Sacred Geometry and Polyhedral Symbolism
triangle. For this reason we can say that this polyhedron reveals the Number 6 already present in latent form in the Tetrahedron. But, more specifically, the truncated Tetrahedron is the three-dimensional equivalent of the regular hexagon. Indeed, the truncated Tetrahedron is the polyhedron obtained from the intersection of two Tetrahedra inverted with respect to each other, although of different dimensions.39 This construction recalls that of the regular hexagon which is obtained as the inner region of two equilateral triangles intersected in the Seal of Solomon or Star of David.
39If the two inverted Tetrahedra were of equal dimension, the Octahedron would be obtained.
Figure 10: Generation of the truncated Tetrahedron in the passage from the Tetrahedron to the dual Tetrahedron. The central point of motion is the Octahedron.
In this sense the truncated Tetrahedron can be associated with the motion of encounter between two "foci" - the ascending or human one and the descending or divine one.
Cubic Symmetry
Cubic symmetry characterizes the Octahedron and the Cube, from which derive 6 other Archimedean polyhedra having the same symmetry: the truncated Octahedron, the Cuboctahedron, the truncated Cube, the Rhombicuboctahedron, the truncated Cuboctahedron and the snub Cube. This symmetry group is composed of 48 distinct elements of which 24 are rotations and, consequently, these numbers permeate all polyhedra influenced by this group.
The uniform polyhedra with cubic symmetry develop the same idea through harmonic numbers that specialize the same concept: the manifestation and realization of the Quaternary or Tetragrammaton. The Numbers involved are those of the development of the Tetragrammaton יהוה, namely first the Number 4 which successively unfolds into the numbers 12, 24, 48. To these are added the Numbers evocative of the Name יהוה: 26, 72 and 144.
In all solids with cubic symmetry the fundamental themes are those of achieved human perfection, wisdom, stability, incorruptibility, eternity, and, finally, the assumption by creation of the uncreated or divine principle.
Octahedron
In the Octahedron the Ternary and the Quaternary are reconciled at each vertex through the union of 4 equilateral Triangles. The Six vertices thus give origin to a Double Pyramid with square base. In it, symbolically, the world, represented by the square base of the Pyramid, is elevated to the point of adhesion or convergence with the Word which, at the two points of the Octahedron, acts first as creator and then as redeemer, as cause and as end or purpose of Creation.
"The numbers corresponding to the whole of the pyramid are 4, 3 and 7, symbol of the bond between Idea and Sign, between Spirit and Matter, between Theory and Practice, to conclude: Realization. The mathematical point at the top (the vertex) of the pyramid gives origin to 4 ideas (the 4 angles). From this unique form (the base) 4 ideas depart and symbolize their 'solidarity'. It is undeniable that the study of pyramids rediscovers the mystery of the Tetragrammaton."40
The Octahedron is obtained as the intersection of two inverted Tetrahedra of the same dimensions. In this sense it is the point of equilibrium of an ascending ascetic motion, represented by the Tetrahedron with point directed upward, and of a descending creative motion, given by the Tetrahedron with point directed downward. The Name associated with this solid is that of the Tetragrammaton יהוה. In fact, summing the number of vertices, edges and faces we obtain $6 + 12 + 8 = 26$; which by gematria equals the sacred Name of 4 letters or Tetragrammaton יהוה, given that the letter yod י equals 10, the hé ה equals 5 and the vav ו equals 6, i.e. $10 + 5 + 6 + 5 = 26$.
Cube
"This solid geometric figure has the same meaning similar to the square. It indicates stability by reason of its flat form and for this serves as the base of thrones. In mystical sense the cube is considered the symbol of moral perfection, wisdom and truth. The Scriptures indicate it as the model of the future Jerusalem, indicated and promised in the Apocalypse in the sense of eternity. [...] The Cubic Stone, called hexahedron, is the masterwork that the archosophical Apprentice must realize through his own efforts."41
The Cube is the symbol of the "cubic stone" or the perfect man who has become God's altar, having passed through the gate of the Cross. By unfolding on a plane the 6 square faces of the cube we obtain, in fact, the form of a Latin cross, symbol of sacrifice and initiatic death realized by Christ on Golgotha. The Cross is also the symbol of singularity, of level rupture, of passage from one condition to another at the point where human horizontal action and divine vertical action meet.
The Cube evokes an idea of perfection, stability and incorruptibility. The number of its edges is Twelve 12, a sacred Number which, in relation to the Quaternary, refers to the permutations of the Tetragrammaton יהוה. The permutations of the Sacred Name are qabbalistically identified with the 12 Faces or aspects of the Presence that manifest the divine Essence, the 12 living Creatures or hayoth haqodesh.
The 12 Names, which were engraved on the 12 stones of the High Priest's "rational", correspond for theurgical practice to the 12 Signs of the Zodiac, to the 12 hours of day and night, to the 12 tribes of Israel, to 12 invocations that must be made at 12 moments of the day, and, finally, to 12 different vocalizations of the Tetragrammaton. Indeed, each of the 12 verses associated with the Permutations of the Name possesses by notariqon the Permutation itself and its letters assume the vocalization of the letters of the verse.42
The Cube is also associated with the idea of harmony and complete development.
40Palamidessi Tommaso. 2012 [1982]. The Spirituality of Sacred Numbers. Pistoia: Archeosofica.
41Palamidessi Tommaso. 2012 [1982]. The Spirituality of Sacred Numbers. Pistoia: Archeosofica.