. EbeneMagazine – RU – Higher mathematics of life: where do Fibonacci numbers occur in nature?. ru

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Every year on November 23, the world remembers the first major mathematician of medieval Europe Leonardo of Pisa, known by the nickname Fibonacci. He opened the decimal Arabic number system for his contemporaries and, in general, enriched their knowledge in the exact sciences.. But his main discovery was a sequence called Fibonacci numbers. She is called amazing for her ability to unexpectedly appear in a variety of areas of life – from biology to painting..

Leonardo of Pisa, best known by the nickname Fibonacci (most often interpreted as « lucky »), was born around 1170 in the Italian city of Pisa. His father was a merchant and visited Algeria for trade, where he brought his son to study mathematics with Arab teachers.. Later, Fibonacci himself traveled to Egypt, Syria, Byzantium and Sicily, where he became even better acquainted with the achievements of ancient and Indian mathematicians. Based on the knowledge gained there, Leonardo wrote a number of mathematical treatises that became revolutionary for medieval Western European science. His most famous work was « The Book of the Abacus » (abacus is an ancient Roman abacus).

« In fact, it was an encyclopedia of mathematics of that time, » says the candidate of physical and mathematical sciences, associate professor of the Kuban State University Eduard Sergeev. – In it, for the first time in Europe, the decimal positional number system of the Arabs was set out. There, for the first time, negative numbers were used as debt. This large book ended with a presentation of algebra and examples of solving practical problems associated with trade.. Its 12th chapter contained the famous rabbit problem. It was thanks to her that the world learned about Fibonacci numbers « .

A problem invented by a medieval mathematician was intended to calculate the offspring of rabbits. According to her condition, two animals were placed in a corral fenced on all sides for breeding. The question is: how many pairs of rabbits can they produce per year if it is known that every month, starting from the second, each pair of rabbits produces one pair? The answer is 233 pairs. To find a solution, the author of the problem deduced a numerical series in which each subsequent term is equal to the sum of the two previous ones. It looks like this: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 and so on ad infinitum. Much later, already in the 19th century, this sequence was called « Fibonacci numbers ».

The date of November 23 for the unofficial holiday of the Fibonacci Day was also chosen based on its sequence. For this, the calendar format adopted in the West was used, in which the numbers first write the month, and then the day.. It turns out 11/23, which repeats the first four numbers from the mathematician’s series: 1, 1, 2, 3.

But even more interesting is that the Fibonacci number series has found application in many areas of mathematics, and to this day it surprises scientists with its versatility. In addition, many phenomena of the surrounding world were also associated with it..

“A periodical magazine is published in Italy called Fibonacci Numbers,” continues Eduard Sergeev. – Authors from all over the world write articles for him related to the Leonardo of Pisa sequence and other properties of numbers. And almost every year they discover something new. In my student years, some properties of Fibonacci numbers were known, but today others have already appeared, including completely unexpected ones.. One of the recently discovered amazing properties of Fibonacci numbers is that the same sequences of the last digits are repeated with a certain frequency.. That is, the growth of this series is not accidental and obey a certain law, which, apparently, is not yet available to our understanding.. This is a really mysterious thing. « .

The amazing properties of the Fibonacci sequence in mathematics are difficult to explain to a person without special knowledge, but a lot can be understood without formulas. One of the main features of this « golden series » is that the ratio of each subsequent member to the previous one is steadily approaching 1.618. It is known to mathematicians as the Phi number, but it also has many other names: the number of God, divine harmony, asymmetric symmetry, the golden ratio (the latter concept was invented by Pythagoras). Phi Constant was named after the ancient Greek sculptor Phidias.. Even the ancient builders knew that when using certain proportions, the building looks as beautiful as possible and, moreover, it turns out to be the most stable.. Briefly, the golden ratio is defined as follows: the smaller part refers to the larger, as the large to the whole. In percentage terms, this corresponds to indicators 62 and 38.

« Leonardo Da Vinci was also a virtuoso of the golden ratio, » says Eduard Sergeev. – This proportion can be found in his famous « La Gioconda » and other paintings. By the same principle, I once gave my students the task of drawing the most beautiful ellipse that is possible.. To do this, you need to calculate the ratio of a large diameter to a smaller one in terms of Phi. This is such a constant to which all recurrent sequences converge in an amazing way « .

A logarithmic spiral is associated with the Fibonacci series and the Phi number in geometry, which unfolds according to the principle of the golden ratio. It can be inscribed in a system of nested « golden » rectangles with aspect ratio equal to Phi, or described around it. And it is surprising that such models are often found in nature.. Shells of mollusks Nautilus pompilius and fossilized ammonites are built in the image of the Fibonacci spiral. Their growth is well described based on the Phi number with a factor of 2.

The ratio of the lengths of three turns of the spiral of the human ear exactly corresponds to Phi and the same parameters are in the shells of some snails. Recently learned that gold and other logarithmic spirals are found in the corneal epithelium of mice..

Even Leonardo da Vinci and the famous German scientist Kepler paid attention to the helical arrangement of leaves in plants, resembling a spiral. Flower petals, sunflower seeds, pine cones, pineapple scales also grow.. In botany, this pattern is called phyllotaxis, and in the formulas for leaf arrangement, there are also Fibonacci numbers located through one. Such properties are determined by genetics, rooted at the cellular and molecular levels.. And the polypeptide chains in the DNA molecule also have a helical arrangement. There is evidence that the ratio of length and width in them carries the formula of the golden ratio.

The same principle is seen in the structure of galaxies. For example, our Milky Way has several arms that grow in a logarithmic spiral with a step of about 12 degrees.. The great poet Goethe, who was also a natural scientist, considered helicity to be one of the characteristic features of all organisms, a manifestation of the innermost essence of life. And maybe it is no coincidence that the spiral symbol was present in the culture of many indigenous peoples of the Earth..
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« Kepler said that God is a good geometer and builds the Universe according to mathematical laws, » continues Eduard Sergeev. – And I agree with that one hundred percent. Getting to know the world around you, you are more and more amazed and surprised. Astrophysicist Mario Livio writes very nicely on these topics.. I read his book “Was God a Mathematician? Gallop across the divine universe with a calculator, caliper and Bradis tables « . He talks there about the spiral of life, and about the structure of DNA and many other phenomena.. Of course, all this is mathematics – and some other kind of mathematics « .

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