Sacred Geometry and the Cube: Kabbalistic Symbolism and Pythagorean Harmonies

twelve gates: above these gates stand twelve angels and names written, the names of the twelve tribes of the children of Israel. [...] He who spoke to me had as measure a golden reed, to measure the city, its gates and its walls. The city is square-shaped, its length is equal to its width. The angel measured the city with the reed: it measures twelve thousand stadia; length, width and height are equal. Symbolically the Cube synthesizes the completion of Creation, which culminates with the realization in itself of the uncreated principle, the creative Word indicated by the Lamb, the Vedic Agni, the Ignis or divine Fire. The measure of 12,000 stadia in length, width and height indicates complete development, whose symbol is the number 12; the fullness of all its possible manifestations in the three worlds is represented by the number 1000, and by all possible dimensions: length, width and height. The Cube evokes an idea of perfection, stability and incorruptibility based on the idea of harmony. To make this connection explicit, let us analyze, through the tools of Geometry, the shape of the cube. The cube is formed by: 8 vertices, 0 dimensional; 12 sides, 1 dimensional; 6 faces; 2 dimensional: The number of vertices, sides and faces is linked by Euler's characteristic, which in the case of regular spherical polyhedra is 2, i.e. vertices $-$ sides $+$ faces $= 2$: $$V - E + F = 2 \quad (2)$$ We can also define the Schläfli numbers $\{p;q\}$ of the polyhedron, i.e. those numbers such that the following relation holds: $p$ faces $= 2$ sides $= q$ vertices; $$p \cdot F = 2 \cdot E = q \cdot V \quad (3)$$ and which, in the case of the cube, are $\{4;3\}$. The vertices, sides and faces of a solid are not random, redundant information, devoid of symbolic significance. This information has topological and algebraic aspects that manifest, in their respective domains, essential characteristics of the solid. Analyzing such data algebraically we quickly derive what the Pythagoreans obtained from their geometric studies. In fact, the ratios derived from the cube are the musical ratios corresponding to Philolaus' Tetrachord and which the Pythagoreans derived from the divine Tetraktys: $$6 : 12 = 1 : 2 \quad \text{(diapason)} \quad (4)$$ $$8 : 12 = 2 : 3 \quad \text{(diapente)} \quad (5)$$ $$6 : 8 = 3 : 4 \quad \text{(diatesseron)} \quad (6)$$ that is, the ratios that regulate the intervals of octave, perfect fifth and perfect fourth. Starting from these ratios, the Pythagoreans obtained the complete scale through the following succession: $$\text{Do} \xrightarrow{\times 2/3} \text{Sol} \xrightarrow{\times 4/3} \text{Re} \xrightarrow{\times 2/3} \text{La} \xrightarrow{\times 4/3} \text{Mi} \xrightarrow{\times 2/3} \text{Si} \xrightarrow{\times 4/3} \text{Fa} \quad (7)$$ Succession (7) can be represented circularly, through the seven-pointed star of index 3 (Fig. 18). The analysis of this cosmic figure is preliminary to its operative use. Meditation or ritual use of geometric figures must always be accompanied by a symbological specification of the operative plane of magical action. Numbers, and their geometric forms, are indeed true anagogical scales, with their respective correspondences on the natural, astral, mental, spiritual and divine planes. Using a geometric figure without specifying the plane on which one intends to operate, is equivalent to having an empty structure, and leads to meditations with vague objectives, when not dangerous. For example, operating in the Judeo-Christian tradition, some numbers and some figures, due to their intrinsic characteristics, give access to meditation on the divine names of the Jewish tradition. The symbolic number of the cube is considered to be the number 4 by virtue of the square polygons given with its faces. Adding the number of vertices, edges and faces we get: $$8 + 12 + 6 = 26 \quad (8)$$ The 26, in the gematria of the Kabbalistic tradition, plays a fundamental role, being the number of the sacred Name of 4 letters or Tetragrammaton יהוה. The Euclidean manifestation of the cube leads to the operation $3^4 = 12$ and 12 is a sacred number particularly important for the Kabbalistic tradition. In fact, there are 12 permutations of the Name יהוה identified with the 12 Faces or divine Essences, the 12 living Creatures or haioth haqodesh: יוההיההויהוה ההיוהיוההוהי וההיויההוהיה ההויהיוההיהו. Eliphas Levi, a master not to be forgotten, in his letters to the Disciple of Wisdom, suggests us to write these names on the edges of the cube. The 12 combinations, traced on the edges of the cube, represent all the harmonies of the Universe and all the balancing combinations of the great magical agent. The resulting figure is, for a just and perfect heart, an instrument that regulates the use of the twelve names and the multiple forces of the sphinx's enigma. A symbolic deepening that we intend to propose in a future work will lead us to the study of the automorphisms of