Platonic Solids and Sacred Geometry

these figures. We note that the tetrahedron with vertices $T=\{(1,1,1), (1,-1,-1), (-1,1,-1), (-1,-1,1)\}$ is contained within the cube with vertices $C=\{(\pm1,\pm1,\pm1)\}$, which, through the duality operation, leads to the octahedron with vertices $O=\{(\pm1,0,0), (0,\pm1,0), (0,0,\pm1)\}$. The icosahedron is derived by dividing the sides of the octahedron in golden ratio, i.e., $\varphi = 1.618\ldots$, obtaining the following vertices $I=\{(0,\pm\varphi,\pm1), (\pm1,0,\pm\varphi), (\pm\varphi,\pm1,0)\}$. Finally, from the icosahedron, through the duality operation, we obtain the vertices of the dodecahedron $D=\{(0,\pm\varphi^{-1},\pm\varphi), (\pm\varphi,0,\pm\varphi^{-1}), (\pm\varphi^{-1},\pm\varphi,0), (\pm1,\pm1,\pm1)\}$. We have thus derived all the Platonic solids from one another, starting from the tetrahedron and ending with the dodecahedron according to a simple analytical generation operated through the duality operation and a division in golden ratio.

Stellated Regular Polyhedra

Among regular polyhedra, besides the five convex ones, there exist 4 stellated ones, usually called Kepler-Poinsot, in honor of the two mathematicians who discovered them in different epochs. The characteristic data of such regular polyhedra are reported in table 2 and, as can be noted, refer to the same data of the icosahedron and dodecahedron to which they can be associated. Symbolically, the 4 stellated regular polyhedra constitute a development of the dodecahedron of which they possess the same symmetry group. These polyhedra find their reason for being in the fact that, in effect, there exist two regular polygons of five sides: the first is the pentagon $\{5\}$ and the second is the pentagram or five-pointed star, indicated in Schläfli notation as $\{5/2\}$. Where Platonic solids use the pentagon, Kepler-Poinsot polyhedra use the pentagram thus obtaining new solutions which, while not being convex, are nevertheless regular.

Archimedean Polyhedra

If it is true that no other regular polyhedra exist besides the 9 solids previously stated, it is also true that there exists a larger number of semiregular polyhedra. In them every vertex is the meeting point of an equal configuration of polygons without these having to be of the same type. Thus the faces of a semiregular polyhedron can be triangles and hexagons, or triangles and pentagons, or squares, hexagons and octagons, etc. The semiregular polyhedra constitute a derivation of regular ones, being reducible to them. However, the union in the same solid of multiple different regular polygons allows a more detailed articulation of those concepts and forces that are synthetically enclosed in the five regular polyhedra. The most famous among semiregular polyhedra are the Archimedean polyhedra, obtainable from regular ones through simple geometric operations, and which are 13 in total: 6 with dodecahedral symmetry, 6 with cubic symmetry and 1 with tetrahedral symmetry.

From Ternary to Polyhedra

The One 1 is the Father of Numbers, as the Point is of geometric figures. From the One originates the Three 3 or Ternary, represented by an equilateral Triangle. From the Ternary then originate all regular polyhedra through the modulation of the numbers of the Sacred Triangle 3, 4 and 5 and also the semiregular ones through a pulsation process that finds its geometric equivalent in the duality operation between polyhedra.

From Unity to Ternary

The delimitation of infinite and adynamic Nothingness is the first operation accomplished by the Conscious Power to bring the Universe into being. In the cosmogonic process, the first act was not of revelation, but of concealment, abstraction. Only in the second act did revelation begin, the unfolding as Creator God, proceeding outside of Himself in that primordial space that is in Him. This is the creation from nothing of Genesis, theophany and theogony in a certain way. Emanation with the addition of a plus, the creative fiat. The Circle in abstract sense represents an act of limitation. It delimits, separates the interior from the exterior, what is and will be from what is not nor will be. The Circle is here the symbol of circumscribed fluid space, the abyss, the primordial Nothingness that will conceive all creation at the rising of the Divine Sun. The Circle is the symbol of Number Zero 0, vivified and dynamized by Number One 1, geometrically represented by the Point. The Point represents arithmetic unity as the philosophical monad, it is the principle of every figure, the source of all geometry. Indeed, to the point corresponds the idea of simplicity or indivisibility, but, at the same time, an active, radiating force corresponds to it. For these reasons the Point can be taken as symbol of the indivisible source of Being, the first manifestation of Being, that is the One 1. When the original Point, concentration of Creative Power and mirror of the Supreme Divine Will, rises on the abyss dynamizing it...