Geometric Analysis and Constructive Module of San Galgano Abbey

visible and invisible to then execute it in form. This completed architectural form we can only analyze and study and we must do so with respect and reverence, trying to move with the same spirit that nourished the mind and heart of these wise architects. Below are the studies and logical reasoning that led to highlighting the constructive module of the Abbey of San Galgano. As already explained in the article on the musical geometries of the Gothic temple (see Mathesis, Numbers 1 and 2), the Master Builders, to graft a precise mystical idea into the proportions of the plan and the rhythm of the elevation, instead of using subjective measures and ratios, used a precise ratio defined by the interaction between numbers of symbolic value. This ratio, called constructive module, represents the physical and metaphysical DNA of the entire building, as it regulates the immanent aspect, that is technical-aesthetic and, at the same time, conveys and expresses the transcendent aspect, or symbolic-theological that lies at the origin of the project, giving the entire construction a plastic harmony to which is added an even more mysterious harmony of spiritual nature. Therefore, in the process that leads to defining the constructive layouts of a temple we have three essential elements: the mystical Idea from which the project originates, the primitive circle theurgically qualified into sacred space and the constructive module that has the purpose of polarizing in purified form the mystical idea inspiring the Temple. We have said that the constructive module was actually translated into a numerical ratio, but this was not calculated but rather applied through particular geometric-symbolic processes that led to generating the Square of the Earth. This is a quadrilateral inscribed in the directing circle that manifests and fixes on the concrete plane the mystical Idea, that is an aspect of the Celestial Will polarized by the module. The four vertices of the Square of the Earth determine the points on the ground where the four columns of the Transept Crossing will be erected. All the nodal points of the construction then originate from this epicenter whose symbolic essence is propagated both in plan and elevation through geometric-musical principles proper to the mathematical structure of the Cosmos. To analyze a symbolic project of a sacred building one should have the drawings, possibly dimensioned. In the absence of these we can reconstruct them by measuring as precisely as possible the main parts that compose the Work to be analyzed. In this case, the surveys and drawings made by Architect Antonio Canestrelli have been of great help, which report the dimensions of the nodal points of the plan and elevation of the monument, many of which were confirmed on site by our measurements. The missing dimensions were then derived from the drawings so they are not free from errors, which, if there were any, are nevertheless of small constructive entity and practically irrelevant in the symbolic process of geometric crystallization of a mystical/theological idea in the complex form of an Abbey or a Cathedral. Let us focus on the measurements of the transept crossing. The rectangle in question has one side of 11.15 meters: the width of the central nave taken on the axis of the great columns. The other side is 8.75 meters: the length of the transept. The ratio of these two measurements equals perfectly (considering a tolerance of a few millimeters) the square root of the golden number, i.e. $\sqrt{\varphi} = 1.2720$. We therefore have that the Transept Crossing reveals a particular right triangle in which the sides are in geometric succession with ratio square root of the golden number. This means that the major leg BC equals the minor leg AB multiplied by $\sqrt{\varphi}$, i.e. $BC = AB \cdot \sqrt{\varphi} = 1.272 \cdot 8.75m$. At the same time, the hypotenuse AC equals the major leg multiplied by the square root of $\varphi$: $AC = BC \cdot \sqrt{\varphi} = 1.272 \cdot 11.15m$. Finally, we have that the diagonal of the transept crossing is in golden ratio with the length of the transept (Fig. 5). $AC = AB \cdot \varphi = 1.618 \cdot 8.75m$ (35). For our investigation on the presumed module of this Abbey, it was fundamental to discover that this particular rectangle can be obtained through the intersection of two circles having radii of 8 and 7 units. In particular, the circle of value 8 units is the foundation circle (defined C1 in the drawing) oriented to the East and on whose circumference is located the center of the second circle (Fig. 6). This ancient method guards and expresses profound symbolic and operative knowledge, here summarized: The two intersection points of the circles are, symbolically, the doors where the two Idea-Forces expressed by the relative Numbers come into contact in form, passing one into the other. This interaction generates a line that expresses the product.