getting is a very brief glimpse into the endless possibilities which any amount of devotion into the study and understanding of Geometry can provide.   The culminations of ideas which have sprouted from this obsession have [for me] been more than eye opening.  It is hoped that at its end, you will be inspired to ask further questions and seek your own light.  All I offer is a glimpse into the rabbit hole.  How deep it goes (as is in most of Masonry) is up to you.

Before we begin this journey together, it is important we take some time to reflect.  Remember back to your initiation. Think back to what it was that brought you to Masonry.  Bring back from the depths of your mind when it was that you first became a Mason.  For what questions did you seek answers? Were you looking for the meaning behind what is seen and experienced? When did you first contemplate the possibility that there might be some “perfect order” to the world as you perceive it?  Close your eyes for an instant and think back to that moment.  Hold that thought in your head, and then reflect back to that very moment in time as we present these ideas. 

It is important that you take what you learn and apply it in a way which fits your path.

What I am offering in this paper is not meant to be taken as law.  It is not intended to impress upon you any unnatural truth or certainties.  Rather, it is meant to bring to your mind those longing questions, that fruitful endeavor you embarked upon when you knocked upon the door of Freemasonry.  It is an attempt to relate to you the importance of your journey founded by the Great Civilizations of Antiquity and shaped by your own thirst for knowledge.

All I ask is that you keep an open and receptive mind.  Remembering that all which is presented to your view is subject to question and interpretation.  The author would not have it any other way. I encourage you, above all else, to question everything..

Into the Rabbit Hole:

When I began researching topics for this paper, the lack of noble topics was in no short supply. Neither was my abundance of questions.  As many things in Masonry often do the more I sought to answer one question, the answer(s) I found would quickly become replaced by even deeper questions.  With a never ending web of questions, my mind began to travel.  I began to reflect back on the various ceremonies of my degrees.  I read and recited lectures and monologues. I poured through article after article. I devoured book after book.  If there was an end to the World Wide Web, then it seemed as if I was determined to find it.

I found myself reading about alternative history (history outside and sometimes contrary to conventional historical accounts).  I read about the various world philosophies and the civilizations of antiquity.  I read about the Egyptians and their mathematical genius who gave way to the pyramids, the obelisk, farming, and irrigation.  I read of the Mayans and their elaborate calendar (which remains the most accurate calendar to date) as well as their architecturally advanced Temples which considered and accounted for the alignment of the stars and planets as part of their design.  The Greeks, the Romans, and the Persians for their ingenuity in the Sciences and Arts as well as their Military and Humanitarian achievements. 

All of these cultures, which were ages ahead of their times and well deserving of in-depth discussion, for the sake of time and ink; this paper makes no attempt to dive into. These topics would be better served in a separate paper of their own.

In all of this however, there seemed to be a question I could begin to wrap my brain around. What did these civilizations have in common, and how did they relate to Freemasonry and my personal quest as a Mason?

It was during my attempt to answer this question when I came across this quote:  “Let none enter here who are ignorant of Geometry.” At first read, this quote seemed to be innocent enough. It was direct and to the point.  I mean, of course! Why would anyone ignorant of Geometry enter into Plato’s Academy anyway? What purpose would it serve to sit in a room where topics were being discussed of which you have no concept and are unable to contribute? I will admit at first glance, it seemed to be pretty cut and dry.  It wasn’t until I came back to this quote that it began to look a bit peculiar.

“Let none enter here who are ignorant of Geometry.”  What was Plato really saying?  There seemed to be something deeper to this than what I was initially reading.  I began to ask myself: where would someone wanting to learn about Geometry go if he wasn’t allowed in the one place that offered this knowledge? 

It seemed as though there must be some significance or some deeper meaning I wasn’t seeing.  How would you have knowledge of a topic as in-depth as Geometry prior to entering the place to receive that light?  Oddly enough, this seemed to have struck a familiar chord with me.  I began to think back to my initiation ceremony into the Craft.

I remembered the questions asked: “Where were you first prepared to be a Mason?”… “In my heart”.  Not in the room adjacent to the lodge, as that was clearly the Second place I was prepared.  It was first “in my heart”.  Was it possible Plato was implying that you must first be a geometrician in your heart before passing through the portal of his academy? If so, then it would stand to reason, Plato was implying an understanding of a philosophical or esoteric element of Geometry, not merely the drawing of shapes and angles.  This seemed to be the most logical path and so I went with it.

Before we go any further in this paper, for the understanding of the reader, it is worth mentioning that at the very earliest appearance of human civilization we observe the presence and importance of Geometry. 

It is clearly evident that Geometry was comprehended and utilized by the ancient Master Builders, who gave the world such masterworks as the megalithic structures of ancient Europe and the Pyramids and temples of Egypt.  It is also evident that Geometry continued to be used throughout the centuries as evident in the cultures and architecture of; China, Central and South America, in pre-Colombian North America, among Native Americans, in Africa, SE Asia, Indonesia, Europe, Rome and of course in classical Greece. Having established the importance of Geometry throughout civil society, let us define Geometry the word before we attempt to define Geometry the idea.

The earliest appearance in history of the use of the term Geometry as we have come to know it was by the ancient Egyptians. The word Geometry itself means “Earth measure” This definition is generally attributed to the fact that the Egyptians would regularly use the concepts of Geometry to resurvey their farmlands, after existing boundaries were buried by the shifting dirt caused by flooding of the Nile river.

I would suggest the possibility that the idea of ‘Earth measure’ applied not only to the local measure of tracts of agricultural land in Egypt, but also on a much larger scale - literally, to the measure of the Earth itself.  When we consider the fact that the Egyptians had a much different understanding of the Earth in which they inhabited, this is quite a significant statement. 

For the sake of time, let’s propose that the Egyptians believed their Earth to have encompassed more than what can be measured by the five senses (or six senses as mystic philosophy would state).  Let’s also concede that the study of Geometry in their eyes was a noble and essential aspect of ascension to God.  We see this within our own Masonic Ritual through the lecture of the Fellow Craft degree.

The Fellow Craft Lecture states:

“If we consider the symmetry and order which govern all the works of creation, we must admit that geometry pervades the universe…By geometry we may curiously trace nature through her various windings to her most concealed recesses; by it we discover how the planets move in their respective orbits and demonstrate their various revolutions; by it we account for the return of the seasons and the variety of the scenes which each season displays to the discerning eye……By it we discover the power, wisdom and goodness of the Grand Artificer of the Universe and view with delight the proportions which connect the vast machine…”

We have all heard this lecture; some of us have even recited it.  The lecture of the Fellow craft teaches us of the origins of Architecture as well as its importance to Freemasons. It defines for us; the relevance of Geometry as a guiding force to union and understanding with the unseen Architect of the Universe. More importantly, we are taught through this (and every other Masonic Lecture) that symbols have the power to conceal as well as reveal their meaning to the individual Mason.  Just as the 24 inch gauge can be used for the measuring and lying of the work, it can also be used as a symbol of dividing our time. It stands to question: If our symbols can conceal as well as reveal then what is to say that our ritual does not?

The Duality of Geometry:

I am sure that most of us tend to think of geometry as a relatively dry, if not altogether boring, subject remembered from our middle school years.  Full of axioms, definitions, postulates, and proofs dating back to the methodology of Euclid’s Elements.  Which, in and of itself, is a brilliant exposition of logical thinking and mental training, but not the most thrilling read you might take up on your Friday night. 

While modern academics takes the approach to the study of geometry as the very embodiment of rationalism and left brain intellectual processes (which indeed it is), it has neglected the right brain; the intuitive, artistic dimension of the subject.  In other words, Geometry as we have come to know it has neglected the esoteric aspect it was intended to include at its inception.  This esoteric aspect of geometry is what is often referred to as Sacred Geometry.

“Let none enter here who are ignorant of Geometry.” Could Sacred Geometry have been what Plato was talking about?  If so, then why haven’t I heard of it? Is this the aspect of Masonic ritual which was concealed? Always intending for the initiate to dive into the deeper meaning of the lecture? Is this what the proficiency is really meant to engage?  So many more questions!

To answer some of these, it is important to first have a very brief understanding of the cultures which have given way to the ritual of Freemasons. There are many works written by scholars on this topic.  I will give you but a brief history.

The Belief that God created the universe according to geometric plans has ancient origins. Historians have shown, dating back to the Egyptians, Man’s acute awareness of God’s hand in the world he encounters.  The Egyptians were well versed in mathematics and used that knowledge to “decode” what they were able to experience.  They believed that there was a power or more accurately a vibration associated with numbers, shapes and colors. 

They observed their five senses and believed they could achieve additional senses, at least one more (the sixth sense), through meditation and prayer.  They studied the symmetry in nature and attempted to replicate it in their architecture. 

This information was considered to be very valuable and as such was privy only to the Pharaoh and his highest priests.  The Egyptians considered this “Sacred Knowledge” so holy that it was punishable by death to divulge these secrets outside of their temples reminiscent of our obligation penalties. We see that this same idea (that of sacred knowledge taught in small guilds) flourishes within Ancient Greece. The understanding of Sacred Geometry is more than evident with the teachings of Pythagoras, Plato, and Euclid. 

Pythagoras described geometry as visual music. Music is created by applying laws of frequency and sound in certain ways. States of harmonic resonance are produced when frequencies are combined in ways that are in unison with universal law.  These same laws can be applied to produce visual harmony (Color and shape). Instead of frequency and sound it is angle and shape that are combined to produce visual symphonies that show the harmonic unification of diversity. Sacred geometric ratios applied to music give it healing powers to harmonize a body that is out of balance. This was believed by the Pythagoreans and now modern science is finding this to have elements of truth.

Masons are perhaps more than familiar with the Pythagorean theorem ( a² + b² = c² ) or the 47th Problem of Euclid, but they rarely understand the significance it plays in what the Greeks referred to as the fifth element or Ether (a concept which quantum physics has come to define as the unified field).  The Greeks also laid the groundwork for the concept of the atom, the four phases of matter, the properties of water, and alchemy which has given the basis for the Scientific Method which is still in use today. 

This information was taught to initiates admitted into Universities, who were progressed through 3 degrees of learning (again another correlation to our modern Masonry). The most important element taught to initiates was that certain ratios and shapes find themselves repeating throughout the universe, these geometric shapes and figures seem to radiate positivity.  An Example of this is the number PHI (or the Golden Ratio)

Leonardo da Vinci's Vitruvian Man

What is the Golden Ratio? In short, the Golden Ratio is 1.61803399. It is no coincidence this numerical value finds itself in many places in nature and the universe.  

We see it in art. Leonardo da Vinci’s Vitruvian Man is the dissection of the human body using the Golden Ratio. 

By studying nature we see that there is so much Sacred Geometry to observe; from the center of a sunflower to the internal structure of a Chambered Nautilus displaying the logarithmic spiral of the Golden Ratio.

The same spiral is found across the universe, with spiraling galaxies using the same ratios.

Examples of the Golden Ration appearing

We also see Sacred Geometry in the Tibetan Buddhist Mandalas, an art that has been carried on for centuries, symbolizing the universe and believed to create a positive environment.  We see Sacred Geometry in religious structures in the planning and construction of churches, temples, mosques. We even see it in the Labyrinth which adorns the floor in Grace Cathedral, San Francisco.

It is all around us, and it must be understood to be part of who we are and bringing to life an extreme truth to the lecture of the Fellow Craft:

…By geometry we may curiously trace nature through her various windings to her most concealed recesses; by it we discover how the planets move in their respective orbits and demonstrate their various revolutions; by it we account for the return of the seasons and the variety of the scenes which each season displays to the discerning eye…

It doesn’t take long to see that there may be more to our ritual than what is merely there.  Through the study of Geometry we are able to understand visible light. We are able to understand the mathematical properties of sound, smell, and touch. Each of these understandings expanding upon the last. 

We experience the concept of 3-5-7.  The concept of 3 yields the Triangle or platonic solid tetrahedron. We get the family trinity (father, mother, baby). We also see our 3 degrees of Masonry, the 3 pillar officers, and the 3 tenants of Freemasonry.  Sacred Geometry finds its way into Alchemy as well.

Through the concept of 5 we receive the pentagon or platonic Solid dodecahedron. We find the 5 elements (Air, Fire, Earth, Water, Ether) which we relate to our 5 senses (Touch, Taste, Sound, Sight, Smell). 

The concept of 7 gives us the 7 Chakras, the 7 liberal arts and Sciences, the 7 notes in a musical Major scale, the seven colours of the visible spectrum (a rainbow) and the 7 days of the week.  We see symmetry in so many aspects of our daily lives and never really give a thought to the role which Geometry may have played.

How does this impact Freemasonry? What does this mean to me and my quest?

Man has, since the dawn of civilization, attempted to find his purpose.  Freemasonry is a modern amalgamation of these ancient truths, secret knowledge, and Sacred teachings.

In the lodges of old, as in Plato’s Academy, the tools of Geometry were simply an unmarked strait edge and a pair of compasses (more recently the Square and Compasses). With those two tools it was possible to draw straight lines and circles, or arcs of circles.  Out of the combination of straight lines and arcs we are able to design or replicate the divine creation we are caused to admire. 

Freemasons have for centuries held the conception of the Universe as the material expression of a hidden reality, an invisible blueprint set down by the hand of the Grand Architect of the Universe [in his Spiritual, Moral and Masonic Trestle Board].

We have learned that the Study of Geometry in past civilizations and founding cultures provided the key and the means to render visible that which is concealed from the undiscerning and untrained eye.  We can see these elements infiltrating our daily lives and lending way to newer and better understandings of ourselves and the world we inhabit. 

In Freemasonry we give these teachings under the guise of lectures and symbols but as Masons we are obliged to seek further light.  We enter into the lodge with a basic understanding of Geometry, as it is part of who we are at our core.  It is where we are initially prepared to be made a Mason.

As I informed you at the beginning of this paper, this is not meant to instill any truths upon you but rather to grant you a glimpse into the esoteric aspects of our ritual.  Sacred Geometry is an immeasurable aspect of who we are as Masons. 

Its origins and implications have ties into many facets of our civilization.  Many of which I propose to delve into in separate papers.  It’s fascinating to see that there’s so much to learn about Sacred Geometry. I hope this paper sheds some light on the topic and that it invokes within you that same desire which brought you to Masonry’s door to start.  I will leave you with this:

From a small handbook frequently given to newly initiated Freemasons we find a valuable elucidation on the meaning of Geometry:

“Geometry is an ‘exact’ science. It leaves nothing to chance. Except for its axioms, it can prove everything it teaches. It is precise. It is definite. By it we buy and sell our land, navigate our ships upon the pathless ocean, foretell eclipses, and measure time. All science rests upon mathematics, and mathematics is first and last, geometry, whether we call its extension ‘trigonometry’ or ‘differential calculus’ or any other name. Geometry is the ultimate fact we have won out of a puzzling universe….There are no ultimate facts of which the human mind can take cognizance which are more certain, more fundamental, than the facts of geometry.” ~ Foreign Countries (1925) Carl H. Claudy

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