Mario Livio. Translated from English: Emmanuel Lotem. Arie Nir Publishing House, 317 pages, NIS 84
Arie is a pharmacist
The hero of the book is the number known as "The Golden Ratio", and its greatness is that it appears in countless unexpected places: in nature, art, geometry and number theory. It is an irrational number (that is, it cannot be expressed as a fraction of two whole numbers), with infinite digits, the first of which are 1.61803. He occupied and occupies the imagination of scientists and creators for a long time, and will continue to occupy them a lot and appear in the most unexpected places.
This number is a relative of an equally famous series, known as the "Fibonacci series". She also stars in the book, and it is actually very difficult to separate these two entities. Therefore, we will start with this aspect of the golden ratio (it is customary to mark it with the Greek letter phi. In Hebrew it is pronounced phi.
The Fibonacci series is named after the Italian mathematician Leonardo Pisano (born in the city of Pisa, 1250-1170), a member of the Bonci family, and he was the first to describe it in his book Liber Abacci. This is the following series of numbers: ...0, each number expressed as the sum of its two predecessors. It is easy to see that division of a number in the previous one is getting closer to the golden ratio, for example 1/1 = 2, 3/5 = 8 and so on. In this context it can be proven that the golden ratio can be obtained by adding one to the root of the number 13, and dividing by two. The Fibonacci series itself has so many surprises and charms that a magazine was founded about thirty years ago - Fibonacci Quarterly - which deals exclusively with magic related to it and new discoveries are made every morning.
One magic is instant: take two numbers in this series, which are two places apart, for example: 13 and 5. Multiply them together. The result is 65. Now take the number in between and square it. The result is 64, one away from 65. It works throughout the series. This phenomenon has the use of various paradoxes and deceptions.
And here is another charming charm presented in the book, and related to another strange combination of Fibonacci numbers with Pythagorean triples. A Pythagorean triple is a triple of whole numbers, so that the square of one of them, plus the square of the second number, form the square of the third number.
For example: the triad 3, 4, 5 is a Pythagorean triad, because three squared plus four squared is twenty-five, that is, five squared. And here, the book gives a magic recipe how to build Pythagorean triples using the Fibonacci series: take four consecutive Fibonacci numbers, for example: 1. Multiply the two outer ones: one times five, and you got five.
This is the first of the Pythagorean trinity that is being built before our wondering eyes. Now multiply the inner two and multiply the result by two: two multiplied by three multiplied by two is twelve. This is the second number in the Pythagorean triple. Now take the sum of the squares of the two inner numbers: two squared plus three squared and the result is thirteen, this is the third number in the Pythagorean triple. Indeed: five squared plus twelve squared is indeed thirteen squared. Here, as promised, is a mysterious, strange and unexpected connection between two different entities in the world of numbers: Pythagorean triples and Fibonacci numbers.
As mentioned, the Fibonacci series is so amazing that there is no point, and it is also impossible to review even a tiny part of its mathematical charms, but we will dwell on an amazing feature of it that I came to know thanks to this wonderful book, and only because of it it is worth reading. This is the first digit feature. If you check the first two thousand numbers in the series, the book says, you will find that the number one appears as the first digit in 30% of the cases, the number two appears as the first digit in 17% of the cases, the number three in 12% of the cases, and so on. And here, a professor named Mark Nigrini studied the populations of about three thousand counties in the US census in 1990. He found that the number one appears in 32% of the cases, the number two appears in 17% of the cases, three in 14% of the cases, Exactly the same distribution we described in Fibonacci numbers. This is again a very strange and mysterious phenomenon.
The physicist Benford generalized this phenomenon to many areas, including the area of river drainage basins. The strangeness and magic of Fibonacci numbers therefore never end. No wonder there are philosophers who claimed that "Fibonacci numbers are the building blocks of God".
It is very possible that this is so, in particular that they do not appear only in mathematics but also in nature, and the beautiful example presented in the book (p. 119) will prove this. The Swiss researcher Charles Bona established the term phyllotaxy, regarding the arrangement of leaves in a tree. For example, the apple, cork oak, and peach grow leaves every 2/5 of a turn (note that 2 and 5 are numbers in the Fibonacci series, with a skip of one), while the pear and weeping willow grow leaves every 3/8 of a turn (again, these They count, in one skip, within the Fibonacci series). How does nature know how to behave like this?
In my humble opinion, an answer to such a phenomenon is not possible. But it is possible to behave differently: it is possible to assume in advance that Fibonacci numbers pop up in every unexpected place in nature and mathematics, and simply look for them. There are also those who claim such an attitude in the behavior of exchanges. The book contains many examples of the presence of Fibonacci numbers in nature, music, art, geometry, economics, and in short - everywhere unexpected.
In the meantime, readers must feel this way, we have neglected the golden ratio, the other hero of the book. what about him Where did he pop up in any unexpected place? do not worry! He also appears in every corner of art, nature and mathematics.
Many art scholars claim that the golden ratio (or the golden ratio) expresses the classical relationship between the length and width of a rectangle and other shapes. They even claim that beauty in nature, for example a beautiful face, contains the golden ratio. The author does not agree with any such statement, and even flaunts it. Similarly, some claim that the dimensions of the great pyramid in Egypt were derived from the golden ratio, and this assertion is also analyzed at length (perhaps a little exaggerated) by the author.
This is an extremely important and interesting book, and the translation by Emanuel Lotem is excellent. On the other hand, there are some disadvantages. On the one hand, the book cites many books and articles but does not indicate the original names in Hebrew, and this makes it difficult for readers to locate the source.
Another disadvantage is the lack of reference to internet sites, which also contain an inexhaustible sea of material. For example, I found the following case on the Internet and it is called the law of Jean-Claude Perez. He studied the sequence of the genetic code in the insulin molecule, which consists of 90 bases ("letters"). The possible letters in the genetic code are: GA, C, T. If you count the number of letters T in the genetic sequence, you get 34 letters, when the sum of the other letters is 34. 56 and 55 are two consecutive Fibonacci numbers.
Furthermore, this is how he found, if you take a subsequence of the genetic code in insulin, this phenomenon repeats itself: in a subsequence of 54 letters, you get a total of 22 letters T (which is close to 21, the Fibonacci number), And the sum of the rest of the letters is 32, which is close to the Fibonacci number that follows 34, 21.
A very simple advice for those interested is to use a search engine and type in the word Fibonacci, or the words golden ratio, and get an inexhaustible abundance of fascinating material.
Another drawback of the book is the too long preoccupation with the question of whether the golden ratio was indeed known to Egypt and Babylon and who discovered it for the first time. But this shortcoming is definitely compensated by the other chapters.
The Golden Ratio Mario Livio
Aryeh Rokeh's articles on Fibonacci numbers were published in "Fibonacci Quarterly" and "Al Ha" - the bulletin for mathematics teachers
https://www.hayadan.org.il/BuildaGate4/general2/data_card.php?Cat=~~~783130595~~~133&SiteName=hayadan
