Sacred Geometry: Regular Polygons and Polyhedra in Initiatic Meditation

of that Energy that regulates corporeal manifestation. Following this path of meditation, which leads to considerations very distant from those that originate from naive reductions of commonly perceived matter to alleged "absolute geometric forms", according to what is attributed to ancient and true initiates by profane historiography, which loves to debate such completely useless questions for those who travel the Way, one intuits that such forms also serve to concentrate subtle consciousness on certain manifestations of forces not completely reducible to it. What is remembered above should be sufficient to justify an in-depth study, at least of the geometry of regular polyhedra, in any initiatic way. In the case of two-dimensional Euclidean plane geometry, a regular polygon is a finite region of the plane bounded by a single broken curve, called the boundary or frontier of $\Omega$, which consists of $l$ segments connecting $l$ vertices $A_1,...,A_l$ and which satisfies the following conditions: 1. all vertices are distinct; 2. each pair $(A_i; A_{i+1})$ identifies a unique segment (with the convention that $A_{n+1}=A_1$); 3. two segments can have only one vertex in common; 4. segments with a vertex in common lie on distinct lines; 5. it has all equal sides 6. it has all equal interior angles. It is not difficult to prove that the interior angles of a regular polygon with $l$ sides all have a measure equal to $\frac{\pi(l-2)}{l}$ and that this determines a property of convexity of the figure. In general, when recourse is made to a notion such as that of convexity, reference is made to an external space, within which a figure is immersed; this, symbolically, indicates that we are working on the aspect of substance, rather than on that of essence. In fact, a portion of space is called convex if it contains every segment that has as endpoints any pair of its points. It is appropriate to remember that the change of viewpoint from the pole of essence to that of substance is a method for finding logical equivalences, which are not metaphysical identities. For example, in the case of regular polygons, wanting to place oneself on the side of substance, there is a characterization of the concept of regular polygon based, precisely, on the idea of convexity. Indeed, it is possible to prove that $\Omega$ is a regular polygon if and only if it satisfies a);b);c);d);e) and moreover it is convex and all segments that connect the midpoints of two consecutive sides have the same length. Let us now try to approach considerations on regular polyhedra while maintaining ourselves from the point of view of substance. We will therefore limit ourselves to considering only convex polyhedra. Given their nature as symbols of acting principles, the pole of essence is nevertheless symbolized, in this theory, through the notion of extremal point of a convex. A point $P$ of a convex domain $\Omega$ is called extremal if it does not contain any segment such that $P$ is in the segment and is not one of its two endpoints; for example, the vertices of a pyramid are its extremal points since for every segment entirely contained in $\Omega$ and passing through one of its vertices $A$ it happens that $A$ is always one of the two vertices of $\sigma$. To understand the nature of polyhedron, using the point of view of substance, it is necessary that the distinction between the notion of interior point of a domain and that of point on the boundary of $\Omega$ be intimately realized. To understand what is important for the purposes of this work it is sufficient to visualize a regular tetrahedron $\Omega$. The boundary $\partial\Omega$ of $\Omega$ consists of 4 equilateral triangles. The interior points of $\Omega$ are the points of $\Omega$ that do not lie in $\partial\Omega$. Therefore any interior point $P$ of $\Omega$ has a minimum distance from the boundary. We can therefore take a solid sphere with center at $P$, which is entirely contained in $\Omega$, and which has no point in common with the boundary. The possibility of inserting spheres disjoint from the boundary with center at a given point is what characterizes the notion of interior point of a domain, when a measure of distance between points of the domain can be assigned. A region of three-dimensional Euclidean space is a convex polyhedron if it is a closed, bounded and convex region having only a finite number of extremal points and with at least one point in its interior part. What is relevant to bring to attention is that the boundary of a convex polyhedron, what restores its shape, is of two-dimensional nature and that in it three basic ideas of plane geometry re-emerge. Indeed, in the boundary we can distinguish