Have you ever heard that mathematics is called the “queen of all sciences”? Do you agree with this statement? As long as mathematics remains for you a set of boring problems in a textbook, you can hardly experience the beauty, versatility and even humor of this science.
But there are topics in mathematics that help make interesting observations about things and phenomena that are common to us. And even try to penetrate the veil of mystery of the creation of our Universe. There are interesting patterns in the world that can be described using mathematics.
Introducing Fibonacci numbers
Fibonacci numbers name the elements of a number sequence. In it, each next number in a series is obtained by summing the two previous numbers.
Example sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…
You can write it like this:
F 0 = 0, F 1 = 1, F n = F n-1 + F n-2, n ≥ 2
You can start a series of Fibonacci numbers with negative values n. Moreover, the sequence in this case is two-way (that is, it covers negative and positive numbers) and tends to infinity in both directions.
An example of such a sequence: -55, -34, -21, -13, -8, 5, 3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
The formula in this case looks like this:
F n = F n+1 - F n+2 or else you can do this: F -n = (-1) n+1 Fn.
What we now know as “Fibonacci numbers” was known to ancient Indian mathematicians long before they began to be used in Europe. And this name is generally one continuous historical anecdote. Let's start with the fact that Fibonacci himself never called himself Fibonacci during his lifetime - this name began to be applied to Leonardo of Pisa only several centuries after his death. But let's talk about everything in order.
Leonardo of Pisa, aka Fibonacci
The son of a merchant who became a mathematician, and subsequently received recognition from posterity as the first major mathematician of Europe during the Middle Ages. Not least thanks to the Fibonacci numbers (which, let us remember, were not called that yet). Which he described at the beginning of the 13th century in his work “Liber abaci” (“Book of Abacus”, 1202).
I traveled with my father to the East, Leonardo studied mathematics with Arab teachers (and in those days they were among the best specialists in this matter, and in many other sciences). He read the works of mathematicians of Antiquity and Ancient India in Arabic translations.
Having thoroughly comprehended everything he had read and using his own inquisitive mind, Fibonacci wrote several scientific treatises on mathematics, including the above-mentioned “Book of Abacus.” In addition to this I created:
- "Practica geometriae" ("Practice of Geometry", 1220);
- "Flos" ("Flower", 1225 - a study on cubic equations);
- "Liber quadratorum" ("Book of Squares", 1225 - problems on indefinite quadratic equations).
He was a big fan of mathematical tournaments, so in his treatises he paid a lot of attention to the analysis of various mathematical problems.
There is very little biographical information left about Leonardo's life. As for the name Fibonacci, under which he entered the history of mathematics, it was assigned to him only in the 19th century.
Fibonacci and his problems
After Fibonacci there remained a large number of problems that were very popular among mathematicians in subsequent centuries. We will look at the rabbit problem, which is solved using Fibonacci numbers.
Rabbits are not only valuable fur
Fibonacci set the following conditions: there is a pair of newborn rabbits (male and female) of such an interesting breed that they regularly (starting from the second month) produce offspring - always one new pair of rabbits. Also, as you might guess, a male and a female.
These conditional rabbits are placed in a confined space and breed with enthusiasm. It is also stipulated that not a single rabbit dies from some mysterious rabbit disease.
We need to calculate how many rabbits we will get in a year.
- At the beginning of 1 month we have 1 pair of rabbits. At the end of the month they mate.
- The second month - we already have 2 pairs of rabbits (a pair has parents + 1 pair is their offspring).
- Third month: The first pair gives birth to a new pair, the second pair mates. Total - 3 pairs of rabbits.
- Fourth month: The first pair gives birth to a new pair, the second pair does not waste time and also gives birth to a new pair, the third pair is still only mating. Total - 5 pairs of rabbits.
Number of rabbits in n th month = number of pairs of rabbits from the previous month + number of newborn pairs (there are the same number of pairs of rabbits as there were pairs of rabbits 2 months before now). And all this is described by the formula that we have already given above: F n = F n-1 + F n-2.
Thus, we obtain a recurrent (explanation about recursion– below) number sequence. In which each next number is equal to the sum of the previous two:
- 1 + 1 = 2
- 2 + 1 = 3
- 3 + 2 = 5
- 5 + 3 = 8
- 8 + 5 = 13
- 13 + 8 = 21
- 21 + 13 = 34
- 34 + 21 = 55
- 55 + 34 = 89
- 89 + 55 = 144
- 144 + 89 = 233
- 233+ 144 = 377 <…>
You can continue the sequence for a long time: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987<…>. But since we have set a specific period - a year, we are interested in the result obtained on the 12th “move”. Those. 13th member of the sequence: 377.
The answer to the problem: 377 rabbits will be obtained if all stated conditions are met.
One of the properties of the Fibonacci number sequence is very interesting. If you take two consecutive pairs from a series and divide the larger number by the smaller number, the result will gradually approach golden ratio(you can read more about it later in the article).
In mathematical terms, "the limit of relationships a n+1 To a n equal to the golden ratio".
More number theory problems
- Find a number that can be divided by 7. Also, if you divide it by 2, 3, 4, 5, 6, the remainder will be one.
- Find the square number. It is known about it that if you add 5 to it or subtract 5, you again get a square number.
We suggest you search for answers to these problems yourself. You can leave us your options in the comments to this article. And then we will tell you whether your calculations were correct.
Explanation of recursion
Recursion– definition, description, image of an object or process that contains this object or process itself. That is, in essence, an object or process is a part of itself.
Recursion is widely used in mathematics and computer science, and even in art and popular culture.
Fibonacci numbers are determined using a recurrence relation. For number n>2 n- e number is equal (n – 1) + (n – 2).
Explanation of the golden ratio
Golden ratio- dividing a whole (for example, a segment) into parts that are related according to the following principle: the larger part is related to the smaller one in the same way as the entire value (for example, the sum of two segments) is to the larger part.
The first mention of the golden ratio can be found in Euclid in his treatise “Elements” (about 300 BC). In the context of constructing a regular rectangle.
The term familiar to us was introduced into circulation in 1835 by the German mathematician Martin Ohm.
If we describe the golden ratio approximately, it represents a proportional division into two unequal parts: approximately 62% and 38%. In numerical terms, the golden ratio is the number 1,6180339887 .
The golden ratio finds practical application in fine arts (paintings by Leonardo da Vinci and other Renaissance painters), architecture, cinema (“Battleship Potemkin” by S. Esenstein) and other areas. For a long time it was believed that the golden ratio is the most aesthetic proportion. This opinion is still popular today. Although, according to research results, visually most people do not perceive this proportion as the most successful option and consider it too elongated (disproportionate).
- Section length With = 1, A = 0,618, b = 0,382.
- Attitude With To A = 1, 618.
- Attitude With To b = 2,618
Now let's get back to Fibonacci numbers. Let's take two consecutive terms from its sequence. Divide the larger number by the smaller number and get approximately 1.618. And now we use the same larger number and the next member of the series (i.e., an even larger number) - their ratio is early 0.618.
Here's an example: 144, 233, 377.
233/144 = 1.618 and 233/377 = 0.618
By the way, if you try to do the same experiment with numbers from the beginning of the sequence (for example, 2, 3, 5), nothing will work. Almost. The golden ratio rule is hardly followed for the beginning of the sequence. But as you move along the series and the numbers increase, it works great.
And in order to calculate the entire series of Fibonacci numbers, it is enough to know three terms of the sequence, coming one after another. You can see this for yourself!
Golden Rectangle and Fibonacci Spiral
Another interesting parallel between the Fibonacci numbers and the golden ratio is the so-called “golden rectangle”: its sides are in proportion 1.618 to 1. But we already know what the number 1.618 is, right?
For example, let's take two consecutive terms of the Fibonacci series - 8 and 13 - and construct a rectangle with the following parameters: width = 8, length = 13.
And then we will divide the large rectangle into smaller ones. Mandatory condition: the lengths of the sides of the rectangles must correspond to the Fibonacci numbers. Those. The side length of the larger rectangle must be equal to the sum of the sides of the two smaller rectangles.
The way it is done in this figure (for convenience, the figures are signed in Latin letters).
By the way, you can build rectangles in reverse order. Those. start building with squares with a side of 1. To which, guided by the principle stated above, figures with sides equal to the Fibonacci numbers are completed. Theoretically, this can be continued indefinitely - after all, the Fibonacci series is formally infinite.
If we connect the corners of the rectangles obtained in the figure with a smooth line, we get a logarithmic spiral. Or rather, its special case is the Fibonacci spiral. It is characterized, in particular, by the fact that it has no boundaries and does not change shape.
A similar spiral is often found in nature. Clam shells are one of the most striking examples. Moreover, some galaxies that can be seen from Earth have a spiral shape. If you pay attention to weather forecasts on TV, you may have noticed that cyclones have a similar spiral shape when photographed from satellites.
It is curious that the DNA helix also obeys the rule of the golden section - the corresponding pattern can be seen in the intervals of its bends.
Such amazing “coincidences” cannot but excite minds and give rise to talk about some single algorithm to which all phenomena in the life of the Universe obey. Now do you understand why this article is called this way? And what kind of amazing worlds can mathematics open for you?
Fibonacci numbers in nature
The connection between Fibonacci numbers and the golden ratio suggests interesting patterns. So curious that it is tempting to try to find sequences similar to Fibonacci numbers in nature and even during historical events. And nature really gives rise to such assumptions. But can everything in our life be explained and described using mathematics?
Examples of living things that can be described using the Fibonacci sequence:
- the arrangement of leaves (and branches) in plants - the distances between them are correlated with Fibonacci numbers (phyllotaxis);
- arrangement of sunflower seeds (the seeds are arranged in two rows of spirals twisted in different directions: one row clockwise, the other counterclockwise);
- arrangement of pine cone scales;
- flower petals;
- pineapple cells;
- ratio of the lengths of the phalanges of the fingers on the human hand (approximately), etc.
Combinatorics problems
Fibonacci numbers are widely used in solving combinatorics problems.
Combinatorics is a branch of mathematics that studies the selection of a certain number of elements from a designated set, enumeration, etc.
Let's look at examples of combinatorics problems designed for high school level (source - http://www.problems.ru/).
Task #1:
Lesha climbs a staircase of 10 steps. At one time he jumps up either one step or two steps. In how many ways can Lesha climb the stairs?
The number of ways in which Lesha can climb the stairs from n steps, let's denote and n. It follows that a 1 = 1, a 2= 2 (after all, Lesha jumps either one or two steps).
It is also agreed that Lesha jumps up the stairs from n> 2 steps. Let's say he jumped two steps the first time. This means, according to the conditions of the problem, he needs to jump another n – 2 steps. Then the number of ways to complete the climb is described as a n–2. And if we assume that the first time Lesha jumped only one step, then we describe the number of ways to finish the climb as a n–1.
From here we get the following equality: a n = a n–1 + a n–2(looks familiar, doesn't it?).
Since we know a 1 And a 2 and remember that according to the conditions of the problem there are 10 steps, calculate all in order and n: a 3 = 3, a 4 = 5, a 5 = 8, a 6 = 13, a 7 = 21, a 8 = 34, a 9 = 55, a 10 = 89.
Answer: 89 ways.
Task #2:
You need to find the number of words 10 letters long that consist only of the letters “a” and “b” and must not contain two letters “b” in a row.
Let's denote by a n number of words length n letters that consist only of the letters “a” and “b” and do not contain two letters “b” in a row. Means, a 1= 2, a 2= 3.
In sequence a 1, a 2, <…>, a n we will express each of its next members through the previous ones. Therefore, the number of words of length is n letters that also do not contain a double letter “b” and begin with the letter “a” are a n–1. And if the word is long n letters begin with the letter “b”, it is logical that the next letter in such a word is “a” (after all, there cannot be two “b” according to the conditions of the problem). Therefore, the number of words of length is n in this case we denote the letters as a n–2. In both the first and second cases, any word (length of n – 1 And n – 2 letters respectively) without double “b”.
We were able to justify why a n = a n–1 + a n–2.
Let us now calculate a 3= a 2+ a 1= 3 + 2 = 5, a 4= a 3+ a 2= 5 + 3 = 8, <…>, a 10= a 9+ a 8= 144. And we get the familiar Fibonacci sequence.
Answer: 144.
Task #3:
Imagine that there is a tape divided into cells. It goes to the right and lasts indefinitely. Place a grasshopper on the first square of the tape. Whatever cell of the tape he is on, he can only move to the right: either one cell, or two. How many ways are there in which a grasshopper can jump from the beginning of the tape to n-th cells?
Let us denote the number of ways to move a grasshopper along the belt to n-th cells like a n. In this case a 1 = a 2= 1. Also in n+1 The grasshopper can enter the -th cell either from n-th cell, or by jumping over it. From here a n + 1 = a n – 1 + a n. Where a n = Fn – 1.
Answer: Fn – 1.
You can create similar problems yourself and try to solve them in math lessons with your classmates.
Fibonacci numbers in popular culture
Of course, such an unusual phenomenon as Fibonacci numbers cannot but attract attention. There is still something attractive and even mysterious in this strictly verified pattern. It is not surprising that the Fibonacci sequence has somehow “lit up” in many works of modern popular culture of various genres.
We will tell you about some of them. And you try to search for yourself again. If you find it, share it with us in the comments – we’re curious too!
- Fibonacci numbers are mentioned in Dan Brown's bestseller The Da Vinci Code: the Fibonacci sequence serves as the code used by the book's main characters to open a safe.
- In the 2009 American film Mr. Nobody, in one episode the address of a house is part of the Fibonacci sequence - 12358. In addition, in another episode the main character must call a phone number, which is essentially the same, but slightly distorted (extra digit after number 5) sequence: 123-581-1321.
- In the 2012 series “Connection”, the main character, a boy suffering from autism, is able to discern patterns in events occurring in the world. Including through Fibonacci numbers. And manage these events also through numbers.
- The developers of the java game for mobile phones Doom RPG placed a secret door on one of the levels. The code that opens it is the Fibonacci sequence.
- In 2012, the Russian rock band Splin released the concept album “Optical Deception.” The eighth track is called “Fibonacci”. The verses of the group leader Alexander Vasiliev play on the sequence of Fibonacci numbers. For each of the nine consecutive terms there is a corresponding number of lines (0, 1, 1, 2, 3, 5, 8, 13, 21):
0 The train set off
1 One joint snapped
1 One sleeve trembled
2 That's it, get the stuff
That's it, get the stuff
3 Request for boiling water
The train goes to the river
The train goes through the taiga<…>.
- A limerick (a short poem of a specific form - usually five lines, with a specific rhyme scheme, humorous in content, in which the first and last lines are repeated or partially duplicate each other) by James Lyndon also uses a reference to the Fibonacci sequence as a humorous motif:
The dense food of Fibonacci's wives
It was only for their benefit, nothing else.
The wives weighed, according to rumor,
Each one is like the previous two.
Let's sum it up
We hope that we were able to tell you a lot of interesting and useful things today. For example, you can now look for the Fibonacci spiral in the nature around you. Maybe you will be the one who will be able to unravel “the secret of life, the Universe and in general.”
Use the formula for Fibonacci numbers when solving combinatorics problems. You can rely on the examples described in this article.
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You are, of course, familiar with the idea that mathematics is the most important of all sciences. But many may disagree with this, because... sometimes it seems that mathematics is just problems, examples and similar boring stuff. However, mathematics can easily show us familiar things from a completely unfamiliar side. Moreover, she can even reveal the secrets of the universe. How? Let's look at Fibonacci numbers.
What are Fibonacci numbers?
Fibonacci numbers are elements of a numerical sequence, where each subsequent one is by summing the two previous ones, for example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... As a rule, such a sequence is written by the formula: F 0 = 0, F 1 = 1, F n = F n-1 + F n-2, n ≥ 2.
Fibonacci numbers can start with negative values of "n", but in this case the sequence will be two-way - it will cover both positive and negative numbers, tending to infinity in both directions. An example of such a sequence would be: -34, -21, -13, -8, -5, -3, -2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and the formula will be: F n = F n+1 - F n+2 or F -n = (-1) n+1 Fn.
The creator of the Fibonacci numbers is one of the first mathematicians of Europe in the Middle Ages named Leonardo of Pisa, who, in fact, is known as Fibonacci - he received this nickname many years after his death.
During his lifetime, Leonardo of Pisa was very fond of mathematical tournaments, which is why in his works (“Liber abaci” / “Book of Abacus”, 1202; “Practica geometriae” / “Practice of Geometry”, 1220, “Flos” / “Flower”, 1225) - a study on cubic equations and "Liber quadratorum" / "Book of Squares", 1225 - problems about indefinite quadratic equations) very often analyzed all kinds of mathematical problems.
Very little is known about the life path of Fibonacci himself. But what is certain is that his problems enjoyed enormous popularity in mathematical circles in subsequent centuries. We will consider one of these further.
Fibonacci problem with rabbits
To complete the task, the author set the following conditions: there is a pair of newborn rabbits (female and male), distinguished by an interesting feature - from the second month of life, they produce a new pair of rabbits - also a female and a male. Rabbits are kept in confined spaces and constantly breed. And not a single rabbit dies.
Task: determine the number of rabbits in a year.
Solution:
We have:
- One pair of rabbits at the beginning of the first month, which mate at the end of the month
- Two pairs of rabbits in the second month (first pair and offspring)
- Three pairs of rabbits in the third month (the first pair, the offspring of the first pair from the previous month and the new offspring)
- Five pairs of rabbits in the fourth month (the first pair, the first and second offspring of the first pair, the third offspring of the first pair and the first offspring of the second pair)
Number of rabbits per month “n” = number of rabbits last month + number of new pairs of rabbits, in other words, the above formula: F n = F n-1 + F n-2. This results in a recurrent number sequence (we will talk about recursion later), where each new number corresponds to the sum of the two previous numbers:
1 month: 1 + 1 = 2
2 month: 2 + 1 = 3
3 month: 3 + 2 = 5
4 month: 5 + 3 = 8
5 month: 8 + 5 = 13
6 month: 13 + 8 = 21
7th month: 21 + 13 = 34
8th month: 34 + 21 = 55
9 month: 55 + 34 = 89
10th month: 89 + 55 = 144
11th month: 144 + 89 = 233
12 month: 233+ 144 = 377
And this sequence can continue indefinitely, but given that the task is to find out the number of rabbits after a year, the result is 377 pairs.
It is also important to note here that one of the properties of Fibonacci numbers is that if you compare two consecutive pairs and then divide the larger one by the smaller one, the result will move towards the golden ratio, which we will also talk about below.
In the meantime, we offer you two more problems on Fibonacci numbers:
- Determine a square number, about which we only know that if you subtract 5 from it or add 5 to it, you will again get a square number.
- Determine a number divisible by 7, but on the condition that dividing it by 2, 3, 4, 5 or 6 leaves a remainder of 1.
Such tasks will not only be an excellent way to develop the mind, but also an entertaining pastime. You can also find out how these problems are solved by searching for information on the Internet. We will not focus on them, but will continue our story.
What are recursion and the golden ratio?
Recursion
Recursion is a description, definition or image of any object or process, which contains the given object or process itself. In other words, an object or process can be called a part of itself.
Recursion is widely used not only in mathematical science, but also in computer science, popular culture and art. Applicable to Fibonacci numbers, we can say that if the number is “n>2”, then “n” = (n-1)+(n-2).
Golden ratio
The golden ratio is the division of a whole into parts that are related according to the principle: the larger relates to the smaller in the same way as the total value relates to the larger part.
The golden ratio was first mentioned by Euclid (the treatise “Elements,” ca. 300 BC), speaking about the construction of a regular rectangle. However, a more familiar concept was introduced by the German mathematician Martin Ohm.
Approximately, the golden ratio can be represented as a proportional division into two different parts, for example, 38% and 68%. The numerical expression of the golden ratio is approximately 1.6180339887.
In practice, the golden ratio is used in architecture, fine arts (look at the works), cinema and other areas. For a long time, as now, the golden ratio was considered an aesthetic proportion, although most people perceive it as disproportionate - elongated.
You can try to estimate the golden ratio yourself, guided by the following proportions:
- Length of the segment a = 0.618
- Length of segment b= 0.382
- Length of the segment c = 1
- Ratio of c and a = 1.618
- Ratio of c and b = 2.618
Now let’s apply the golden ratio to the Fibonacci numbers: we take two adjacent terms of its sequence and divide the larger one by the smaller one. We get approximately 1.618. If we take the same larger number and divide it by the next larger number after it, we get approximately 0.618. Try it yourself: “play” with the numbers 21 and 34 or some others. If we carry out this experiment with the first numbers of the Fibonacci sequence, then such a result will no longer exist, because the golden ratio "doesn't work" at the beginning of the sequence. By the way, to determine all Fibonacci numbers, you only need to know the first three consecutive numbers.
And in conclusion, some more food for thought.
Golden Rectangle and Fibonacci Spiral
The “Golden Rectangle” is another relationship between the golden ratio and Fibonacci numbers, because... its aspect ratio is 1.618 to 1 (remember the number 1.618!).
Here is an example: we take two numbers from the Fibonacci sequence, for example 8 and 13, and draw a rectangle with a width of 8 cm and a length of 13 cm. Next, we divide the main rectangle into small ones, but their length and width should correspond to the Fibonacci numbers - the length of one edge of the large rectangle should equal to two lengths of the edge of the smaller one.
After this, we connect the corners of all the rectangles we have with a smooth line and get a special case of a logarithmic spiral - the Fibonacci spiral. Its main properties are the absence of boundaries and changes in shape. Such a spiral can often be found in nature: the most striking examples are mollusk shells, cyclones in satellite images, and even a number of galaxies. But what’s more interesting is that the DNA of living organisms also obeys the same rule, because do you remember that it has a spiral shape?
These and many other “random” coincidences even today excite the consciousness of scientists and suggest that everything in the Universe is subject to a single algorithm, moreover, a mathematical one. And this science hides a huge number of completely boring secrets and mysteries.
However, this is not all that can be done with the golden ratio. If we divide one by 0.618, we get 1.618; if we square it, we get 2.618; if we cube it, we get 4.236. These are the Fibonacci expansion ratios. The only missing number here is 3,236, which was proposed by John Murphy.
What do experts think about consistency?
Some might say that these numbers are already familiar because they are used in technical analysis programs to determine the magnitude of corrections and extensions. In addition, these same series play an important role in Eliot's wave theory. They are its numerical basis.
Our expert Nikolay is a proven portfolio manager at the Vostok investment company.
- — Nikolay, do you think that the appearance of Fibonacci numbers and its derivatives on the charts of various instruments is accidental? And is it possible to say: “Fibonacci series practical application” takes place?
- — I have a bad attitude towards mysticism. And even more so on stock exchange charts. Everything has its reasons. in the book “Fibonacci Levels” he beautifully described where the golden ratio appears, that he was not surprised that it appeared on stock exchange quote charts. But in vain! In many of the examples he gave, the number Pi appears frequently. But for some reason it is not included in the price ratios.
- — So you don’t believe in the effectiveness of Eliot’s wave principle?
- - No, that’s not the point. The wave principle is one thing. The numerical ratio is different. And the reasons for their appearance on price charts are the third
- — What, in your opinion, are the reasons for the appearance of the golden ratio on stock charts?
- — The correct answer to this question may earn you the Nobel Prize in Economics. For now we can guess about the true reasons. They are clearly not in harmony with nature. There are many models of exchange pricing. They do not explain the designated phenomenon. But not understanding the nature of a phenomenon should not deny the phenomenon as such.
- — And if this law is ever opened, will it be able to destroy the exchange process?
- — As the same wave theory shows, the law of changes in stock prices is pure psychology. It seems to me that knowledge of this law will not change anything and will not be able to destroy the stock exchange.
Material provided by webmaster Maxim's blog.
The coincidence of the fundamental principles of mathematics in a variety of theories seems incredible. Maybe it's fantasy or customized for the final result. Wait and see. Much of what was previously considered unusual or was not possible: space exploration, for example, has become commonplace and does not surprise anyone. Also, the wave theory, which may be incomprehensible, will become more accessible and understandable over time. What was previously unnecessary will, in the hands of an experienced analyst, become a powerful tool for predicting future behavior.
Fibonacci numbers in nature.
Look
Now, let's talk about how you can refute the fact that the Fibonacci digital series is involved in any patterns in nature.
Let's take any other two numbers and build a sequence with the same logic as the Fibonacci numbers. That is, the next member of the sequence is equal to the sum of the previous two. For example, let's take two numbers: 6 and 51. Now we will build a sequence that we will complete with two numbers 1860 and 3009. Note that when dividing these numbers, we get a number close to the golden ratio.
At the same time, the numbers that were obtained when dividing other pairs decreased from the first to the last, which allows us to say that if this series continues indefinitely, then we will get a number equal to the golden ratio.
Thus, Fibonacci numbers do not stand out in any way. There are other sequences of numbers, of which there are an infinite number, that as a result of the same operations give the golden number phi.
Fibonacci was not an esotericist. He didn't want to put any mysticism into the numbers, he was simply solving an ordinary problem about rabbits. And he wrote a sequence of numbers that followed from his problem, in the first, second and other months, how many rabbits there would be after breeding. Within a year, he received that same sequence. And I didn't do a relationship. There was no talk of any golden proportion or divine relation. All this was invented after him during the Renaissance.
Compared to mathematics, the advantages of Fibonacci are enormous. He adopted the number system from the Arabs and proved its validity. It was a hard and long struggle. From the Roman number system: heavy and inconvenient for counting. It disappeared after the French Revolution. Fibonacci has nothing to do with the golden ratio.
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THE HIGHEST PURPOSE OF MATHEMATICS IS TO FIND THE HIDDEN ORDER IN THE CHAOS THAT SURROUND US.
Viner N.
A person strives for knowledge all his life, trying to study the world around him. And in the process of observation, questions arise that require answers. The answers are found, but new questions arise. In archaeological finds, in traces of civilization, distant from each other in time and space, one and the same element is found - a pattern in the form of a spiral. Some consider it a symbol of the sun and associate it with the legendary Atlantis, but its true meaning is unknown. What do the shapes of a galaxy and an atmospheric cyclone, the arrangement of leaves on a stem, and the arrangement of seeds in a sunflower have in common? These patterns come down to the so-called “golden” spiral, the amazing Fibonacci sequence discovered by the great Italian mathematician of the 13th century.
History of Fibonacci numbers
For the first time I heard about what Fibonacci numbers are from a mathematics teacher. But, besides, I didn’t know how the sequence of these numbers came together. This is what this sequence is actually famous for, how it affects a person, I want to tell you. Little is known about Leonardo Fibonacci. There is not even an exact date of his birth. It is known that he was born in 1170 into a merchant family in the city of Pisa in Italy. Fibonacci's father often visited Algeria on trade matters, and Leonardo studied mathematics there with Arab teachers. Subsequently, he wrote several mathematical works, the most famous of which is the “Book of Abacus,” which contains almost all the arithmetic and algebraic information of that time. 2
Fibonacci numbers are a sequence of numbers that have a number of properties. Fibonacci discovered this number sequence by accident when he was trying to solve a practical problem about rabbits in 1202. “Someone placed a pair of rabbits in a certain place, fenced on all sides by a wall, in order to find out how many pairs of rabbits would be born during the year, if the nature of rabbits is such that after a month a pair of rabbits gives birth to another pair, and rabbits give birth from the second months after your birth." When solving the problem, he took into account that each pair of rabbits gives birth to two more pairs throughout their lives, and then dies. This is how the sequence of numbers appeared: 1, 1, 2, 3, 5, 8, 13, 21, ... In this sequence, each next number is equal to the sum of the two previous ones. It was called the Fibonacci sequence. Mathematical properties of the sequence
I wanted to explore this sequence, and I discovered some of its properties. This pattern is of great importance. The sequence is slowly approaching a certain constant ratio of approximately 1.618, and the ratio of any number to the next one is approximately 0.618.
You can notice a number of interesting properties of Fibonacci numbers: two neighboring numbers are relatively prime; every third number is even; every fifteenth ends in zero; every fourth is a multiple of three. If you choose any 10 adjacent numbers from the Fibonacci sequence and add them together, you will always get a number that is a multiple of 11. But that's not all. Each sum is equal to the number 11 multiplied by the seventh term of the given sequence. Here's another interesting feature. For any n, the sum of the firstn terms of the sequence will always be equal to the difference between the (n+ 2)th and first terms of the sequence. This fact can be expressed by the formula: 1+1+2+3+5+…+an=a n+2 - 1. Now we have the following trick at our disposal: to find the sum of all terms
sequence between two given terms, it is enough to find the difference of the corresponding (n+2)-x terms. For example, a 26 +…+a 40 = a 42 - a 27. Now let's look for the connection between Fibonacci, Pythagoras and the “golden ratio”. The most famous evidence of the mathematical genius of mankind is the Pythagorean theorem: in any right triangle, the square of the hypotenuse is equal to the sum of the squares of its legs: c 2 =b 2 +a 2. From a geometric point of view, we can consider all the sides of a right triangle as the sides of three squares constructed on them. The Pythagorean theorem states that the total area of squares built on the sides of a right triangle is equal to the area of the square built on the hypotenuse. If the lengths of the sides of a right triangle are integers, then they form a group of three numbers called Pythagorean triplets. Using the Fibonacci sequence you can find such triplets. Let's take any four consecutive numbers from the sequence, for example, 2, 3, 5 and 8, and construct three more numbers as follows: 1) the product of the two extreme numbers: 2*8=16; 2) the double product of the two numbers in the middle: 2* (3*5)=30;3) the sum of the squares of two average numbers: 3 2 +5 2 =34; 34 2 =30 2 +16 2. This method works for any four consecutive Fibonacci numbers. Any three consecutive numbers in the Fibonacci series behave in a predictable way. If you multiply the two extreme ones and compare the result with the square of the average number, the result will always differ by one. For example, for the numbers 5, 8 and 13 we get: 5*13=8 2 +1. If you look at this property from a geometric point of view, you will notice something strange. Divide the square
8x8 in size (64 small squares in total) into four parts, the lengths of the sides being equal to the Fibonacci numbers. Now from these parts we will build a rectangle measuring 5x13. Its area is 65 small squares. Where does the extra square come from? The thing is that an ideal rectangle is not formed, but tiny gaps remain, which in total give this additional unit of area. Pascal's triangle also has a connection with the Fibonacci sequence. You just need to write the lines of Pascal's triangle one under the other, and then add the elements diagonally. The result is the Fibonacci sequence.
Now consider a golden rectangle, one side of which is 1.618 times longer than the other. At first glance, it may seem like an ordinary rectangle to us. However, let's do a simple experiment with two ordinary bank cards. Let's place one of them horizontally and the other vertically so that their lower sides are on the same line. If we draw a diagonal line in a horizontal map and extend it, we will see that it will pass exactly through the upper right corner of the vertical map - a pleasant surprise. Maybe this is an accident, or maybe these rectangles and other geometric shapes that use the “golden ratio” are especially pleasing to the eye. Did Leonardo da Vinci think about the golden ratio while working on his masterpiece? This seems unlikely. However, it can be argued that he attached great importance to the connection between aesthetics and mathematics.
Fibonacci numbers in nature
The connection of the golden ratio with beauty is not only a matter of human perception. It seems that nature itself has allocated a special role to F. If you inscribe squares sequentially into a “golden” rectangle, then draw an arc in each square, you will get an elegant curve called a logarithmic spiral. It is not a mathematical curiosity at all. 5
On the contrary, this remarkable line is often found in the physical world: from the shell of a nautilus to the arms of galaxies, and in the elegant spiral of petals of a blooming rose. The connections between the golden ratio and Fibonacci numbers are numerous and surprising. Let's consider a flower that looks very different from a rose - a sunflower with seeds. The first thing we see is that the seeds are arranged in two types of spirals: clockwise and counterclockwise. If we count the clockwise spirals, we get two seemingly ordinary numbers: 21 and 34. This is not the only example where Fibonacci numbers can be found in the structure of plants.
Nature gives us numerous examples of the arrangement of homogeneous objects described by Fibonacci numbers. In the various spiral arrangements of small plant parts, two families of spirals can usually be discerned. In one of these families the spirals curl clockwise, while in the other they curl counterclockwise. The numbers of spirals of one and another type often turn out to be adjacent Fibonacci numbers. So, taking a young pine twig, it is easy to notice that the needles form two spirals, going from bottom left to top right. On many cones, the seeds are arranged in three spirals, gently winding around the stem of the cone. They are located in five spirals, winding steeply in the opposite direction. In large cones it is possible to observe 5 and 8, and even 8 and 13 spirals. Fibonacci spirals are also clearly visible on a pineapple: there are usually 8 and 13 of them.
The chicory shoot makes a strong ejection into space, stops, releases a leaf, but this time is shorter than the first one, again makes an ejection into space, but with less force, releases a leaf of an even smaller size and is ejected again. The impulses of its growth gradually decrease in proportion to the “golden” section. To appreciate the enormous role of Fibonacci numbers, you just need to look at the beauty of the nature around us. Fibonacci numbers can be found in quantities
branches on the stem of each growing plant and in the number of petals.
Let's count the petals of some flowers - iris with its 3 petals, primrose with 5 petals, ragweed with 13 petals, cornflower with 34 petals, aster with 55 petals, etc. Is this a coincidence, or is it a law of nature? Look at the stems and flowers of yarrow. Thus, the total Fibonacci sequence can easily interpret the pattern of manifestations of “Golden” numbers found in nature. These laws operate regardless of our consciousness and desire to accept them or not. The patterns of “golden” symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and cosmic systems, in the gene structures of living organisms, in the structure of individual human organs and the body as a whole, and also manifest themselves in the biorhythms and functioning of the brain and visual perception.
Fibonacci numbers in architecture
The “Golden Ratio” is also evident in many remarkable architectural creations throughout human history. It turns out that ancient Greek and ancient Egyptian mathematicians knew these coefficients long before Fibonacci and called them the “golden ratio”. The Greeks used the principle of the “golden ratio” in the construction of the Parthenon, and the Egyptians used the Great Pyramid of Giza. Advances in construction technology and the development of new materials opened up new opportunities for twentieth-century architects. American Frank Lloyd Wright was one of the main proponents of organic architecture. Shortly before his death, he designed the Solomon Guggenheim Museum in New York, which is an inverted spiral, and the interior of the museum resembles a nautilus shell. Polish-Israeli architect Zvi Hecker also used spiral structures in his design for the Heinz Galinski School in Berlin, completed in 1995. Hecker started with the idea of a sunflower with a central circle, from where
All architectural elements are diverging. The building is a combination
orthogonal and concentric spirals, symbolizing the interaction of limited human knowledge and the controlled chaos of nature. Its architecture imitates a plant that follows the movement of the Sun, so classrooms are illuminated throughout the day.
In Quincy Park, located in Cambridge, Massachusetts (USA), the “golden” spiral can often be found. The park was designed in 1997 by artist David Phillips and is located near the Clay Mathematical Institute. This institution is a renowned center for mathematical research. In Quincy Park, you can stroll among “golden” spirals and metal curves, reliefs of two shells and a rock with a square root symbol. The sign contains information about the “golden” ratio. Even bicycle parking uses the F symbol.
Fibonacci numbers in psychology
In psychology, turning points, crises, and revolutions have been noted that mark transformations in the structure and functions of the soul in a person’s life path. If a person successfully overcomes these crises, then he becomes capable of solving problems of a new class that he had not even thought about before.
The presence of fundamental changes gives reason to consider life time as a decisive factor in the development of spiritual qualities. After all, nature does not measure out time generously for us, “no matter how much it will be, so much will be,” but just enough for the development process to materialize:
in body structures;
in feelings, thinking and psychomotor skills - until they acquire harmony necessary for the emergence and launch of the mechanism
creativity;
in the structure of human energy potential.
The development of the body cannot be stopped: the child becomes an adult. With the mechanism of creativity, everything is not so simple. Its development can be stopped and its direction changed.
Is there a chance to catch up with time? Undoubtedly. But for this you need to do a lot of work on yourself. What develops freely, naturally, does not require special efforts: the child develops freely and does not notice this enormous work, because the process of free development is created without violence against oneself.
How is the meaning of life's journey understood in everyday consciousness? The average person sees it this way: at the bottom there is birth, at the top there is the prime of life, and then everything goes downhill.
The sage will say: everything is much more complicated. He divides the ascent into stages: childhood, adolescence, youth... Why is this so? Few are able to answer, although everyone is sure that these are closed, integral stages of life.
To find out how the mechanism of creativity develops, V.V. Klimenko used mathematics, namely the laws of Fibonacci numbers and the proportion of the “golden section” - the laws of nature and human life.
Fibonacci numbers divide our lives into stages according to the number of years lived: 0 - the beginning of the countdown - the child is born. He still lacks not only psychomotor skills, thinking, feelings, imagination, but also operational energy potential. He is the beginning of a new life, a new harmony;
1 - the child has mastered walking and is mastering his immediate environment;
2 - understands speech and acts using verbal instructions;
3 - acts through words, asks questions;
5 - “age of grace” - harmony of psychomotor, memory, imagination and feelings, which already allow the child to embrace the world in all its integrity;
8 - feelings come to the fore. They are served by imagination, and thinking, through its criticality, is aimed at supporting the internal and external harmony of life;
13 - the mechanism of talent begins to work, aimed at transforming the material acquired in the process of inheritance, developing one’s own talent;
21 - the mechanism of creativity has approached a state of harmony and attempts are being made to perform talented work;
34—harmony of thinking, feelings, imagination and psychomotor skills: the ability to work ingeniously is born;
55 - at this age, provided the harmony of soul and body is preserved, a person is ready to become a creator. And so on…
What are the Fibonacci Numbers serifs? They can be compared to dams along the path of life. These dams await each of us. First of all, you need to overcome each of them, and then patiently raise your level of development until one fine day it falls apart, opening the way to the next one for free flow.
Now that we understand the meaning of these key points of age-related development, let’s try to decipher how it all happens.
B1 year the child masters walking. Before this, he experienced the world with the front of his head. Now he gets to know the world with his hands—an exceptional human privilege. The animal moves in space, and he, by learning, masters the space and masters the territory in which he lives.
2 years- understands the word and acts in accordance with it. It means that:
the child learns a minimum number of words - meanings and modes of action;
has not yet separated itself from the environment and is fused into integrity with the environment,
therefore he acts according to someone else's instructions. At this age he is the most obedient and pleasant to his parents. From a sensual person, a child turns into a cognitive person.
3 years- action using one's own word. The separation of this person from the environment has already occurred - and he learns to be an independently acting person. From here he:
consciously opposes the environment and parents, kindergarten teachers, etc.;
realizes its sovereignty and fights for independence;
tries to subjugate close and well-known people to his will.
Now for a child, a word is an action. This is where the active person begins.
5 years- “age of grace.” He is the personification of harmony. Games, dancing, deft movements - everything is saturated with harmony, which a person tries to master with his own strength. Harmonious psychomotor behavior helps bring about a new state. Therefore, the child is focused on psychomotor activity and strives for the most active actions.
Materialization of the products of sensitivity work is carried out through:
the ability to display the environment and ourselves as part of this world (we hear, see, touch, smell, etc. - all senses work for this process);
ability to design the external world, including oneself
(creation of second nature, hypotheses - do this and that tomorrow, build a new machine, solve a problem), by the forces of critical thinking, feelings and imagination;
the ability to create a second, man-made nature, products of activity (realization of plans, specific mental or psychomotor actions with specific objects and processes).
After 5 years, the imagination mechanism comes forward and begins to dominate the others. The child does a tremendous amount of work, creating fantastic images, and lives in the world of fairy tales and myths. The hypertrophied imagination of a child causes surprise in adults, because the imagination does not correspond to reality.
8 years— feelings come to the fore and one’s own standards of feelings (cognitive, moral, aesthetic) arise when the child unmistakably:
evaluates the known and the unknown;
distinguishes moral from immoral, moral from immoral;
beauty from what threatens life, harmony from chaos.
13 years— the mechanism of creativity begins to work. But this does not mean that it is working at full capacity. One of the elements of the mechanism comes to the fore, and all the others contribute to its work. If in this age period of development harmony is maintained, which almost constantly rebuilds its structure, then the youth will painlessly reach the next dam, unnoticed by himself will overcome it and will live at the age of a revolutionary. At the age of a revolutionary, a youth must take a new step forward: separate from the nearest society and live a harmonious life and activity in it. Not everyone can solve this problem that arises before each of us.
21 years old. If a revolutionary has successfully overcome the first harmonious peak of life, then his mechanism of talent is capable of performing talented
work. Feelings (cognitive, moral or aesthetic) sometimes overshadow thinking, but in general all elements work harmoniously: feelings are open to the world, and logical thinking is able to name and find measures of things from this peak.
The mechanism of creativity, developing normally, reaches a state that allows it to receive certain fruits. He starts working. At this age, the mechanism of feelings comes forward. As the imagination and its products are evaluated by the senses and the mind, antagonism arises between them. Feelings win. This ability gradually gains power, and the boy begins to use it.
34 years- balance and harmony, productive effectiveness of talent. The harmony of thinking, feelings and imagination, psychomotor skills, which are replenished with optimal energy potential, and the mechanism as a whole - the opportunity to perform brilliant work is born.
55 years- a person can become a creator. The third harmonious peak of life: thinking subjugates the power of feelings.
Fibonacci numbers refer to the stages of human development. Whether a person will go through this path without stopping depends on parents and teachers, the educational system, and then - on himself and on how a person will learn and overcome himself.
On the path of life, a person discovers 7 relationship objects:
From birthday to 2 years - discovery of the physical and objective world of the immediate environment.
From 2 to 3 years - self-discovery: “I am Myself.”
From 3 to 5 years - speech, the active world of words, harmony and the “I - You” system.
From 5 to 8 years - discovery of the world of other people's thoughts, feelings and images - the “I - We” system.
From 8 to 13 years - discovery of the world of tasks and problems solved by the geniuses and talents of humanity - the “I - Spirituality” system.
From 13 to 21 years - the discovery of the ability to independently solve well-known problems, when thoughts, feelings and imagination begin to work actively, the “I - Noosphere” system arises.
From 21 to 34 years old - discovery of the ability to create a new world or its fragments - awareness of the self-concept “I am the Creator”.
The life path has a spatiotemporal structure. It consists of age and individual phases, determined by many life parameters. A person masters, to a certain extent, the circumstances of his life, becomes the creator of his history and the creator of the history of society. A truly creative attitude to life, however, does not appear immediately and not even in every person. There are genetic connections between the phases of the life path, and this determines its natural character. It follows that, in principle, it is possible to predict future development on the basis of knowledge about its early phases.
Fibonacci numbers in astronomy
From the history of astronomy it is known that I. Titius, a German astronomer of the 18th century, using the Fibonacci series, found a pattern and order in the distances between the planets of the solar system. But one case seemed to contradict the law: there was no planet between Mars and Jupiter. But after the death of Titius at the beginning of the 19th century. concentrated observation of this part of the sky led to the discovery of the asteroid belt.
Conclusion
During the research, I found out that Fibonacci numbers are widely used in the technical analysis of stock prices. One of the simplest ways to use Fibonacci numbers in practice is to determine the time intervals after which a particular event will occur, for example, a price change. The analyst counts a certain number of Fibonacci days or weeks (13,21,34,55, etc.) from the previous similar event and makes a forecast. But this is still too difficult for me to figure out. Although Fibonacci was the greatest mathematician of the Middle Ages, the only monuments to Fibonacci are a statue in front of the Leaning Tower of Pisa and two streets that bear his name: one in Pisa and the other in Florence. And yet, in connection with everything I have seen and read, quite natural questions arise. Where did these numbers come from? Who is this architect of the universe who tried to make it ideal? What will be next? Having found the answer to one question, you will get the next one. If you solve it, you'll get two new ones. Once you deal with them, three more will appear. Having solved them too, you will have five unsolved ones. Then eight, thirteen, etc. Do not forget that two hands have five fingers, two of which consist of two phalanges, and eight of three.
Literature:
Voloshinov A.V. “Mathematics and Art”, M., Education, 1992.
Vorobyov N.N. “Fibonacci Numbers”, M., Nauka, 1984.
Stakhov A.P. “The Da Vinci Code and the Fibonacci Series”, St. Petersburg format, 2006
F. Corvalan “The Golden Ratio. Mathematical language of beauty", M., De Agostini, 2014.
Maksimenko S.D. "Sensitive periods of life and their codes."
"Fibonacci numbers". Wikipedia
