Geometric Modules in Gothic Cathedral Construction

translates from a constructive point of view, into the cathedral's module. The module is, before anything else, the geometric incarnation of the idea and inspirational force that form the basis of the construction. From a practical point of view, the module is a symbolic arithmetic ratio - visualized and applied through special geometric forms - on the basis of which the entire cathedral will be harmonized. The arrangement of the columns of the transept and naves, the shape of the apse, the fundamental lines of the elevation, all these architectural elements, together with many others, have positions and proportions that are wisely defined and regulated by the Master Builder through the rhythm dictated by the geometric expansion of the base module, the DNA that unites the construction into a whole-unity. In Gothic cathedrals, this module directly enters into the tracing of what is called the square of the earth, consisting of the fourth of the six ritual phases associated with the foundation of the Temple. The square of the earth is a quadrilateral that will determine, with its vertices, the position of the four large columns of the transept crossing. It is important to specify that in most cases, the name square is geometrically improper, since the four columns of the crossing generally delimit a rectangle and only in some constructions, especially of Cistercian origin, these columns identify a true square, thus constituting the known ad quadratum module. Although improper from a strictly geometric point of view, the name perfectly accords with the symbolic meaning of the geometric construction. The relationship that exists between the square of the earth and the square of the sky traces the relationship existing between the Material Universe and the Divine Archetypal Plan, a relationship that must be expressed, enhanced and made dynamic in the Temple itself, thanks to the power of symbols and analogical correspondences used by the priest. One could indeed say that precisely in making explicit, dynamizing and protecting this harmony between the square of the sky and that of the earth, the main function of the Temple is resolved. Symbolically, the square of the earth is an incarnation of the square of the sky. While the square of the sky represents the regulatory and ordering powers of nature in a celestial and universal aspect, the square of the earth realizes them by creating the space for the material substantiation of the cathedral, thus allowing the incarnation of the celestial principle, which is polarized and conveyed by the geometric module. In Gothic cathedrals, from an architectural point of view, the square of the earth is often obtained by intersecting the fundamental circle, the directing circle of orientation, with two other circles having their centers at the points indicated by the vertices of the celestial square, and a specific radius in relation to the constructive module chosen for that cathedral. Classically, the Temple's module translates geometrically into a simple ratio between the diameter of the fundamental circle and that of the two new circles (see Fig. 14). For example, a $\frac{6}{5}$ module, which is often hypothesized to be the module of Reims Cathedral, will lead to establishing the diameter of the two circles in a ratio of $5$ to $6$ with respect to the fundamental one. The intersections of the three circles determine the position of the crossing columns, that is, the width of the main nave and transept. The Plan of Reims Cathedral The module represents the magical proportion capable of linking heaven to earth, allowing the assimilation and terrestrial substantiation of the celestial principle. The celestial will is symbolized by the action of the Sun which, through the Column, descends to earth and indicates the four cardinal points. Once the symbolic accord between the square of the earth and the square of the sky is found, mediated by the constructive module, this fundamental harmony is then amplified through a series of concentric circles rhythmed on particular geometric-musical harmonies, circles which, together with the parallels of the cardo and decumanus, determine all the nodal points of the cathedral's plan. From the heart of the cathedral, represented by the square of the earth, therefore depart all those operations that will lead to the structuring of the sensitive points of the cathedral's plan and to the definition of the rhythm of the wall elevation, all through a golden amplification and according to the ratios of the musical scale. To give a concrete form to this discourse, and