The "absolute value" of the proportion

Article written by Serena Nardoni

Towards the end of the Middle Ages, at the dawn of Humanism and the Renaissance, philosophical and artistic research began to show the need for a return to the concept of perfection of bodies, which tended to lead every element of the natural world (above all the human body) to geometric forms that were an extreme expression of symmetry. It is the so-called phenomenon of the rediscovery of the Platonic solids, that is, 5 regular polyhedra distinguished by characteristics absent in other geometric solids:

• They are primarily formed by regular polygons and, in particular, by equilateral triangles, pentagons or squares;
• The solid angles enclosed between two or more sides of the same figure are all homogeneous with each other;
• They are the only convex polyhedra inscribed in a sphere with the vertices touching the surface of the sphere.

The recurrence of these elements conforms the peculiar character that has made them the object of study and design particularly sought after not only by mathematicians, but also by philosophers, who have imaginatively sought them in cosmic representations (Pythagoras) or in the fundamental elements, namely fire, earth, water and air (Plato).

In his Timaeus, Plato says:

Before that time, in truth, all these things were in a state devoid of reason or measure, but when the work of setting in order this Universe was being undertaken,fire and water and earth and air, although possessing some traces of their own nature, were yet so disposed as everything is likely to be in the absence of God; and inasmuch as this was then their natural condition, God began by first marking them out into shapes by means of forms and numbers.

But it is with the Renaissance that the Platonic solids are projected into the world of art. Paolo Uccello, for the flooring of the basilica of S. Marco in Venice, juggles between perspective and geometric inspiration, with the design of an unprecedented starry dodecahedron.

The Timaeus was undoubtedly the text on which the studies of another immense expert of perspective of his time were focused: Piero della Francesca, who dedicated his "De quinque corporibus regularibus" (1482-1492) to the five polyhedra.

In the text, the artist tries to demonstrate how every known physical element can be traced back to volumes and shapes apparently complex, always decomposable and traceable to the Platonic solids, as basic forms, eternal and perfect.

A little later are the reflections carried out by an undisputed genius of the history of art: Leonardo da Vinci. The artist illustrates with wonderful watercolors the "De divina proportione" (1509), a mathematical treatise in three volumes by the scholar Luca Pacioli.

Leonardo's representations express the lightness, elegance and perfection of these figures, hanging from a thin wire and titled with a cartouche bearing the Latin name of the solid.

Luca Pacioli with the concept of "divine proportion", presented in the first volume, means the golden ratio, that is that specific value, expressed with the greek symbol Φ (PHI) and corresponding to the irrational number 1,6180339887, resulting from the ratio between two positive quantities of which the one with the greater value is the average proportional between the smaller quantity and the sum of the two.

To simplify, let's assume two quantities A and B, where A is greater than B.

If the two quantities are such that they form a proportion for which A is the average proportional to both B and the sum A+B, then the ratio between the two quantities will give as value the golden number.

If A>B;

If (A+B):A = A:B

Then A/B = Φ

Why "divine?" We can identify a number of unique characteristics:

1. Because it consists of different, non-periodic digits;

2. Because it is triune, being made up of three terms;

3. Because it is indefinable, being irrational;

Because it is invariable.

In the second volume, Pacioli applies his reflections both to proportions applied to the world of architecture and to the human body, while the third seems to be a vernacular translation of Piero della Francesca's work ("De Corporibus regularibus") another illustrious spokesman of the concepts of proportion and perspective (see his "De prospectiva pingendi", of 1460-1480, the first systematic treatise of perspective entirely illustrated).

Although betraying intuitions that were not entirely his own, Pacioli cannot be denied the effort he made to lead every field of knowledge back to mathematical ratios, as an expression of God's action. The golden ratio would be, therefore, a manifestation of the divine that surrounds us.

In order to go beyond the boundaries of mathematical research and enter the world of art, Pacioli found an excellent interlocutor in Da Vinci, a scholar of sculpture and architecture, botany, zoology, anatomy, astronomy, physics (in particular optics), chemistry, even hydraulics and mechanics.

However, Leonardo always remained faithful to a vision of perspective in a less "mathematical" key, preferring an artistic result attentive to the atmospheric characteristics that influence vision. What will remain most to Leonardo will be "De Divina Proportione" in the study of the relationships between the individual parts of the human body and the natural elements. An example is the illustration of the Vitruvian man, in which the distance between the navel and the ground and the entire height of the man are in a golden ratio.

Opening our reflections to the European panorama, the illustrations of "De Divina Proportione" were also decisive in the study of perspective in Europe. Among them, Dürer came into contact with Luca Pacioli during a stay in Bologna that saw the mathematician engaged in teaching activities at the University of Bologna around 1506. In a letter of the same year, Dürer says that he wanted to go to "Bologna for love of the secret art of perspective that someone is willing to teach me". And this someone could have been Pacioli himself. From this meeting the painter acquired the rudiments of perspective, guided in the reading of "De prospectiva pingendi" by Piero della Francesca and in "De viribus quantitatis" by Pacioli, in which geometric solids are traced back to planets and celestial bodies (so-called "Magic Squares"). Albrecht Dürer, in Melencolia 1 (1514), inserts precisely a magic square above the head of the angel symbolizing melancholy. The angel, with the sixth in his hand and a ruler at his feet, stares astonished at the horizon with his head lazily resting on his left hand. Around his clothes the floor is strewn with symbolic objects: a sphere, a saw knife, a table, nails, etc.. The magic square is the link between the planets and the moods of men.

Geometric shapes find further space in those repetitive patterns called "fractals", of which Escher was one of the greatest representatives.

The fascination of fractals is mainly linked to their self-similarity, that is, a system composed of the repetition, at smaller and smaller scales, of the same structure it has at larger scales, creating a pattern that can be replicated indefinitely (like the infinite branches of a tree).

This system of repetition suggests a sense of perfection, pleasantness and harmony, generating the perfect mix of predictability and surprise. Our brains naturally seek patterns in everything around us and in the way we process thought, so an art capable of restoring this sense of order and proportion delights the mind and soothes the spirit.

Is it perhaps in these patterns, which mathematicians and scholars of all times have written about, that the purest essence of beauty can be sought?