The Kabbalah – the tetractys seen differently
Ten sephirot in four worlds, ten "numbers without anything" in the Book of Creation, and 231 gates from 22 letters that form a 22-gon simplex.
Anyone who knows the basics of the Kabbalah knows the tree of life with its ten sephirot in four worlds. In numbers: ten sephirot spread over four worlds, like the tetractys, 1 + 2 + 3 + 4 = 10. The tree shows how ten principles emanate from one divine cause, pour forth and take shape.
The tree of life with the ten sephirot. The colours assign them to the four worlds, as the tetractys can be read in it: 1 in Atziluth, 2 in Beriah, 3 in Yetzirah, 4 in Assiah.
The Kabbalah asks how the human being is bound to God. Three foundational writings belong to it: the Sefer Yetzirah, the “Book of Creation”, the oldest, the Zohar, the “Book of Splendour”, and the teaching of the Merkabah, the divine throne-chariot from the vision of Ezekiel. Many later books on the mysticism of letters and numbers, on magic or Christian Kabbalah contradict each other on important points and should be read with caution. Anyone looking for firm ground in this thicket finds it in number and figure, because they can be checked.
Two sides of one coin
The Sefer Yetzirah and the Pythagorean tetractys are two sides of the same coin: both claim to describe the nature of creation. As early as 1494 Johannes Reuchlin, in De verbo mirifico (On the wonder-working word), compared the four-letter name of God YHWH with the tetractys. In some depictions, for example by Papus, the Tetragrammaton is indeed arranged in a triangle, exactly like the triangle of ten points: Y, YH, YHW, YHWH, ten letters in all.
Ten numbers without anything
The Sefer Yetzirah describes creation through “thirty-two hidden paths of wisdom”: ten numbers “without anything” (the sephirot) and the 22 letters of the Hebrew alphabet, divided into three mothers, seven doubles and twelve simples. The text stresses the ten emphatically: ten and not nine, ten and not eleven. It compares them with the ten fingers, five opposite five. Their measure is ten, yet they are without limit. Their end is bound to their beginning like the flame to the coal. And it calls on the reader: know, count and draw.
This is no mere coincidence. The geometry of the circle as the idea of creation, from the spiritual into the material, has a long tradition in the secret teachings. Simplexes and fractal polygons, crystal lattices and sphere packings: this is the geometry of perfect symmetry, of circle and sphere. In their original form, so the thesis goes, these connections also stood godparent to the Kabbalah. The spatial lattice of the densest sphere packing probably also lies behind the creation account of the Sefer Yetzirah (see Gap-free filling of space): there first a tetrahedron, then an octahedron and finally a cube become visible.
The 32 paths in infinite space
On first reading, the account of creation in the Sefer Yetzirah seems naive, almost meaningless. Read it several times and it becomes clear that directions in space are being staked out, with geometric precision, and that only afterwards are correspondences from the world of experience assigned to the positions so fixed. Without space, no room for the world. Anyone who studies the geometry seriously and patiently recognises in it the crystal lattice of the densest sphere packing, Buckminster Fuller’s vector matrix, a fractal structure of tetrahedral and octahedral holes that describes the “anatomy of space”. Depending on the viewing angle its faces can only be triangles and squares, and the simplest solid of four points is the tetrahedron.
In brief, following Lazarus Goldschmidt’s translation (1894): God establishes his throne and creates from himself three further beings, spirit from spirit, water from spirit and fire from water. “Dimensions” are mentioned here. God makes room for himself with the tetractys of point, line, surface and solid, the four together corresponding to the four-letter name of God, YHWH. He then seals the six directions of space, each with a different permutation of the three letters Y, H and W:
| Number | Direction | Seal |
|---|---|---|
| 5 | height, upwards | Y H W |
| 6 | depth, downwards | Y W H |
| 7 | east, forwards | H Y W |
| 8 | west, backwards | H W Y |
| 9 | south, to the right | W Y H |
| 10 | north, to the left | W H Y |
It is remarkable that the sealing begins only at 5 and ends with 10. If six directions are to fit harmoniously into a tetrahedron, they form an octahedron: its six vertices lie exactly at the midpoints of the six tetrahedron edges. The octahedron with six vertices is thus added to the tetrahedron with four, 4 + 6 = 10. “These are the ten numbers without anything.” The tetractys again: from 1 + 2 + 3 + 4 comes the ten.
The 'primal fractal': a tetrahedron whose six edge midpoints form an octahedron. 4 + 6 = 10 vertices, as a lattice or as ten spheres.
This lattice can be continued to infinity, and that is exactly what the text stresses more than once: their measure is ten, yet they are without limit. Read as a sphere packing, the ten vertices correspond to ten spheres (see Space filling without gaps). The closeness to the Tree of Life with its ten sefirot is striking: in the centre lies the octahedral hole, seen as a hexagram, like Tiferet at the centre of the tree. The four vertices of the tetrahedron correspond to the four worlds. This “particle of ten” is, as it were, the initial spark, and one may suspect that the Tree of Life itself is just a diagram of this structure. A 19th-century book cover on the origin and content of the Kabbalah shows, above the circle with a point in the centre, the symbol of creation, an octahedron in hexagram form.
The centre, the image of the Creator, is transformed as well: from the triangle with its three vertices to the octahedron with six vertices and the centre as the seventh. An even more convincing link to the holy seven is provided by the “seven doubles”.
Seven doubles and twelve simples
Every node of this lattice, imagined as infinite, every sphere, admits two kinds of straight lines into space.
The first kind are seven lines that pass alternately through faces and vertices of the cells, that is through duality itself. Four of them alternate through triangular faces (female) and vertices (male), three alternate through square faces and vertices. And indeed the text describes the seven doubles as duality par excellence. The corresponding figure is the stella octangula, the spatial hexagram: four lines join the eight vertices of its cube, three join the six vertices of its octahedral core. Together it has 8 + 6 = 14 vertices, seven pairs, the “seven doubles”.
The stella octangula: 8 cube vertices and 6 octahedron vertices, 14 vertices in seven pairs. The four space diagonals and the three axes through the octahedron vertices are the seven doubles.
The second kind are the lines to the immediate neighbours. Twelve spheres fit exactly around a thirteenth, and their centres form a cuboctahedron. From its centre twelve lines lead simply and directly, as the crow flies, into infinite space: the “twelve simples”. The Sefer Yetzirah expressly assigns them to twelve edges, each joining two directions, north-east, south-east, upper east, lower east and so on, and says they extend to infinity and are the arms of the world.
Twelve spheres around a thirteenth. Their centres form a cuboctahedron, the twelve directions to the neighbours are the twelve simples.
All this can be illustrated in simplified form with a cube, as has often been published. The catch: that way one never learns what the seven doubles and twelve simples actually are. Building the lattice models oneself, from toothpicks for instance, is highly recommended.
The simple tetrahedron at the beginning corresponds to the three mothers, because it is the first to span the three dimensions, through 1 + 2 + 3 + 4, point, line, surface and solid, or vertex, edge and face that make the solid. From these three mothers come the seven doubles and the twelve simples, 3 + 7 + 12 = 22 letters.
The breaking of the vessels
A remarkable parallel to this fractal geometry of space is found in the Kabbalah of Isaac Luria (1534–1572), as presented for instance by the Viennese Judaic scholar Klaus Samuel Davidowicz. Luria asked: if God is perfect and everywhere, there can be no empty space. How then could anything be created from nothing? His answer is the tzimtzum: God withdrew into himself and so freed a primordial space in which creation could take place. A trace of the divine light, reshimu, remained there like the residue in an empty bottle. The first ray of light into this space created the primordial man Adam Kadmon, from whom the sefirot streamed forth. Four worlds arise, Atziluth, Beriah, Yetzirah and Assiah, assigned to the four letters of the divine name.
Vessels were to receive the light. The vessels of the three highest sefirot held, those of the seven lower ones broke under the force of the light, the breaking of the vessels. Sparks of light clung to the shards and sank down. From the shards came the other side, in which the holy sparks await redemption. Gathering them and restoring the world, tikkun, is also the task of human beings, and every commandment kept contributes to it.
The word fractal was coined by Benoît Mandelbrot from the Latin fractus, broken. Fractal structures shape the whole of nature. The breaking of the vessels as an image of a space that divides itself again and again is an attractive reading. Two views of the same lattice, once as a tetrahedron, once as a pyramid with a square base, show that both are parts of the same structure. After exactly four subdivision steps of the tetrahedron, the duality of stella octangula and cuboctahedron appears. Old depictions of the Shield of David with the twelve signs of the zodiac around it, the twelve simples, also show smaller hexagrams around the large one (see Pyramids, knights’ crosses, space lattices).
The beard of the Ancient One
Some texts hint at the right geometry only in veiled form. The Zohar says, in substance, that the highest dignity, the first sefira Keter, depends on seven formations of the skull, and that the beard of the venerable Ancient One is divided into thirteen parts. The cuboctahedron with its twelve spheres around a thirteenth fits exactly. Thought of in space, it contains four hexagons of seven spheres each, nested on four planes. The central sphere is the meditator himself, who imagines the twelve around him, like Jesus and his twelve disciples. Thirteen stands for transformation, and that is precisely the aim of the exercise. Whoever wants to internalise this geometry should meditate on models (see Kissing numbers and 24).
In summary
This spatial structure was understandably equated with the infinite universe and furnished with correspondences:
- the three mothers: the three dimensions and the three basic elements,
- the seven doubles: the seven planets,
- the twelve simples: the twelve signs of the zodiac.
The lattice consists only of triangles and squares, and so it also contains the hermetic duality of spirit and matter, 3 + 4 = 7. The products 3 × 4 and 4 × 3 correspond to the twelve disciples and the 24 elders of the Bible, for all symmetries of this lattice give 12 or 24. One may suppose that these biblical images arose from knowledge of the lattice.
What is special about the 22 lines is that they lead to infinity without bend or detour, through infinitely many nodes. From every point of this order 22 paths lead into infinity, so every point is a throne of God. It fits that from the centre of a symmetry view the cross and the hexagram become directly visible. Whoever can hold these lines in mind, engraving and permuting them, feels connected with infinity, the macrocosm. Magic circles mostly have the same 12- or 24-fold symmetry, only they are drawn on the floor as a protective circle with the magician standing in the middle. Pentagonal geometry is not part of this lattice, which is why the pentagrams stand outside the circle as protective signs (see The throne of God and the number 24).
Sense and nonsense of number mysticism
The geometric correspondences in the Sefer Yetzirah show that this is not about nebulous number mysticism and letter magic whose meaning must remain veiled. Nor does it make sense to cling to “mystical numbers” without really knowing them. Put plainly: the earlier a number stands on the number line, and the more so if it is a prime, the more important it is by nature, because all later numbers are made of their predecessors and even later primes are put in their places by them. Those who call this mystical often simply mean: not understood. If number mysticism has any relevance, it lies in the combinatorics that necessarily follow from the first numbers, and that leads straight to number theory, primes and their geometric counterparts, to what the Platonists called the tetractys and the Kabbalists the four-letter name of God.
In Papus’ book on the Kabbalah, the name YHWH is given right at the start with the numerical values 10 + 5 + 6 + 5, and these values are then pressed into ever new schemes and permuted until hardly any meaning is left. What significance the numbers have in number theory remains unclear. The start figures of the tetractys, by contrast, show four cases of how a number relates to a divisor (see The key to the tetractys):
- polygon: the basic case,
- star polygon: the number is not divisible by the divisor,
- multiple polygon: the number is divisible by the divisor,
- multiple star polygon: not divisible, but with a common divisor greater than 1.
Papus also interprets the four letters: the first three stand for the trinity of the act of creation, the fourth for visible creation itself, which has no meaning of its own any more. This matches the tetractys astonishingly well: four is no longer a prime. Tracing the parallels in detail is left to each reader.
Elsewhere Papus explains that the three highest sefirot are, as it were, the thrones of the divine persons, but everything is carried out by the other seven, which the Kabbalists therefore call the “sefirot of construction”. This agrees with the Sefer Yetzirah: YHWH creates three ministering angels and then seals the six directions, in whose centre, the seventh point, he builds his holy palace. Into the space of the tetrahedron, the Tetragrammaton, the octahedron with six vertices and the centre as the seventh is fitted, constructed as it were. With each further subdivision step, 12- and 24-fold symmetries arise around every centre.
Space has its counterpart in number space. Up to 1 × 2 × 3 × 4 = 24 every number of the form 6n ± 1 is prime, 5, 7, 11, 13, 17, 19 and 23, and the twin primes are present without a gap. Only 25 = 5 × 5 breaks the series. The first numbers 1, 2 and 3 secure the rhythm of six of the twin primes to infinity (see The order of the primes). This corresponds to the fact that plane and space can only be filled with triangles, squares and hexagons, which form 12- and 24-fold symmetries around a centre.
Papus also shows a depiction in which the word YHWH is formed of 24 spheres with three prongs each, the 24 elders of the Apocalypse. 24 × 3 gives the 72 letters from which practical Kabbalah forms the names of the 72 genii. They come from three verses in Exodus (14:19–21), each of which has exactly 72 letters in Hebrew. This diagram, however, has no direct connection with the lattice structure described in the Sefer Yetzirah.
The golem and the 231 gates
Besides the tree of life there is a second Kabbalistic practice linked to the idea of creation: the making of a golem, a servant being of clay. The traditions describe a strictly regulated ritual: the adept works with at least two companions, purifies himself in body and mind, wears white garments and works at night, when the darkness recalls the time before creation. The golem is formed from untouched earth and fresh spring water. For bringing it to life there are highly contradictory instructions, with names of God of four, twelve, 24, 48 or 72 letters. The best-known golem legend is that of Rabbi Loew in Prague.
To bring the individual limbs to life the adepts drew on the Sefer Yetzirah. It says the 22 letters were engraved, hewn, weighed, permuted and combined. Because Hebrew writes no vowels, the verse can also be read as a command: engrave them, hew them, permute them! The Sefer Yetzirah calls chaos the primal substance of creation. Into this chaos it places 231 gates, “in a circle like a wall”: all possible pairings of two of the 22 letters, without repetition.
The ritual consisted in visualising each letter (engraving), separating it from other thoughts (hewing) and joining it with all the others (permuting): first aleph with every letter, then bet with every letter, and so on round the whole circle, sung, quickly and without pause. Each letter belongs to a part of the body, and limb by limb the golem was meant to come into being.
The diagram for this practice is nothing other than a 22-gon simplex:
The 22-gon simplex: 22 points, each joined to every other. That makes 231 lines, the 231 gates of the Sefer Yetzirah, and 231 is the triangular number T(21).
A tongue-in-cheek interpretation: the dutiful adept will intone, engrave and permute until he pauses and reflects. If he is awake in mind, he recognises that the combinatorics of this practice contain a deep truth. He need only take seriously what the Sefer Yetzirah says: know, count and draw! Then he arrives at the insights that matter here: number theory + geometry = philosophy.
A traditional Kabbalistic amulet combines pentagram, hexagram and octagram, figures that play a special part in this geometry.