Sacred Triangle and Geometric Symmetries in Initiatic Geometry
to eternal realities. Conversely, the Sacred Triangle or Pythagorean Triangle is a scalene triangle. The quadratic expansion underlying its use allows us to symbolize the action of the Ternary on manifestation. In this case, to grasp the participation of its parts in a unique whole, it is necessary to know intimately what the quadratic expansion symbolizes. It is appropriate to distinguish between transformations and geometric symbols that are invariant with respect to them. Symmetric geometric figures manifest ontological superiority over the geometric transformations with respect to which they are invariant, precisely because of their invariance. It is not by the arbitrariness of mathematicians that the study of transformation groups uses geometric forms that are invariant with respect to their action. Man's instinctive attraction toward symmetric realities is the reflection, not always conscious, of the sense of eternity exercised by the world of essences on that of phenomena. Therefore, to be able to orient a being toward the Good through the operations of Symbolurgical Science, it is crucial to know how to make conscious use of symmetry, given that, often, a symmetric figure directs the consciousness that exercises itself upon it, toward an idea of perfection directed to the world of essences. On the contrary, geometric figures that are manifestly asymmetric or incomplete, meditated upon or drawn during ritual actions, can easily direct consciousness toward forces, entities or realities that disintegrate the psycho-physical composition of a being. If such operations are undergone unconsciously, it is not so difficult to understand that, in this case, forces are at work that aim to chain a being, to make them passively undergo fears and stimuli devoid of true life. Some General Principles Having made these general premises, we can formulate the following fundamental principle of Initiatic Geometry: Every spiritual or manifested reality has its own internal hierarchical structure. In particular, it is a geometric object. Although many non-manifested realities have also been symbolized over the centuries with specific geometric figures, one need only think of the figure of the Triangle as a symbol of the Divine Trinity, in the present work we only want to help clarify the relationships between the informal manifestation and the evolution of the formal one. In particular, the latter is subject to transformations and through them we will see the action of such relationships. However, it is fundamental to remember that, on the contrary, in esoteric doctrine, God does not evolve, but is. Let us now formulate a second principle of Initiatic Geometry: A symmetric figure evokes a spiritual reality of a higher degree than that of the domain containing the figure. If we then consider that the presence of a symmetry indicates invariance with respect to a transformation and that a transformation is a symbol of an action on the domain, here is that the search for symmetric figures can be symbolically connected with the search for a superior spiritual reality and law. To be able to express ourselves clearly we will use mathematical tools. Let $D$ be a domain and $\Omega$ a geometric object of such domain. Intuitively the reader can conceive $\Omega$ as a subpart of $D$. Let $f: D \rightarrow D$ be a transformation of domain $D$; that is, the datum of a law that associates to every element $d$ of $D$ an element $d'$ of $D$; we will write this in the following way: $d' = f(d)$. For expository simplicity we will suppose that $f: D \rightarrow D$ is a bijective function, that is, that for every element $d$ of $D$ there exists a unique element $d'$ belonging to $D$ such that $d' = f(d)$. Now suppose that for the transformation $f: D \rightarrow D$ the geometric object $\Omega \subset D$ remains globally equal to itself, that is $$f(\Omega) = \Omega \qquad (1)$$ i.e., for every element $\omega$ that is in $\Omega$ it holds that $f(\omega)$ is in $\Omega$, then $\Omega$ is said to be invariant with respect to $f$. For example, let us consider as domain $D$ the Euclidean plane $\mathbb{R}^2$, as transformation $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ the rotation of angle $180°$, that is $\pi$, and as $\Omega$ we take a subset of points of $\mathbb{R}^2$. Being invariant for the action of $f$ means that $\Omega$ is symmetric with respect to the axis of abscissas, i.e., the line $\{y = 0\}$. In this case the geometric object represented by $\Omega$ manifests at least binary symmetry. If, instead, $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ were a rotation of angle $2\pi/3$, then $\Omega$ would manifest ternary symmetry, if $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ were a rotation of angle $2\pi/5$, we would have pentagonal symmetry, if of angle $2\pi/6$ the symmetry would be hexagonal and so on. If $f$ is an infinitesimal rotation then condition (1) implies that $\Omega$ is a circumference. In the case of the circumference indeed we no longer have discrete symmetry of finite cardinality as in the case of polygonal symmetry, but we have continuous symmetry of infinite continuous cardinality. The symmetry groups with this characteristic that unite the algebraic properties of groups with the geometric ones of differential varieties are called Lie groups, and will be treated