Geometric Singularities and Sacred Symbolism

a symbolism of a much more occult nature. The geometric notion that allows us to symbolize domains in which folds or special points are present is given by the notion of singularity. Essentially a geometric domain has a place where it is singular if the variety of virtualities present in it is of orders of magnitude greater than that of the virtualities present in a general point of it. A domain is said to be non-singular or smooth if there are no places where it is singular. Without too much difficulty one can demonstrate that the variety of virtualities present in vertex $O$ of a cone is of dimension $3$ while the variety of virtualities present in any other point of it is of dimension $2$: the cone is, therefore, a singular domain. The sphere is a perfectly homogeneous domain and is therefore non-singular. It is fundamental to conceive that such dimensions do not depend, in themselves, on conceiving the domain $D$ immersed in a larger domain, as in the case of a cone immersed in ordinary three-dimensional space. Let us take the case of a line $r$, whether it be in a Euclidean plane or in ordinary three-dimensional space. Each of its points $p$ identifies a unique directionality, which, if fully unfolded, restores the line itself. The dimension of the virtualities present in a point $p \in r$ is therefore always equal to $1$, whether $r$ is immersed in a Euclidean plane or in ordinary three-dimensional space. Let us consider a circumference $C$. Again each of its points $p$ identifies a unique directionality, which, fully unfolded, can be symbolized by the tangent line at $p$ to the circumference itself. In a deeper symbolism and more oriented towards the pole of essence, the place $T_C$ can be realized as a surface in a three-dimensional space, more precisely as a cylinder that expands indefinitely from a circumference taken as the source origin of the cylinder, identical image of the circumference $C$. The image of the circumference and of the lines tangent to it represents, instead, a symbolism of a nature more linked to the pole of substance. The geometric symbol par excellence of a singular domain is constituted, as we have already seen from the Cross. In the center $O$ of it, the variety of virtualities in $O$ has dimension $2$, and we know that this does not depend on representing the Cross internally to a Euclidean plane, although, in such a basic case all the symbols referable to it resonate with one another. All the directions emanating from $O$ are virtually in $O$. The operation of spatial expansion applied to $O$, in its completeness, produces a circumference $C$, in place of $O$, and two lines $\ell$, $\ell'$ which, symbolically, can be arranged on a vertical plane orthogonal with respect to the plane, which we will call $\Pi$ that contains $C$. The spatial expansion, which we have already treated when we constructed the surface $S$ through the visualization of the place swept by a line, constitutes a symbol of universal manifestation, whose full understanding occurs by leaving Cartesian three-dimensionality. We can realize the line $\ell$ so that it crosses the plane $\Pi$ of the circumference $C$ orthogonally and so as to pass through the north pole of $C$. We denote with $N$ the unique point of intersection between $\ell$ and $\Pi$. Similarly, we now denote with $S$ the unique point of intersection between $\ell'$ and $\Pi$. The ascent to the Father through the descent of the Son is realized in the Love of the circularity of the Holy Spirit: the one-dimensionality of the ascent and of the descent are well symbolizable by the two points $N$ and $S$, extremes of a spiritual circularity that animates a two-dimensionality orthogonal to both directions, the filial one and that of paternity. The understanding and integral realization of the Way are symbolizable in this symbolic operation, in which all the virtualities present in $O$ are instantaneously realized in the circumference $C$, through the transfiguration that we have symbolically realized in the four-dimensionality of the mind oriented by the heart. The Holy Spirit manifests itself as Spirit in the World symbolized by $\Pi$ and the vectors tangent to $C$ as place in $\Pi$ and applied respectively in $N$ and in $S$ symbolize such principial beneficent action. The well-being, essential meaning of the word swastika, and the symbolism itself of the swastika find an easy symbolic derivation from what we have meditated. The singularity of point $O$, with respect to the geometry of the arms of the Cross, is a symbol of a radiating luminosity, concentrated in order to be realized through transfiguration. The mystery is here well symbolized by the four-dimensionality of the symbol, which conceals itself from a vision not inflamed by love. The apparent simplicity of the singularity present in point $O$ of the Cross must not deceive and poses the problem of the symbolic meaning associable with other types of singularities. We will develop, if we are given the opportunity...