Seduced, exasperated, we cannot
make our harmonious relationship.
You believe that it is the body which apeteces.
Worm, are numbers!
Love with rage, hate vehemently,
5 is to 8, 5 is to 8… quick, fast,
Let’s make music and madness.
You danced, golden section!”
A murdered mathematician
The history goes back to the time of the ancient Greeks: the mathematical Hippasus of Metaponto would have been launched from a ship to the waters of the Mediterranean members of the Pythagorean school, making it the first martyr of science. The reason? Hippasus would have revealed the secrets of the Golden number to members of another mathematical school…
This story is referred to in several historical sources (some more serious than others), always in a different way. Some pointed out that, in reality, the revealed number would have been the square root of 2. in others, that Hippasus was not released to water, they would have committed suicide by drowning (force?). Finally, there are those who point out that everything was due to the zeal of the same Pythagoras, who envied the talent of Hippasus. Thus, revealed number would have been just an excuse to justify the crime (because, indeed, reveal the nature of irrational numbers was a real crime to the Pythagoreans…)
Several centuries later, none of this was already a mystery, because it was masterfully displayed by Euclid in his elements (one of the most important books in the history of science, comparable only with works by the likes of the mathematical principles of philosophy born in Newton or the origin of species Darwin). Euclid spoke “Division in mean and extreme right”, alluding thus to one segment of any cut in two pieces so that the ratio of the lengths of the total segment and the longer section is equal to this and the short stub. The Hellenic wise revealed, moreover, how to make this cut using the unique technology of the time: a rope (ancestor of the already forgotten rule and compass of the school). In addition, ruled that the Court is given in a point, at an equal proportion to
Since the beginning of the 20th century, this number would be denoted with the Greek letter φ and called aureus or Golden number. But let us follow in order.
Another of the great advances of Euclid is the layout of a Pentagon regular (i.e., of equal sides and equal angles), something about what previous civilizations had failed. It happens that the Pentagon is directly related to the number of gold, as the ratio of the lengths of its diagonal and his side is necessarily equal to φ.
By the way, after the advance of Euclid had to wait more than two thousand years for the German genius Karl Gauss revealed the construction of regular polygons of 17, 257 and 65537 sides. Along with of 3, 4 and 5 sides, as well as their combinations (for example, of 15 sides, because 15 = 3 x 5), these are the only ones known to date. But this is another story…
Lost knowledge
Fire marks a milestone at the dawn of the European dark ages. Indeed, the burning of the library of Alexandria and its subsequent decline (including the destruction of the library Serapeo sister) remains until today the greatest cultural tragedy of history. Although there is an official version (no clear dates) for what has happened, seems to be that the instigation Christian (Coptic) played an important role both in this and in the murder of Hypatia, the greater genius of humanity by that then (by the way, is a (female, but the rule forces to name it in the masculine when compared also with colleagues from this sex…). However, some myth in everything there is also this. Carl Sagan masterfully gathers some fragments of the history of Hypatia and the library in his Cosmos series. As he mentions, “only we know attractive titles of the books that were destroyed; in the majority of cases, we ignore the titles and authors”.
Probably the most visionary work of Archimedes, “The method” was lost in this destruction. A copy of this was found in Constantinople at the beginning of the 20th century in the form of a palimpsest (i.e. had been deleted to write about her other text – in this case, one religious-). To decrypt it, it was found that the basic ideas of integral calculus, conceived by the genius of Syracuse nearly two thousand years before Leibniz and Newton were there deployed. Let me dream: If “Method” has not been lost, most likely the photograph of the first black hole would have captured it a couple of centuries ago, and we would be already venturing into space colonization. But this is another story…
To the 5th century, much of classical knowledge had been banished from Europe. For centuries, science moved to India, Persia and the Arab world. When the latter was extended to cover the current Andalusia (the Al – Andalus era), began a curious cultural flow: the ancient texts returned to Europe through retraducciones made (from the Arabic into latin) from books containing annotations and extensions of sages and the Juarismi (the inventor of algebra). Among them was the secret of Euclid Division in mean and extreme reason, of which only a pale record was kept.
The Golden or divine proportion
It was the mathematician and Franciscan Friar Luca Pacioli, who, during the Renaissance, granted a sacred connotation to the Euclid division, and expressed this idea in his famous work the divine proportion (available here). It is very likely that this is the most beautiful of the history of mathematics book, because it was illustrated by his friend Leonardo da Vinci. Include drawings of the book down, than a perfect truncated icosahedron (figure based on a current soccer ball…). Why include it there? Because, as Pacioli, showed in the Polyhedra regular and semirregulares – especially the icosahedron – Golden proportions appear everywhere between different components, so there always tends to be a hidden pentagonal geometry.
By the way, the five regular polyhedrons were objects venerated by the ancient Greeks, who saw in them mystical properties. Thus, when Johannes Kepler proposed its first model of solar system erred deeply, because he was so imbued with this geometric mysticism that he thought that the hitherto known planets had to fit in accordance with the forms of such objects, as is illustrated below. It was not until he had access to the data of the movement of Mars who understood their mistake. It desdijo and, incidentally, formulated his famous laws of planetary motions. Finally, let us not forget nor who corroborated these laws, that he “which genius outperformed all mankind” (Isaac Newton), was – as well as physicist and mathematician – a regular Alchemist…
The number of gold and art
The name “Golden number” or “Golden” for the divine section is much later. She was born in a book in 1931 of the mathematical and Romanian Prince Matila Ghyka, who picked up the theory of the German philosopher Adolf Zeising according to which this number is an aesthetic rule universal, respected also by nature. It then began a frantic search for Golden proportions in all the works of art of ancient Greece. Was it also in paintings of da Vinci, who allegedly used it as a secret code (here is the origin of the title cleverly chosen by Dan Brown to his best-selling book). And, of course, appeared in the measurements of the Gothic cathedrals, including that of Notre Dame (in its facade) and Amiens (on the ship). The first thing would have to be confirmed by Eugène Viollet-le-Duc, the architect who restored the Temple of Paris in the 19th century and became erect needle that succumbed to the fire of a few days ago.
Notre Dame
A beautiful and sympathetic account of all this appears in Donald in the country of the math film, produced by Walt Disney and winner of an Oscar Award in 1959 (see here, minute 2:43).
The cultural penetration
Celebrities from the world of culture of early 20th century succumbed to the charms of the Golden proportions. Without going any further, Le Corbusier expressed them in a theory of aesthetic rules comprising the creation of a universal system of measurement units: the Modulor. If you notice reproduced below 10 Swiss francs banknote, you can see the famous circular sketches of the theory of the great Swiss architect.
Ghyka ideas had wide acceptance in France once his book was praised by Paul Valéry. In a letter to the Romanian – which was transformed into the introduction of the book, the famous French poet wrote among many other accolades: “what poem analysis of φ!”.
Can the numbers be poetry? As none other than Albert Einstein stated that mathematics is the poetry of logical ideas…
In the end, poems for φ abound in Spanish. For example, Rafael Alberti dedicated his sonnet “the golden ratio”. And the poem on this article header? It is a fragment of “Venus in el Pudridero”, of our dear Eduardo Anguita. And the numbers 5 and 8 are named in it? Because 8/5 = 1.6 is one of the first approaches of φ which delivers the famous sequence of numbers of other Leonardo of Pisa, better known as Fibonacci. Incidentally, this is the same sequence of numbers that is sketched in the Modulor of Le Corbusier.
There is (more than) a Chilean in the Middle
Anguita is not the only Chilean in this golden story In 1984, Gastón Soublette published the star of Chile, work which embodied his aesthetic and symbolic analysis of several national emblems. The most notable es referral to the flag that was signed the independence of Chile, which is preserved in the national historical Museum (a curious fact is that, for those years, the flag was “kidnapped” by the MIR…) Soublette argues that various proportions in this emblem are auras, which registers it in the most noble of the European aesthetic tradition and, at the same time, merges it with the mapuche tradition through the eight pointed star which carries embroidered at the center of the Pentagram.
Four years ago I did a work of reconstitution of a geometric model of the flag which leads to the Golden proportions. Certainly there was unavoidable inconsistencies in what was postulated by Soublette, but corrected once flowed an unquestionably elegant object. Part of this can be read here.
At the end of last year, a review on this story I published in the country of Spain. The impact of this note in the local media was particularly regrettable: swept away the old and clumsy chauvinism of always, advertised it as “the definitive scientific proof that the flag of Chile is the most beautiful of the world”. Simply, a disgrace…
A divino-aureo Sham – Golden?
As you might expect, the golden section had wide implications in the architecture of the 20th century. Proof of this are the designs of the building of the United Nations in New York or Toronto CN Tower. But the exegetes of the Golden reason soon fell into claims that, despite its obvious nonsense, have been difficult to banish. For example, said that it is nothing less than the Egyptian pyramids (which, as the pseudo-arqueologos, not would have been able to be built without extraterrestrial help…) present. It is also used to justify certain practices eugenic and racist (one whose bodily proportions are not gold belongs to an inferior race). Finally, internet browsing I find no less than with a manual for use of the golden ratio in dentistry…
In particular, it is impossible to say with certainty the relevance of the golden ratio in the architecture of the old if there is no written document that support it. However, continues to be amazing – though, for the more skeptical, not conclusive – see her appear almost irrevocable way in numerous works: in addition to the cathedrals of Notre Dame and Amiens are counted the Farnese Palace in Italy, in France Maulnes Castle , the forts of Vauben in France and Spain, numerous temples in Burma (as sketched below on the left below), and the Taj Mahal of the India (reproduced right). For many, this is the most beautiful monument on the face of the Earth.
In accordance with the above, it is also impossible to confirm the presence of the golden ratio in our flag of independence if there is no official document about its origin. In this regard, a task for historians would seek in the archives of the Military Academy or Naval Academy any record of our first emblem, it is known that his clothing in the original proportions spread until the end of the 19th century (proof of this they are the flags preserved in the historical Museum and weapon of Arica and Valparaiso National Maritime Museum).
On the inherent aesthetic value of the golden ratio, this isn’t scientifically corroborating. While in this regard are carried out several experiences (starting by Gustav Fechner in the 19th century), nothing is totally conclusive. There is consensus, however, that a statement of this type should be field of study for the Neurosciences in the not-too-distant future.
Finally, about the presence of the golden ratio in nature, it is impossible to deny. Numerous processes occur following patterns hopelessly flowing in these proportions, which can be explained through mathematical models ad hoc. Now, as points out the famous Harold Coxeter Canadian geometer, it is more of a trend, and not an exclusive law.
And the number of gold, is really important in math?
Contrary to what you think skeptics of everything that revolves around the golden section, the number if has importance in mathematics, as it appears in the most diverse situations (“magically”). For concrete, I will try to explain just one.
As it is well known, the square root of 5 is an irrational number, and therefore so is this it means that he can not be written as division a/b between integers a and b. However, may be approximate, for example, by the Fibonacci ratios (if you don’t know the rule of the sequence below, then discover it!):
2/3, 3/5, 5/8 (anguita), 8/13, 13/21, 21/34, 34/55…
Now, any approach by fractions of an irrational number leads to a “cost”: the difference between the original number and its approximant a/b is close to 1/b2. What’s so close? Then depends on the “level of irrationality” number. ” A theorem proved in 1891 by Adolf Hurwitz says that, in this ranking of irrationality, none surpasses φ: this is the most irrational of all numbers! Certainly, Pacioli had intended this as a confirmation of the Supreme and divine character of the golden ratio.
To be convinced of the scientific topicality of the above, refer to the plenary lecture by the great Brazilian mathematician Carlos Gustavo Moreira in the last International Congress of mathematicians, that is, by far, the most important event of the mathematical world (see from min 42:10). Finally, if all this seems too complicated, I leave the formula most beautiful number of gold for simple contemplation:
While the mystery persists
Beyond its questionable technical and scientific relevance, the myth of the Golden Division seems having triumphed by make vibrate a crucial strand of the human being: his aspiration to a higher reality. Pacioli was that reality in the paradise of God; Paul Valéry and Le Corbusier were it in the aesthetics and forms; Newton sought her through the universal language of mathematics.
Regardless of its Catholic roots, the Cathedral of Notre Dame is also a symbol of this aspiration, and the presence of the reason Golden therein (whether genuine or forced) is nothing but a manifestation of this. The flames were now part of its structure and its treasures; Fortunately, they cannot be deleted them forever — and, unfortunately, the flames of Alexandria did with his library-.
Repair of a work like this is arguably costly, although certainly required. But obviously, with equal intensity should have mobilized hundreds of other recent losses in the world (Nineveh, Palmira, the Buddhas of Bamiyan, the National Museum of Brazil…), as well as other closer: what we do against the reverse held our do forests of araucarias – mapuche paradise, paraphrasing Soublette-?; could already be time to restore the symbolic nature of our original Golden flag?; Why not allocate more resources to repair several of our own heritage buildings abandoned for years?
