Sacred Geometry: Regular Polygons and Platonic Solids

Numbers act and, specifically, according to the number of spatial dimensions of that environment. For example, when Number 4 acts in a one-dimensional space it can take the form of 4 distinct points; in a two-dimensional space it can take the form of a square or a cross; in three-dimensional space it acts through the cube, the tetrahedron, or through other polyhedra, etc. Given that in this evolutionary phase human consciousness is bound to a three-dimensional spatial perception, the only geometric forms of interest from a practical point of view are two-dimensional and three-dimensional ones. Regular two-dimensional figures are characterized by a purity of expression of the Number used, while three-dimensional ones allow the harmonic fusion of multiple Numbers among themselves. For example, the Pentagram perfectly expresses the action of Number 5 in pure form, while the Dodecahedron harmonically conjugates and modulates Number 5 with Number 12. In the two-dimensional geometric space there exists an indefinite number of regular figures, which are the regular polygons. These are formed by an equal number of vertices and sides and thus resonate perfectly and maximally with the Number corresponding to them. Every natural number has one or more corresponding regular polygons. In the three-dimensional geometric space, the situation is different because there exist only 9 regular figures of which 5 are the convex regular polyhedra or Platonic solids, while 4 are the stellated regular polyhedra or Kepler-Poinsot solids. The differences between the two geometric spaces don't end here because, while in two-dimensional space every regular convex polygon carries its own symmetry, in the three-dimensional world all regular polygons can be enclosed in three fundamental symmetry groups: tetrahedral symmetry, octahedral or cubic symmetry, and icosahedral or dodecahedral symmetry. To be clear, therefore, we can say that in three-dimensional geometric space there exist 5 convex regular polyhedra, 4 stellated regular polyhedra, and finally, all of these fall into only 3 symmetry groups. Indeed, the triad 3, 4, and 5 governs all regular forms of the three-dimensional world. The Five Regular Convex Polyhedra In every regular polygon, whether stellated or convex, the vertices are indistinguishable from one another. When we say that a regular polygon is symmetric, we mean that by substituting one vertex for any other we obtain no alteration in the figure. In regular polygons each vertex is obtained from the conjunction of two sides with a predetermined angle $\alpha$. The number of vertices of a regular polygon, indicated with the letter $n$, together with the value of the angle between two adjacent sides, totally characterizes the polygon in question. In other terms, the pair $\{n;\alpha\}$ univocally identifies the regular polygon of our interest and for this reason it is said to constitute a complete classification. Analogously to the two-dimensional case, also in the three-dimensional case the vertices of regular polyhedra are indistinguishable from each other and constitute the meeting point of a number $q$ of regular polygons each having $p$ sides. The pair $\{p,q\}$, which we can obtain from the study of the vertex, univocally identifies the regular polyhedron and thus constitutes a complete classification. However, unlike the case of regular polygons where it is always possible to construct regular polygons of any number of sides $n$, in the three-dimensional case of convex regular polyhedra the possible values of $p$ and $q$ are very restricted. In fact we know that: 1. every vertex of the solid must be a vertex of at least three faces constituted by identical polygons, therefore $q \geq 3$; 2. the sum of angles between respective sides adjacent to each vertex must be strictly less than $360°$; 3. regular polygons of six or more sides have only angles of $120°$ or more, therefore the common face can only be a triangle, a square or a pentagon, i.e. $p \in \{3,4,5\}$; 4. If $p = 3$, i.e. in the case of triangular faces, every vertex of an equilateral triangle is $60°$, therefore a form can have 3, 4, or 5 triangles meeting at a vertex, i.e. $q \in \{3,4,5\}$.