Geometric Actions and the Symbolism of the Cross

We activate, is that which places itself in perpendicular contrast with the action of the identical. It shares with the latter the same central focus. It is that which manifests according to the influence $X: \mathbb{R}^2 \rightarrow T\mathbb{R}^2$ describable in the following way: $X: (x,y) \rightarrow ((x,y), (-y,x))$. On the point $p = (x,y)$ the virtuality $v := (-y,x)$ is activated. The cyclic effect of perpendicular dragging operates perpendicular to the intentionality realized with the identity of self with self. The virtuality of permanence in the identical tends to maintain the same radial direction, while the integral curve of the orthogonal dragging action is the circumference passing through the point $p = (x,y)$ with radius of length equal to $r := \sqrt{x^2 + y^2}$. In the previous actions the origin $O$ is present within the considered virtuality. In both cases the fact that the acting virtuality is null in $O$ is a symbol of the compossibility at $0$, for the action $X$, of all directions exiting from $O$ and which all manifest as soon as one moves away from $O$. An action, quite simple to internalize and which we often activate for stretches that are not always brief compared to the duration of our earthly life, is given by the action of unitary radial virtuality; that is, by the influence $X: \mathbb{R}^2 \setminus \{O\} \rightarrow T\mathbb{R}^2$ which is exercised in $\mathbb{R}^2$ in the following way: $X: (x,y) \rightarrow \left((x,y), \left(\frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}}\right)\right)$. This action is similar to the law of the identical, but at every point different from $O$, a virtuality of total intensity always constant, equal to unity, acts. We note that this action is prevented from operating on $O$. Given any point $p = (x,y)$ different from $O$ and considering the corresponding virtuality $v_p := \left(\frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}}\right) \in T_p\mathbb{R}^2$ activated by $X$, it is easy to see that the evolutionary line through $p$ still has as its image in the Euclidean plane the half-line exiting from $O$ and passing through $p$. However, the law of traversal is now of the following type: $(x,y): \mathbb{R} \ni t \rightarrow \left(x \frac{t+\sqrt{x^2+y^2}}{\sqrt{x^2+y^2}}, y \frac{t+\sqrt{x^2+y^2}}{\sqrt{x^2+y^2}}\right)$. Unlike the virtuality given with the law of the identical, for the unitary radial virtuality each evolutionary line through $p$ can be extended to $O$. However, if such passage through $O$ were governed by a Flow law we would have in $O$ the simultaneous activation of all distinct directions exiting from $O$, and each with its own direction. In the domain $\mathbb{R}^2$ this cannot be realized in $O$. Each evolutionary line arrives at $O$ with velocity of unitary intensity and direction individuated by the segment from initial point $O$ to final point at $p$. The impossibility in $T\mathbb{R}^2$ of realizing in a single law such simultaneous directional multiplicity resides in the impossibility of extending to all $D = \mathbb{R}^2$ the law $X: (x,y) \rightarrow \left((x,y), \left(\frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}}\right)\right)$. The indefiniteness of the points of the unit circumference centered at $O$ is unmanifestable in $O$, if we remain within the domain $\mathbb{R}^2$. The simultaneous realization of all directions exiting from $O$ is symbolically realizable through circumferences centered at $O$. However, such symbolism also indicates an extraneousness of $O$ with respect to each of them. The symbolism of the circumference centered at $O$ is a basis for the simultaneous realization of all directions exiting from $O$ only partially. The instantaneous realization of multiple directions in self-luminous simultaneity symbolizes a level break that surpasses, by superordinating it, the law of Necessity referred to an action $X$ on domain $D$. The Tradition, which in the land of Italy, from Pythagoras to Virgil and from Virgil to Dante was never interrupted, indicates the way to exit from the Great Flow of Becoming. The limits that circumscribe the action of the Law of the Great Flow are surpassed by means of the Universal Symbol of Total Realization: the Cross. The symbol of the Cross, theophanic support par excellence, privileged support in the eternal instant of Liberation, is, in its essential simplicity, the luminous heart of Initiation, or rather of the voluntary overcoming of the Law of Necessity. We, for our part, can only indicate here that all modern Geometric and Algebraic theory of singularities is an unheeded commentary on the symbolism of the Cross. The Pythagorean way leads us to meditate on the Symbolism of the Cross taking the theory of singular varieties as support to be able to symbolize the overcoming of the law of the Great Flow. In this meditation, which follows the two preceding ones: geometry and limitations to the law of karma and the law of karma and the great flow, we will realize how geometries, very different from those that provide support for meditating on the uniqueness of the evolutionary way, allow us a visualization and understanding of the possibilities of overcoming the Law of Karma. Essentially, it is the uniqueness of the integral curve passing through a point...