The Holy Tetraktys, Pythagoreanism & Freemasonry

The Tetraktys of the Pythagoreans consists of four tiers or stages, each having both geometrical and meta-geometrical significance. This dynamic is in keeping with the operative and speculative denominations of Freemasonry, as well as the Craft’s veneration for Geometry, which is defined in Masonic ritual as “the first and noblest of sciences, [and] the basis upon which the superstructure of Masonry is erected.” In its Masonic context, the Tetraktys may be encountered everywhere from the old Prestonian lectures to its appearance on the jewels and in the attendant symbolism of certain (primarily Continental) jurisdictions.

What is the Tetraktys?

The Tetraktys (Greek: τετρακτύς) is a mystical figure first utilized in the teachings of the Craft’s “ancient friend and Brother” Pythagoras of Samos. The philosopher and his followers held the symbol in such high regard that they deemed it “holy” and were said to have taken their oaths upon it. [Mackey, An Encyclopedia of Freemasonry and its Kindred Sciences, Everts & Co., 1884; Entry: “Tetraktys”]

Bless us, divine number, thou who generated gods and men! O holy, holy Tetraktys, thou that containest the root and source of the eternally flowing creation! For the divine number begins with the profound, pure unity until it comes to the holy four; then it begets the mother of all, the all-comprising, all-bounding, the first-born, the never-swerving, the never-tiring holy ten, the keyholder of all. [Dantzig, Number: The Language of Science, Macmillan, 1930, p. 42]

The Tetraktys consists of ten regularly spaced points arranged in an upward-pointing equilateral triangle, having one point on the top tier, two on the second, three on the third and four at the bottom. It is a model illustrating the organization of space through the emanations of the Grand Geometer: from a point, to a line, to a superficies and, finally, to a solid. We will presently turn our attention to the first cosmogonical stage of this enigmatic figure: the point.

The Point.

From the geometrical (operative) perspective, the point is the beginning, or origin, of all constructions. It describes a position without dimension; a location in space without magnitude. A point is located in space by reference to the three cartesian coordinates x, y and z, of which it is the theoretical vertex, or origin. The abscissa (x axis) runs side-to-side; the ordinate (y axis) runs front-to-back; and the applicate (z axis) runs up-and-down. Taken together, the three Working Tools of the Fellowcraft – the plumb, the level and the square – may be seen to ‘try’ each of these dimensional coordinates. The plumb proves the applicate in that it tries a vertical line; the level proves the abscissa in that it tries the horizontal; and the square proves the ordinate, in that it bisects the x axis at the origin, thereby forming a 90° angle, which is of course a square.

Metaphysically – or, more precisely, meta-geometrically – the point is the singularity from whence spatial dimension comes into being – this being the essence of cosmogony, illustrating the origin of the very cosmos. In the Pythagorean tradition, the point, or Monad, symbolizes God, as it was called the “First,” the “Seed” and the “Builder.” We are also reminded of the famous quote from the 12th century theologian, Alain de Lille (though Rabelais attributed it to Hermes Trismegistus), who said, “God is an intelligible sphere, whose center is everywhere and whose circumference is nowhere.” In Freemasonry, we encounter this concept in reference to the ‘Point within a Circle’ diagram, wherein the point represents the individual Mason.

The Line.

In the domain of geometry, a line is described as a one-dimensional, so-called ‘primitive object’ having no width or depth – it is simply defined by length. Technically, a line is infinite and has no curvature; however, in the context of the Tetraktys, we are specifically dealing with a line segment connecting two points, therefore representing the Dyad.

It is important to note that the Pythagorean worldview was/is holistic and integrated in terms of quantity and quality. Numbers are not merely quantitative counting integers – they also retain the numinous, qualitative element. Yes, one could say something like “Jane has two apples in her basket,” in reference to the quantity two, but the number two also denotes a particular meta-numerical dyadic quality.

In the Pythagorean tradition, the Dyad was symbolically used to convey the concepts of “twoness” and “otherness.” Whereas the point represented God, the Monad, the Pythagoreans associated the Dyad with matter and the general concept of duality. This would apply to all manner of binary opposites, such as: light & darkness, active & passive, positive & negative, abstract & concrete, objective & subjective, general & particular, cosmos & chaos, etc.

The quantity/quality dynamic is mirrored by the operative and speculative denominations of the Craft. Whereas the stonemason operatively utilizes the working tools in the building of a physical temple according to the quantitative measurements of a given design; the Freemason speculatively applies these implements in the building of a Metaphysical Temple according to (and, to some degree, shaping) the quality of his character. It would be more accurate to say that the work of the Freemason is holistic and integrated – in the manner of the Pythagoreans – as it encompasses both the abstract and the concrete.

Freemasons labor in their own sort of quarries, raising and squaring good men who will, through the calibrating application of the working tools, be fitted as living stones for “that house not made with hands, eternal in the Heavens.” This is a project – perhaps the central project of the Craft – that necessarily unites the quantitative and the qualitative; the abstract and the concrete; the operative and the speculative; and all manner of dyadic opposites.

*Pythagoras, the Tetrahedron and his Theorem.*

The Plane.

Geometrically, a plane is described as a flat, Euclidean, two-dimensional surface, or superficies. If a third point is positioned anywhere in relation to the two points of the line (other than on their axis), a plane is constructed. The simplest plane consists of three points, therefore representing the Triad.

According to Pythagorean number symbolism, the Triad is the oldest and ‘first born’ number because the equilateral triangle (which consists of three points) is the first polygon to emerge from the womb-like vesica piscis (this is, of course, illustrated in Euclid’s 1st Proposition in Book I of his Elements). The Pythagoreans also noted that the number three is the only number that is equal to the sum of the two preceding numbers. To them, the Triad was considered perfect (consisting of a beginning, a middle and an end; a past, a present and a future; a youth, a manhood and an old age) and symbolized proportion, harmony and perfection.

As we consider the three points that form a plane from a symbolic and meta-numerical perspective, we are reminded of both the Neoplatonic and Christian Hypostases. According to the philosophical doctrines of the Neoplatonists, the higher spiritual principles were stratified into the hypostasis of the One, the Intellect and the Soul, as laid out in the work of Plotinus. This model was adopted and modified in Christian theology as the Father, the Son and the Holy Ghost.

Speculatively, or metaphysically, we encounter this triadic distinction in what has been called the “triune nature of man.” This doctrine differentiates the body, mind and spirit as being distinct components of being. In ancient Greek philosophy, we see this triad represented by the soma, psyche and pneuma (Latin: corpus, anima and spiritus). The body is, of course, the physical vessel; the mind is the numinous product of the brain, consciousness and thought; and the spirit is the animating life principle, as well as the seat of the emotions and character.

This triad also has a correlative in alchemical theory. Paracelsus, in his Liber Paragranum (1565), was looking for an alternative to the humoral theory of Hippocrates and Galen, which was popular in the medicine of his day. As an alternate, he described with what is known as the Tria Prima (“three primes”): salt, sulfur and mercury, representing the Three Philosophical Principles of spagyric alchemy. Salt corresponds to the body; sulfur to the soul; and mercury to the spirit.

*Alchemy’s Three Philosophical Principles.*

The Solid.

When the point of the Tetraktys is joined by another, a line segment is formed. Adding a third point creates a two-dimensional plane. Four points, joined together by their vertices (corners), are necessary to produce the simplest regular, convex polyhedron: the tetrahedron, or triangular pyramid. This is the segue into solid geometry, which is the geometry of three-dimensional space. The introduction of the fourth point adds depth to the length and width of planar geometrical figures. Concepts such as volume – the measurement of occupied space – now become applicable.

A solid is generally divided into two types: polyhedra and non-polyhedra. Polyhedra necessarily have flat faces (superficies) and include shapes such as cubes, pyramids and prisms. Whereas non-polyhedra have at least one face that is not flat; this includes shapes such as spheres, cylinders and cones.

In the philosophical context, the most significant convex, regular polyhedra are the five Platonic solids: the tetrahedron, the hexahedron (cube), the octahedron, the dodecahedron and the icosahedron. Though they are referred to as being “Platonic,” the Neoplatonic philosopher Proclus credits their discovery to Pythagoras. These shapes have been studied by geometers for thousands of years. Plato, in his Timaeus, equates each of the solids to a classical element: fire to the tetrahedron; earth to the hexahedron; air to the octahedron; water to the icosahedron; and, finally, aether (the quintessence, or “fifth element”) to the dodecahedron. These fit neatly into Aristotle’s cosmological model, which posited the four Empedoclean elements in the sphere of the Earth (the sublunary sphere, “below the Moon”) and aether as being the composition of the stars and planets, due to their unchangeability and incorruptibility.

Again, the Tetraktys, as a cosmological model, illustrates how things come into being, or how they manifest intelligibly. Bringing this notion into the sphere of Masonry, we note that the operative Mason studies the two-dimensional plans from the architect’s drafting table and executes them in the building of a three-dimensional edifice. Analogously, the speculative Freemason contemplates the metaphysical designs on the Grand Geometer’s trestleboard and actualizes them in the building of his individual mnemonic Temple (through memory work) as well as the egregorical Temple (through fraternalization and labor) to which all Freemasons energetically contribute.

Throughout this article, we have contemplated the great Tetraktys and used it to further investigate the operative and speculative denominations in Freemasonry. Most importantly, we have shown that the Pythagorean marriage of the quantifiable and the qualifiable, as an actionable project, did not wholly end over 2,000 years ago with the dissolution of their sects; this notion periodically surfaces when a fulcrum is reached between the arts and the sciences, such as was achieved during the early Renaissance, and it survives in societies such as Freemasonry.