Figure 1     As well as the Tree of Life known to Kabbalists for hundreds of years, which may be called the ‘outer form’ of this blueprint for holistic systems embodying the divine paradigm, there is an ‘inner form’ of this geometrical object. It consists of two sets of seven regular polygons: triangle, square, pentagon, hexagon, octagon, decagon & dodecagon. One set is the mirror image of the other. They are joined at the ‘root edge.’ The plane in which they lie passes through the vertical Pillars of Mercy and Severity but not through the central Pillar of Equilibrium, because the outer form of the Tree of Life is really three-dimensional, not two-dimensional, as usually depicted in books on Kabbalah. This means that the Sephiroth Chokmah, Chesed, Netzach, Binah, Geburah & Hod are located at corners of the triangles and hexagons but that Tiphareth and Daath do not coincide with the endpoints of the root edge —only their projections onto the plane of the polygons do.

Figure 2     The number 137 is embodied in the blueprint of the inner Tree of Life. Its (7+7) enfolded polygons have 94 sectors. When each sector is divided into three tetractyses, 1370 yods are generated, i.e., the number of yods in 137 tetractyses. This proves beyond reasonable doubt that the number 137 is a basic structural parameter of the Tree of Life, in keeping with its central status in physics as a number which determines one of the fundamental constants of nature — the fine-structure constant, whose magnitude sets the scale of the energies of electrons in atoms.

Figure 3         Ten overlapping Trees of Life map the 10-dimensional space-time of superstrings. With their triangles turned into tetractyses, there are 248 (red & violet) yods up to Chesed of the fifth tree — its first Sephirah of Construction. There are a further 248 (blue) yods up to, but not including, Chesed of the tenth tree. Each yod denotes a particle, a gauge field of E8 or E8'. The Godname EL of Chesed with number value 31 prescribes the dimension 248 of E8 because there are 31 emanations up to Chesed of the fifth tree.

Figure 6     An octagon whose sectors are transformed into 2nd-order tetractyses has 496 hexagonal yods. Each yod symbolizes a particle involved in the transmission of the superstring force. The number of tetractyses is 80. This is the number value of Yesod, the Sephirah immediately above Malkuth in the Tree of Life. This archetypal pattern is prescribed by the Godnames. For example, it is prescribed by the Godname YAH with number value 15 because each sector of the octagon has 10 tetractyses with 15 corners. There are 576 yods surrounding the centre of the octagon, where 576 = 242 = 12×22×32×42. 33 yods are corners of 1st-order tetractyses, where 33 = 1! + 2! + 3! + 4!. These are examples of how the integers 1, 2, 3 & 4 symbolized by the four rows of the Pythagorean tetractys express the properties of archetypal patterns and holistic systems possessing sacred geometry.

Figure 10 As 132   – 1 = 168 and a parallelogram constructed from two 2nd-order tetractyses has 13 yods along each side, a 10-pointed array of 10 parallelograms has 1680 yods surrounding its centre. They comprise 90 yods in each inner 2nd-order tetractys and 78 yods in the remainder of a parallelogram. This reproduces the gematria number values of the two Hebrew words in Cholem Yesodoth, the Kabbalistic name of the Mundane Chakra of Malkuth:

Figure 12     The semi-regular polyhedra are divided into two groups of 13 (15, if enantiomorphs are included). They are called the Archimedean solids and the Catalan solids (their duals, in which vertices & faces are interchanged). The Catalan solid with the most faces is the disdyakis triacontahedron. It has 62 vertices, 180 edges & 120 triangular faces. Each edge can be thought of as the base of an interior triangle with a corner at the centre of the polyhedron. If these triangles are divided into their sectors, it can be calculated that 1680 vertices, edges & triangles surround an axis passing through any two diametrically opposite vertices. This is the number of turns made in each helical whorl of the E8×E8' heterotic superstring as it revolves five time around its axis of spin. It is an indication that the disdyakis triacontahedron is the polyhedral version of the Tree of Life, which encodes the same structural parameter of superstrings.