Where Mathematics Twinned With Beauty

The Great Secret of Creation is an idea that remains the only eternal power to reveal the creation of the Like One from the One Like !

- I don't contradict you anymore! I will even add the other characteristic of the golden ratio, that of showing that there is a possibility to measure beauty. The Greek philosophers were thinking of this golden ratio when they said that beauty means "well balanced proportion". And, even if they didn't call it "section aurea", like Leonardo da Vinci, all the ancient statues, not only those from the century of Pericles, but also from later, were sculpted with this golden section in mind.

Similarly, Vitruvius said that "symmetry derives from proportion - what the Greeks call analogy - the consonance between each part of the whole. Leonardo da Vinci argued that the golden ratio should be the law that dominates architecture, so that proportions can be established through it between the different parts of a building, as well as between the built volume and the free one, because only the observance of this ratio makes an architectural ensemble pleasing to the eye". That is why, in Matila Ghyka's book: Aesthetics and the theory of arts, numerous measurements are mentioned that were carried out in relation to the "dimensions and proportions of the temples of Hellas" from which it appears that, in the beauty and harmony spread by the Parthenon, the Propylaea and other Greek temples, the golden ratio also intervenes.

- You are right. Greek architecture presents another aspect of beauty, I could say, diametrically opposed to that which Egyptian architecture inspires, for if the majesty of the pyramids or the Sphinx brings to mind the thought of death and eternity, Greek architecture pulsates with life. It is enough to look at the Temple of Athena Nike, its elegant and harmonious columns that carry a pediment in the form of an isosceles triangle, everything being calculated to give the impression of grace and elegance.

- But even in the statues, the Greek sculptors did not use live models for their deities, but made them taking into account a certain system of proportions that had the purpose of producing the impression of harmony of everything. From the Renaissance to the present day, all artists have taken into account this division of segments. I only remind you of Kepler, the astronomer, mathematician and, I could say, the esthetician of the 16th century, because he wrote, in addition to the other well-known works, a treatise on aesthetics with the title: Harmonics of work (Harmony of the world), in which expresses itself in an entirely original form about the golden ratio: "This geometrical proportion, I believe, is an idea which remains the only eternal power to reveal the creation of the Like from the Like!"

Can you form a certain perception of what "the beautiful from divine creation" is, so that you follow the coordinates of a single purpose: to generate a single vision of a science that firmly assumes an object of study?

- If I were to choose the beautiful from the mathematics left by the ancient Greeks, I can honestly say that I don't know what I should leave out. I have the impression that the Greek philosophers, who were also mathematicians, wove all mathematical problems with threads of beauty, harmony or delight. For example, the theorem: "all angles inscribed in a semicircle are right", which was established and proved by Thales. It is said that it impressed him so much that he sacrificed an ox to the gods.

And only Thales was neither at his first scientific discovery, nor a naive one, who let himself be caught in the trap of pride. He was one of the seven wise men of Greece, had traveled through Egypt, and had astonished Pharaoh by showing him that he knew how to measure the height of the pyramid by the shadow it cast alone. And yet, in this theorem, which brings no practical use, he saw a beauty worthy of admiration, a beauty that delighted him and for which he felt the need to thank those he considered to be the masters of wisdom.

- But this theorem is still so beautiful today, although we pass over it with great ease! The fact that it is enough to take any point on the semicircle and join it with the extremities of the diameter to form a right angle by itself, can delight you like a song, or like a poem, or like a painting! Equally remarkable is the Pythagoreans' admiration for number itself, number abstract and devoid of any practical meaning. In order to isolate it from any kind of application that could be attributed to it in social and economic life, the Pythagoreans first, and then the other Greek mathematicians, invented a new science, called Arithmetic, a science that would deal only with the properties of numbers abstract, of course wholes, for only wholes were regarded as numbers.

The other concrete numbers, which could also be fractional or even irrational, were considered as quantities and the calculations with them belonged to another science, called logistics, a science within the reach of merchants, slaves and neglected by philosophers. Arithmetic did not have any practical applications, it was studied only as a pleasure, and Socrates defines it in Charmides or on wisdom as: "the science of what is even and odd, of the differences between numbers and of the relations between them".

Can you express what you perceive as "beautiful" within a similarity between a discovery that is made only to the soul and through the soul, and an empirical discovery that is constantly confirmed in all situations of life?

- That's right. For Pythagoras, Arithmetic was the most beautiful of the sciences and even the most advantageous, because only in this field, "things appear in the form of numbers"!

- As if only Pythagoras saw Arithmetic as the most beautiful science? Didn't Gauss say that Mathematics is the queen of sciences, and Arithmetic is the queen of mathematics?

- He gave his hand to Gauss to say it, because he had the science already cultivated by the Pythagoreans and all those who followed them for more than twenty centuries, while Pythagoras was stirring the harmony that would charm future generations. There were things that no one had thought of before, and which he then strung like shining beads on a thread. Let us think only of the division of numbers into primes and composites. From a practical point of view this division serves no purpose. But how much delight have these numbers created through their properties in the souls of mathematicians, starting from the time of Pythagoras and up to today?

I am thinking, for example, of the demonstration that Euclid gives, in Book IX, about the infinity of prime numbers.

- You thought well, because this demonstration delighted many mathematicians, from then until today, and they encouraged them to add other new demonstrations. Should we take the Elements to read, right there, how Euclid formulated this discovery of the Pythagoreans?

The Great Secret of Creation aims at a discovery that is not necessarily based on observation and mathematical calculations, but on belief in a world that has been ordered by a supreme intelligence.

Where mathematics has twinned with the beautiful, the obligation is established to recognize the transfer of the responsibility of divine creation to man and to follow the advice of the Creator: "Everything you do for science, you also do for Me. And everything you discover through the power of faith, is a invitation to explore the beauty in you as a reflection of the beauty around you".

Returning to mathematics, here is what Wikipedia tells us: Among the more recent works on the golden section, which are the basis of Matila Ghyka's thinking, we should also mention those of Heinrich Emil Timerding who, analyzing how the principle of the golden section was applied, insists on the fact that "the golden section is only a particular case of a more general rule, that of the recurrence of the same proportions in the elements of a whole".

* Note: Campan, Florica - Stories with proportions and symmetries, Albatros Publishing House, 1985.