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. 
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)
Freemasons 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. 
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. 
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.
