The golden section in mathematics

The form of the subject, based on the correlation of the golden proportion, as psychologists and mathematicians have proved, is perceived by man as beauty and harmony. The golden section in mathematics is such a division into parts of a segment, when the entire segment refers to the greater part, as large refers to the smaller part.

It is believed that the concept of a golden section was first introduced by Pythagoras. There is a suggestion that he took from the Babylonians and Egyptians his knowledge of what a golden section in mathematics and not only in it, but also in architecture, painting, art and many other things, was. In fact, the proportions of the temples, the pyramid of Cheops, some objects of everyday life indicate that the masters from Egypt used the proportions of the golden proportion in their construction and fabrication.

Plato also knew about the golden section. In his dialogue "Timaeus" he considers questions concerning the aesthetic and mathematical aspects of the Pythagorean school, including the problems of the golden proportion.

In the proportions of the facade of the temple of the Parthenon, there is a presence of golden division. During the excavations of this temple, circuses were found, which were used by sculptors and architects of Ancient Greece. In the circus found in Pompeii, which is now in the museum in Naples, these divine proportions are also laid.

The first mention of the golden division in ancient literature, which has come down to us, can be found in the "Elements" of Euclid, where the construction of the golden section is given geometrically.

In medieval Europe, the secrets of the golden section were kept in strict secrecy, carefully guarded. They could only be known to the initiated.

During the Renaissance, interest in gold division is growing. The brilliant artist and scientist Leonardo da Vinci, of course, could not have been unaware of the divine proportion and used it in his works. In addition, he began to write a book on geometry, where he wanted to show the wonders of the golden proportion, but he was ahead of the monk and great mathematician of Italy Luca Pacioli, who in Venice in 1509 published the book "Divine Proportion."

Mathematician of the Middle Ages Leonardo of Pisa (born around 1170 - died about 1250), better known as Fibonacci, was one of the famous scientists of the time. For the first time in Europe, he used Arabic instead of Roman numerals and opened a sequence of numbers in mathematics, later named Fibonacci. It looks like this: 1,1,2,3,5,8,13,21, ... and so on. The sequence of such numbers is called sometimes Fibonacci numbers. The golden section can be seen here. You can see that in this sequence of numbers, each of the following is formed if you combine the two previous ones. If we divide each successive term of this remarkable sequence by the previous one, we obtain a gradual approximation to the Fibonacci number (Φ = 1.6180339 ...). This is the golden section of Fibonacci, expressed by the number F. This number, like the famous number Pi = 3.1415 ..., does not have an exact meaning. After the decimal, the number of digits is infinite. So the golden section in mathematics shows itself. Thus, mathematical and not only miracles begin. If we divide any member of the sequence into the next one, then we get the number 0, 6180339 ... Miracles happen again - after the decimal the digits repeat exactly all the digits of the number Ф, only the comma is not 1, but 0. There are a lot of mathematical paradoxes . And this is only the beginning. The golden section in mathematics and not only in it simply works wonders, only we sometimes do not notice it.

It exists in architecture, music, mathematics, poetry, economics, in the structure of plants, in the stock market, in the proportions of the human body and animal bodies, in the spiral of the cochlea, in the macro- and microworld, in the Universe, and so on, up to Infinity ...

So, one can make the assumption that the Golden Section (the Golden Proportion, the Divine Proportion) is present at all levels of the universe.