A Child’s Garden of Tessellae

The far-reaching symbolism of the Tetractys (The Tetracys, Part 2)

In addition to a more-than-passing familiarity with Pythagoras (Πῡθαγόρᾱς) and mysticism in the ancient Mediterranean world, I am also well versed in hexagons. You might be asking why I say hexagons when the tetractys is triangular. Well, the lattice for the equilateral triangle and the hexagon are the same—as a regular hexagon is made up of six equilateral triangles and the lattice points for both define a hexagonal nucleus; put another way, a hexagon is a truncated triangle—and the Pythagorean symbol is just a discrete set of points within this lattice:

I know the pattern well. It was one that you’d get by staring at the tessellae of a midcentury bathroom floor—a favorite activity of mine. When you move out from the basic tile, you get a triangle, then a rose; a hex made of seven hexes (i.e. six corners and a center).

This rose was also the shape of Honeycomb cereal, which I would painstakingly nibble to make stars, triangles, individual cells. From the hex rose, if you add three corners, you’re back to a triangle, another three and a flower with more pronounced petals, or what I’d later come to know as a Star of David.

When I was seven, I unknowingly encountered the tetractys at a friend’s cub scout meeting, made of 10 pennies and presented together with the fiction that it was a squadron of jets in formation that needed to reverse course and resume the same formation while only changing the positions of three planes. To me, the pennies were simply inexact representations of the hexagons I loved to play with, so of course I knew what to do.

It appeared in other places too, bowling alleys, real honeycombs, cut paper snowflakes, Chinese Checkers, rock candy, chicken wire, the quartz crystals in my brother’s rock collection. Of these the tenpins pattern and the colored corners of the Chinese Checkers board are true examples of the tetractys, as is the baryon decuplet (the Chinese Checkers field can be thought of as another set of six tetractys pointing inward and defining a hexagon). And when I got into strategy board games, there were those bathroom tiles again, now overlaying terrain maps. And then there were Japanese decorative motifs where the hexagon represents a scute from a tortoise’s carapace (亀甲, kikkō).

And now Eco tells me this is a sacred symbol of the Pythagoreans:

The Tetraktys is the symbolic figure by which Pythagoreans swore their oaths, and it represents a perfect and exemplary reduction of the numerical to the spatial and of the arithmetical to the geometrical. Each side of this triangle is formed by four points and at its center there stands a sole point, unity, from which all other numbers are generated.

Unity is one of Pythagoras’ influential principles of numbers, in this case, the number one. It also represents deity, which has no parts. That is, it is indivisible. It also echoes the “one” at the center of the Adonai Square, and, indeed, that figure is related to this one via the dissemination of Pythagorean ideas throughout the Mediterranean, so much so that the tetractys emblazoned with the Tetragrammaton has become a Kabbalist symbol as well. In addition, one is the origin of all things, as Eco mentions. Each of the three corners can also be thought of as representing this same unity, which allows us to overlay the upsilon (Υ, ὖ ψιλόν). This letter is known as the Pythagorean or Samian letter (Samian as Pythagoras hailed from the island of Samos, Σάμος), symbolizing the branching path that leads to earthly or divine wisdom—the path begins (at whichever corner) and branches at the center point:

The influential principles continue, counting across the rows, where two is diversity, and therefore disorder, the principle of strife and all evil. This should not be mistaken for in any way being about race, or anything else like that, but reflected as in the Berber saying, “A devil takes one and makes two; a saint takes two and makes one.” The next row is three, which is perfect harmony, or the union of unity and diversity. Both principles reflect again the upsilon symbology. One can see that the image below, DaVinci’s representation of a “tetrahedron with empty planes” in perspective closely resembles the upsilon tetractys. This set of numbers makes up the triangle itself and also symbolizes the Pythagorean idea of a threefold god: the beginning, middle, and end of all things. This older concept of a divine trinity can also be seen in the Hindu Trimūrti, wherein there is a triad of deities, Brahma the creator, Vishnu the preserver, and Shiva the destroyer, all ultimately aspects of a single avatar, Dattatreya. Furthermore, the soul has three vehicles: the ethereal, which is luminous and celestial, in which the soul resides in a state of bliss in the stars; the luminous, which suffers the punishment of sin after death; and between those two, the terrestrial, which is the vehicle it occupies on this earth.

The final line of the tetractys is four—Eco continues:

Four thus becomes synonymous with strength, justice and solidity; the triangle formed by the series of four numbers is and remains a symbol of perfect equality.

As an influential principle, four represents perfection, also expressed as cosmos. One of the ideas most central to the symbol is that the sum of these first four numbers is ten (1 + 2 + 3 + 4 = 10), the basis of all numbers. Four is also the first square (2 x 2 = 4).

These rows further represented geometrical ideas as points: the first row being a single point has zero dimensions. The second is a pair of points that define a line, the third is a plane—a two dimensional figure requiring three points. The fourth line of four points creates the simplest solid: a tetrahedron, and the tetrahedron is the essential form of the caltrop in this site’s icon. These four lines further symbolize the four classical elements: fire, air, water, and earth, and therefore a whole series of associations: the four seasons, the four cardinal directions, the set of simple bodies (tetrahedron, octahedron, icosahedron, cube), the ages of man, etc. Also note that the ancient symbols for the elements were a set of triangles and inverted triangles.

Further, the rows can be read musically as ratios: 1:1—the fundamental, 2:1—the octave, 3:2—the fifth, and 4:3—the fourth. These are the basic intervals of the Pythagorean scales, and also form the basis of the concept of the music of the spheres. Also known as musica universalis, this is the idea that the proportions and movements of celestial bodies create a kind of divine mathematical harmony—not, as is often mistakenly thought, literal, audible music.

Turning again to Eco:

The sum of the points that form the triangle is the number ten, and with the first ten numbers all possible numbers can be expressed. If number is the essence of the universe, then the Tetraktys (or decade) represents a condensation of all universal wisdom, all numbers, and all possible numerical operations.

And this echoes the Aristotle (Ἀριστοτέλης) quote from Part 1, of which, we can be sure Eco was aware.

The tetractys has found its way into art and architecture down through the ages, some even claim it to be the basis of the Masonic symbol depicted on the reverse of the Great Seal of the United States: an incomplete pyramid surmounted by the Eye of Providence. It also forms the basis of a layout for Tarot card readings, as well as a syllabic poetic form. The syllabic values for the lines are 1, 2, 3, 4, 10. Here’s an example penned by Ray Stebbing, the form’s creator:

us all greatly.
Volatile, big-bodied tots are selfish.

I’ll leave you with one final fun fact: in the gematria, the value of the word τετρακτύς yields the value 1,234.

Read Previous Articles in this Series

Part 1: Eco, Pythagoras, and the Mystic

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