Geometric Transformations and Group Theory: Identity, Symmetry and Invariance Principles

in other specific articles. We note that for every geometric object, the identical transformation of domain $D$, that is the law $\text{id}_D: D \to D$, such that $x = \text{id}_D(x)$, also transforms $\Omega$ into itself. Every geometric object is symmetric or invariant with respect to this transformation. This law symbolizes the presence of Unity in everything. Let us now consider a transformation $f: D \to D$ that is not the identical one and with respect to which the geometric object nevertheless remains invariant, i.e., $f(\Omega) = \Omega$. As soon as such a law exists for $\Omega$, it immediately follows that $\Omega$ is also invariant with respect to the inverse transformation. We recall that the inverse transformation $f^{-1}: D \to D$ is that transformation such that if $d' = f(d)$ then $d = f^{-1}(d')$. In particular, if $\omega' = f(\omega)$ where $\omega$ lies in $\Omega$ and $\Omega$ is $f$-invariant, we have that then $\omega'$ lies in $\Omega$ and, obviously, also $\omega = f^{-1}(\omega')$ lies in $\Omega$. We therefore have $f^{-1}(\Omega) = \Omega$. Let us now consider the law obtained by composing the law $f$ with itself; that is, consider the function $f^2: D \to D$ given by $z = f^2(x) = f(f(x))$. It is evident that $f^2(\Omega) = f(f(\Omega)) = \Omega$. Indeed, for the function obtained by composition of the law $f$ with itself an arbitrary number $n \in \mathbb{N}$ of times, that is for the law $f^n: D \to D$, we will still have $f^n(\Omega) = f(f(f(\Omega))) = \Omega$ and the same thing holds for the inverse, that is $(f^{-1})^n(\Omega) = \Omega$. In mathematical terms we can say that $f: D \to D$ generates a cyclic group $G$ of transformations, for which the geometric object is invariant. This method allows us to intellectually identify an infinity of possible laws. We note, meanwhile, that, just as repeating twice the same rotation of angle $\theta$ we obtain the identical transformation, so it can easily happen that there exists a number $n \in \mathbb{N}$ such that $f^n = \text{id}_D$. We will have, in particular, the law of the binary, when $f^2 = \text{id}_D$ and the law of the ternary, when $f^3 = \text{id}_D$. The study of the law of the binary, for example, includes the study of all phenomena given by mutual opposition. Such study can be approached through the symbolism associable to the finite group of order 2. This algebraic group has many different realizations. It can in fact be commonly realized by the set $\{1, -1\}$ with the usual law of multiplication, i.e., $\begin{array}{c|cc} \cdot & 1 & -1 \\ \hline 1 & 1 & -1 \\ -1 & -1 & 1 \end{array}$ Or the same algebraic group can be realized by the set $\{0, 1\}$ with the binary law of addition, i.e., $\begin{array}{c|cc} + & 0 & 1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \end{array}$ Or the same algebraic group can find a realization with the set $\{\text{id}, \sigma\}$ where $\sigma$ is a rotation of 180° and $\begin{array}{c|cc} & \text{id} & \sigma \\ \hline \text{id} & \text{id} & \sigma \\ \sigma & \sigma & \text{id} \end{array}$ At a more principal level, compared to what we want to maintain in this work, we should have spoken of Duality as the principle to which the law of the binary can be reduced. It is, in fact, easy to see that the three different realizations have something in common and that they can be obtained one from the other by means of simple formal substitutions. This invariance is of a higher level compared to the invariance that we have treated so far and this is not the place to approach such study, we can only hint that it is based on Duality. In terms drawn from contemporary mathematics, it is customary to write that the three algebraic structures presented previously are isomorphic, while, from the esoteric point of view, we can simply say that they are all different manifestations of principal Duality. In the same way one can proceed with the study of the law of the Ternary, which is manifested through the cyclic group of order 3. In a further work we will make explicit the difference between laws expressed through a prime number and those reducible to non-prime numbers. Here we can only observe that there exist two groups of order four that explicate the metaphysical principle of double Duality and that this too will be the subject of future work. We believe that an attentive reader has understood that the study of the geometric object is intimately connected with that of the group of transformations that leave $\Omega$ invariant, usually indicated with $\text{Aut}(\Omega)$. In reality both $\Omega$ and $\text{Aut}(\Omega)$ are nothing but distinct aspects or manifestations of the same numerical entity. Starting from the second principle of Initiatic Geometry and by analogy between the various domains of reality, we can formulate the following two methodological criteria: 1. Every transformation of a domain $D$ is a manifestation of realities of higher degree than that of domain $D$. 2. To the overall resulting action on $D$ produced by such realities of higher degree, an algebraic group is associable. We develop some consequences by means of the study of the action of a single transformation $f: D \to D$ acting on $\Omega$ and of the group $G$ generated by it. We have that only two situations are possible: The first possibility is that the transformation $f$ repeated a number