If you ask math people which is the most beautiful of all the numbers, many will choose the following:

Strange, but the truth is that this number (known as *golden number*) appears in numerous scenarios: numerical series, geometric figures, forms of nature … Both Leonardo da Vinci and Le Corbusier always respected these proportions in their works. Also the followers of occult currents have incorporated it into the iconography of their organizations as a symbol of sacredness, and the flags of several countries rely on the golden number in their design.

In geometry, we find φ in a pentagon of equal sides, exactly in the ratio between the length of the diagonal and that of the pentagon side. In arithmetic, it appears in the Fibonacci series, in which each term is obtained by adding the two previous ones ( starting with 1, 1, and then 1 + 1 = 2, 2 + 1 = 3, 3 + 2 = 5, 5 + 3 = 8, 13, 21, 34, 55, …). If you divide each element by the previous one, you get: 2/1 = 2, 3/2 = 1,5, 5/3 = 1,666 …, 8/5 = 1,6, 13/8 = 1,625, 21/13 = 1,615, 34/21 = 1,619 … Well, these quotients are getting closer and closer to φ.

The number φ is irrational, that is, it is not the division of two integers. In this aspect, again φ stands out, because it is the number *more irrational* of all. Indeed, there is a concrete way to measure how far is an irrational number from being a fraction and, in this ranking, φ surpasses all irrationals.

On the other hand, the proportions of various parts of living beings tend to be golden, that is, equal to φ. In humans, this is true, for example, for the distances from the feet to the navel and from there to the top of the head, the lengths of the forearm and hand, or the distances from the tip of the nose to the chin. and from there to the lips.

Already the ancient Greeks unveiled some of the mathematical mysteries of the golden ratio. For example, they proposed the first exact construction of a regular pentagon using ruler and compass. They were also sensitive to the aesthetic properties of φ, and implemented it in the design of decorations and constructions. In fact, the Greek letter φ (fi) was coined in honor of the famous sculptor Phidias, who followed golden patterns in the elaboration of the sculptures that decorated the Parthenon of Athens.

As for flags, the proportion between the length and the height of Togo's is equal to φ However, the displays the golden proportions at their maximum splendor It is an almost forgotten flag, the one with which Chile signed its independence 200 years ago.

Like several countries in Latin America, the Chilean independence process was inspired by the libertarian and enlightenment ideas of France and the United States. collected, among others, by Masonic lodges. The colors chosen for the flag of the nascent country, white, blue and red, are an evident sample of this. Even more, its perfect geometric design reveals a very deep connection. In it, φ appears in various proportions: between the lengths of the white and blue fields between the height of the blue part and the diameter of the circumference in which the star is inscribed, and between the parts of the blue side delimited by the projections of the axes of the star.

The last property is achieved thanks to the blue sector corresponds to a special rectangle, in which an angular configuration of 36 is displayed^{or}, 72^{or} and 108^{or}, very similar to the one that appears in a regular pentagon.

Several mysteries surround this design. Who did it? Why was no record saved? Who possessed this knowledge and taught it until the end of the 19th century? Why did no one give testimony of the presence of the golden reason for almost 200 years, until the philosopher and aesthetic expert Gaston Soublette found it?

Incredibly, all these questions remain unanswered. On the design, it is supposed that this would have been partly devised by the liberator Bernardo O'Higgins, corresponding the technical implementation to a military engineer. Here the opinion of the historians is divided: some posit Antonio Arcos and others Gregorio de Andía and Varela. A curiosity: both were of Spanish origin, but intervened in favor of Chile during the war.

The Flag of the Independence is conserved in the National Historical Museum of Santiago. In 1912, the proportions of the Chilean flag were modified (and simplified), thus losing much of the symbolism of this emblem. It was a true geometric sacrilege.

**Andrés Navas** He is a professor at the University of Santiago de Chile and a visiting researcher at the National Autonomous University of Mexico.

*Coffee and Theorems* is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share points of contact between mathematics and other social and cultural expressions and remind those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: "A mathematician is a machine that transforms coffee into theorems".

Editing and coordination: **Agate Rudder** (ICMAT)