Euclidean and Spherical Geometry: Mathematical Symbolism and Masonic Principles

Euclidean geometry. Let us focus our mind on the planar nature of the triangle in Euclidean geometry. With our mind we can generate the Euclidean plane through a multiplicity of points arranged in a form having zero curvature. It is well known that Euclid's fifth postulate is equivalent to the assertion that the sum of the interior angles of a plane triangle equals the angle of a half-turn ($180°$), just as the value that the symbolism of the number $3$ and the triangle has in the operative works of a Blue Lodge is well known. The three pillars of Liberty, Equality and Fraternity compose the vertices of a triangle, generally equilateral, whose horizontal planar expansion is, among others, the symbol of the action of justice that the Masonic Order proposes to the world. We think that the trinomial Liberty, Equality and Fraternity does not belong to the origins of Traditional Masonry. However, we believe that such absence only increases the didactic value of our example. If we traverse the triangle along the sides we return to the same vertex but in a different direction, and even the symbolism of the half-turn angle indicates an opposition between positive and negative polarity that must be kept in mind. Does not all this indicate that only in the simultaneous expansion of the triad is there hope of bringing justice to the world? While, operating differently, it is easy to obtain results opposite to those desired? If we flip the triangle with respect to the base, inverting the direction given to the axis of the triangle's base, we obtain a regular rhombus, whose sum of interior angles equals that associated with a complete turn, i.e., equals $360°$. Symbolically, the overturning of the triangle, in the passage of the vertex from top to bottom, indicates the active action of a principle on matter. However, in this case, the polarity that has been created is not fully harmonic, the stability, symbolically associated with the number $4$ thus obtained, fades considerably, observing, for example, that such a rhombus, while evoking with the sum of its interior angles a complete turn, cannot be inscribed in the planar symbol of perfect circularity: the circumference. It is to harmonize the effect of spiritual action with the necessary reaction of which we must always take account, that it is not with the regular quadrilateral obtained by oppositive doubling of the triangle, but it is with Solomon's seal, obtained by rotating the triangle $180°$ around its center that we must operate. This geometric symbol of just action and the geometry associated with it provide a universal language of understanding among men, so much so that, not by chance, many esoteric traditions base some of their ritual actions on it. Here we limit ourselves to observing that to traverse this symbol along its sides, in addition to the $6$ vertices, where direction changes twice, there are another $6$ points where direction changes $4$ times. Furthermore, it is immediate to verify that starting the path from the top vertex we can well complete the entire circuit, only after having traversed the sides of the remaining $5$ external triangles, and then the six sides constituted by the six bases of the external triangles and finally close the path by traversing the last side, that of the triangle from whose top vertex we had started. In such an operation it is symbolically indicated that to perform an action externally, in a just and perfect way, and $12$ is then the symbolic number associable with it, it is first necessary to have completed the corresponding interior action, whose symbolic number is the number $6$, which is reflected in it. The double movement of expansion and contraction is also well symbolized by the blooming outward, like petals of a rose, of the six equilateral triangles internal to the regular hexagon, each having a vertex at its center. Triangles and spherical geometry. We have mentioned some possible constructions in the Euclidean plane. What happens in varieties of points arranged in forms having non-zero curvature? Let us focus our mind on a spherical arrangement of points. If $S(r)$ is the spherical surface of radius length $r$ and center $O$, then we know from geometry, but also from common tactile experience, that $S(r)$ is a space of positive curvature, actually equal to the inverse of the radius length, i.e., $\frac{1}{r}$. The fact that the spherical surface has positive curvature requires considering triangles of a nature quite different from those of Euclidean geometry; this latter fact well known to anyone minimally expert in open sea navigation. Indeed it is not difficult to discover...