Symbolic Meditation on Cusps and Singularities

the possibility, some more specific aspects of the symbolism of singular varieties theory in subsequent works. We conclude with a symbolic meditation about an easily traceable architectural symbol, which completes our interior listening to the Word in the figured Symbol. We observe the form of the onion dome that often surmounts church bell towers, both of Orthodox and Catholic rite, which see a cross rising above the cusp. The welcome that Unity reserves for Duality is symbolized in the harmony of the cusp's form. The heart of true welcome lies in the tangential meeting of the two branches, rigorously distinct, and symmetrically consonant, which merge into one with the unique tangent direction. In the architectural symbolism here recalled, the symbol of Duality in Unity manifests with extreme clarity, also thanks to the integration of the tangential support axis of the cusp that extends into the vertical arm of the cross surmounting the dome. If we bring such symbolism back to its primordial essentiality, the heart guides the mind to visualize a planar curve in the form of a cusp $C$, contained in the plane domain $\mathbb{R}^2$. We immediately note that the place $O$ where the two wings of the cusp rest tangentially upon each other has a singular nature with respect to the remaining points $p$ of the cusp $C$. The cusp is the most elementary type of singularity, therefore the most essential, after the singularity manifested by the Cross. We operate in $O$ the directional expansion $S$ of $D$, which we have already treated when we constructed the surface $S$ through the visualization of the place swept by a line, initially placed in the horizontal plane and which, while ascending vertically, simultaneously rotates around the vertical axis passing through $O$. Through the symbol expressed with the application $\pi: S \to \mathbb{R}^2$, we see that in $S$ there is a unique curve $C_0$, which is tangent at only one point $A$ to the circumference $\Gamma$, contained in $S$, which parametrizes the directions exiting from $O$ and $C_0$ is such that the application $\pi$ outside of $A$ behaves in a way to perfectly identify $C_0 \setminus \{A\}$ and $C \setminus \{O\}$. The singular nature of $O$ for $C$ allows us to overturn half of the parabola transforming it into a cusp (Fig. 15). Symbolically we have operated through inverse analogy and we have united the branches of the cusp transforming the cusp into a parabola tangent to $\Gamma$ at $A$. The two wings of the cusp, which manifest in the Euclidean plane as well distinct, for those who have not realized the instantaneous unity of the plane directions exiting from $O$, are united to form a single smooth curve, a parabola, for those who have risen to a higher level. This is possible in the place that permits the manifestation in a total and absolute symbolic unity of all the virtualities of the plane activatable in $O$: what appears distinct under certain limitative conditions is a special manifestation of what is One.