Mathematical Symbolism of the Cross and Cone: Geometric Meditation on Dynamic Transformations

Following $X: (x,y) \mapsto ((x,y), (y,x))$: (59) The reader who has not read together with us geometry and limitations to the law of karma, may suspend this meditation and resume this work starting from the subsection that deals with the symbolism of the Cone, a few lines below. It is immediate to see that if $p_0 = (x_0, y_0)$ is a point that does not belong to the Cross, which is manifested algebraically by the algebraic relation $xy = 0$ (in which the role that in the geometry of the Cross is played by the origin $O$ is algebraically played by the product), then the unique integral line passing through $p_0$ relative to the law (59) of dynamization of the identical has support constituted by the appropriate branch of the hyperbola contained in $\mathbb{R}^2$ and described by the equation $xy = c_0$, where $c_0 = x_0 y_0$. In terms of the actual integral line, we have that the integral line passing through $p_0$ is given by: $$p_0: \mathbb{R} \to \mathbb{R}^2; \quad t \mapsto (x_0 e^t, y_0 e^{-t})$$ It is appropriate that the reader, in a purely mental way, brings to mind the entire family of possible equilateral hyperbolas that constitute the integral lines of the action, as the point $p_0$ varies. It is also easy to see that if $y_0 > 0$ and $x_0 = 0$, then $p_0: \mathbb{R} \to \mathbb{R}^2$ is the vertical half-line traversed in the direction from $O$ upward; if $y_0 < 0$ and $x_0 = 0$, then $p_0: \mathbb{R} \to \mathbb{R}^2$ is the vertical half-line from $O$ downward. If $y_0 = 0$ and $x_0 > 0$, then $p_0: \mathbb{R} \to \mathbb{R}^2$ is the horizontal half-line traversed in the direction from right toward $O$; if $y_0 = 0$ and $x_0 < 0$, then $p_0: \mathbb{R} \to \mathbb{R}^2$ is the horizontal half-line traversed in the direction from left toward $O$. The integration of the law (59) of dynamization of the identical elevates in the sense of exaltation and is centripetal in the sense of amplitude: the vertical axis symbolizes the Essence that animates all the dynamism we have before our eyes. However, in $O$ no virtuality seems to act: $X: (0,0) \mapsto ((0,0), (0,0))$. These types of action constitute a meditative basis also for all the initiatic symbolism of Non-Action of the Universal Principle, explored in certain extreme oriental initiatic ways. We see that, even in the case of the Law of dynamization of the identical, no integral line actually reaches $O$. Furthermore, it is easy to visualize how the limits of the integral lines tend to reach the arms of the Cross according to the four modes described previously. The Symbolism of the Cone The mathematical symbolism that has led us to see and realize that while at every point $p$, different from $O$, the tangent line to the integral line uniquely determines the evolution, constraining it to develop along the integral line itself, at point $O$, instead, there are two activatable directions, those along the arms of the Cross, should be meditated upon with deep and serene calm. This is a symbol that in $O$ the law of Universal Flow has ceased to determine the evolution of a being that has been initiated and therefore rendered capable of realizing in itself the totality of directions of $O$. To deepen our meditation about points where a greater totality of directions is condensed than that manifested in the general point of a geometric domain $D$, we propose to visualize the cross on a frontal plane, but rotated by $\pi/4$, that is, 45 degrees. The rotation with respect to the vertical axis allows us to visualize before us a double circular cone $C$, which widens in increasingly extended circumferences as one rises or descends along the vertical axis (Fig. 14). In this meditation we observe that, unlike those proposed on the Law of Karma and the Great Flow, the geometric symbol of the cone manifests clearly a special point, namely again point $O$, center of a three-dimensional Cartesian reference of ordinary three-dimensional space, vertex at the top of the cone that widens downward, vertex at the bottom of the cone that widens, raising us toward the top. Every point $p$ of $C$, which is different from $O$, admits a tangent space: the space of virtualities activatable at $p$ by the fact that $p$ is a point of $C$ is isomorphic to a Euclidean plane. Every tangent plane $T_p C$ cuts on $C$ the entire directrix line of the cone, which passes through two points, $p$ and $O$. In a certain sense all the planes of ordinary space that are tangent at a point $p$ of $C$ are also tangent at $O$. The center $O$ can easily symbolize a luminous source that radiates along the lines contained in the cone $C$. All the luminous lines of $C$ pass through $O$. Proceeding, by means of symbolic analogy, in the same way we realized the Riemannian surface $\Sigma$, and distinctly perceiving in $O$ a place of luminous concentration greater than in the other points of $C$, we expand the...