Fibonacci number series. Fibonacci sequence illustrated by nature

Leonardo Fibonacci is one of the most famous mathematicians of the Middle Ages. One of his most important achievements is the number series, which defines the golden ratio and can be traced throughout the nature of our planet.

The amazing property of these numbers is that the sum of all previous numbers equal to the following number (check for yourself):

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610… - Fibonacci series

It turns out that this sequence has many interesting properties from a mathematical point of view. Here's an example: you can split a line into two parts. The ratio of the smaller part of the line to the larger one will be equal to the ratio of the larger part to the entire line. This proportionality ratio, approximately 1.618, is known as the golden ratio.

The Fibonacci series could have remained only a mathematical incident, if not for the fact that all researchers of the golden ratio find this sequence in the entire plant and animal world. Here are some amazing examples:

The arrangement of leaves on a branch, sunflower seeds, pine cones manifests itself as the golden ratio. If you look at the leaves of such a plant from above, you will notice that they bloom in a spiral. The angles between adjacent leaves form a regular mathematical series known as the Fibonacci sequence. Thanks to this, each individual leaf growing on a tree receives the maximum available amount of heat and light.

At first glance, the lizard has proportions that are pleasing to our eyes - the length of its tail is related to the length of the rest of the body as 62 to 38.

The scientist Zeising did a tremendous amount of work to discover the golden ratio in the human body. He measured about two thousand human bodies. The division of the body by the navel point is the most important indicator of the golden ratio. The proportions of the male body fluctuate within the average ratio of 13: 8 = 1.625 and are somewhat closer to the golden ratio than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 = 1.6. The proportions of the golden ratio also appear in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.

During the Renaissance, it was believed that it was precisely this proportion from the Fibonacci series, observed in architectural structures and other forms of art, most pleasing to the eye. Here are some examples of the use of the golden ratio in art:

Portrait of Mona Lisa

The portrait of Monna Lisa has attracted the attention of researchers for many years, who discovered that the composition of the picture is based on golden triangles, which are parts of a regular star-shaped pentagon, which is built on the principles of the golden ratio.

Parferon

Golden proportions are present in the dimensions of the facade of the ancient Greek temple of the Parthenon. This ancient structure with its harmonious proportions gives us the same aesthetic pleasure as it did our ancestors. Many art historians, who sought to uncover the secret of the powerful emotional impact that this building has on the viewer, sought and found the golden proportion in the relationships of its parts.

Raphael - "Massacre of the Babies"

The picture is built on a spiral that follows the proportions of the golden ratio. We do not know whether Raphael actually drew the golden spiral when creating the composition “Massacre of the Innocents” or only “felt” it.

Our world is wonderful and full of great surprises. An amazing thread of connection connects many everyday things for us. Golden ratio legendary for the fact that it united seemingly two completely different branches of knowledge - mathematics, the queen of precision and order, and humanitarian aesthetics.

Hello, dear readers!

Golden ratio - what is it? Fibonacci numbers are? The article contains answers to these questions briefly and clearly, in simple words.

These questions have been exciting the minds of more and more generations for several millennia! It turns out that mathematics may not be boring, but exciting, interesting, and fascinating!

Other useful articles:

What are Fibonacci numbers?

The amazing fact is that when dividing each subsequent number in a numerical sequence by the previous one the result is a number tending to 1.618.

A lucky guy discovered this mysterious sequence medieval mathematician Leonardo of Pisa (better known as Fibonacci). Before him Leonardo da Vinci discovered a surprisingly repeating proportion in the structure of the human body, plants and animals Phi = 1.618. Scientists also call this number (1.61) the “Number of God.”

Before Leonardo da Vinci, this sequence of numbers was known in Ancient India And Ancient Egypt . Egyptian pyramids were built using proportions Phi = 1.618.

But that's not all, it turns out laws of nature of the Earth and Space in some inexplicable way they obey strict mathematical laws Fidonacci number sequences.

For example, both a shell on Earth and a galaxy in Space are built using Fibonacci numbers. The vast majority of flowers have 5, 8, 13 petals. In sunflowers, on plant stems, in swirling vortices of clouds, in whirlpools and even in Forex exchange rate charts, Fibonacci numbers work everywhere.

Watch a simple and entertaining explanation of the Fibonacci sequence and the Golden Ratio in this SHORT VIDEO (6 minutes):

What is the Golden Ratio or Divine Proportion?

So, what is the Golden Ratio or Golden or Divine Proportion? Fibonacci also discovered that the sequence that consists of the squares of Fibonacci numbers is an even bigger mystery. Let's try graphically represent the sequence in the form of an area:

1², 2², 3², 5², 8²…

If you fit a spiral into graphic image sequence of squares of Fibonacci numbers, then we get the Golden Ratio, according to the rules of which everything in the universe is built, including plants, animals, the DNA spiral, the human body, ... This list can be continued indefinitely.

Golden ratio and Fibonacci numbers in nature VIDEO

I suggest watching a short film (7 minutes) that reveals some of the mysteries of the Golden Ratio. When thinking about the Fibonacci law of numbers, as the primary law that governs living and inanimate nature, the question arises: Did this ideal formula for the macrocosm and microcosm arise on its own or did someone create it and successfully apply it?

What do YOU ​​think about this? Let's think about this riddle together and maybe we will get closer to it.

I really hope that the article was useful to you and you learned what is the Golden Ratio * and Fibonacci Numbers? See you again on the blog pages, subscribe to the blog. The subscription form is below the article.

I wish everyone many new ideas and inspiration for their implementation!

Fibonacci sequence, known to everyone from the film "The Da Vinci Code" - a series of numbers described in the form of a riddle by the Italian mathematician Leonardo of Pisa, better known by the nickname Fibonacci, in the 13th century. Briefly the essence of the riddle:

Someone placed a pair of rabbits in a certain enclosed space in order to find out how many pairs of rabbits would be born during the year, if the nature of rabbits is such that every month a pair of rabbits gives birth to another pair, and they become capable of producing offspring when they reach two months of age.

The result is a series of numbers like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 , where the number of pairs of rabbits in each of the twelve months is shown, separated by commas. It can be continued indefinitely. Its essence is that each next number is the sum of the two previous ones.

This series has several mathematical features that definitely need to be touched upon. It asymptotically (approaching more and more slowly) tends to some constant ratio. However, this ratio is irrational, that is, it is a number with an infinite, unpredictable sequence of decimal digits in the fractional part. It is impossible to express it precisely.

Thus, the ratio of any member of a series to the one preceding it fluctuates around the number 1,618 , sometimes exceeding it, sometimes not achieving it. The ratio to the following similarly approaches the number 0,618 , which is inversely proportional 1,618 . If we divide the elements by one, we get numbers 2,618 And 0,382 , which are also inversely proportional. These are the so-called Fibonacci ratios.

What is all this for? Thus we are approaching one of the most mysterious phenomena nature. The savvy Leonardo essentially did not discover anything new, he simply reminded the world of such a phenomenon as Golden Ratio, which is not inferior in importance to the Pythagorean theorem.

We distinguish all the objects around us by their shape. We like some more, some less, some are completely off-putting. Sometimes interest can be dictated by the life situation, and sometimes by the beauty of the observed object. The symmetrical and proportional shape promotes the best visual perception and evokes a feeling of beauty and harmony. A complete image always consists of parts of different sizes that are in a certain relationship with each other and the whole. Golden ratio- the highest manifestation of the perfection of the whole and its parts in science, art and nature.

To use a simple example, the Golden Ratio is the division of a segment into two parts in such a ratio that the larger part is related to the smaller one, as their sum (the entire segment) is to the larger one.

If we take the entire segment c for 1 , then the segment a will be equal 0,618 , segment b - 0,382 , only in this way will the condition of the Golden Ratio be met (0,618/0,382=1,618 ; 1/0,618=1,618 ) . Attitude c To a equals 1,618 , A With To b 2,618 . These are the same Fibonacci ratios that are already familiar to us.

Of course there is a golden rectangle, a golden triangle and even a golden cuboid. The proportions of the human body are in many respects close to the Golden Section.

Image: marcus-frings.de

But the fun begins when we combine the knowledge we have gained. The figure clearly shows the relationship between the Fibonacci sequence and the Golden Ratio. We start with two squares of the first size. Add a square of the second size on top. Draw a square next to it with a side equal to the amount sides of the previous two, third size. By analogy, a square of size five appears. And so on until you get tired, the main thing is that the length of the side of each next square is equal to the sum of the lengths of the sides of the previous two. We see a series of rectangles whose side lengths are Fibonacci numbers, and, oddly enough, they are called Fibonacci rectangles.

If we draw smooth lines through the corners of our squares, we will get nothing more than an Archimedes spiral, the increment of which is always uniform.

Doesn't remind you of anything?

Photo: ethanhein on Flickr

And not only in the shell of a mollusk you can find Archimedes’ spirals, but in many flowers and plants, they’re just not so obvious.

Aloe multifolia:

Photo: brewbooks on Flickr

Photo: beart.org.uk
Photo: esdrascalderan on Flickr
Photo: manj98 on Flickr

And now it’s time to remember the Golden Section! Are some of the most beautiful and harmonious creations of nature depicted in these photographs? And that's not all. If you look closely, you can find similar patterns in many forms.

Of course, the statement that all these phenomena are based on the Fibonacci sequence sounds too loud, but the trend is obvious. And besides, she herself is far from perfect, like everything in this world.

There is an assumption that the Fibonacci series is an attempt by nature to adapt to a more fundamental and perfect golden ratio logarithmic sequence, which is almost the same, only it starts from nowhere and goes to nowhere. Nature definitely needs some kind of whole beginning from which it can start; it cannot create something out of nothing. The ratios of the first terms of the Fibonacci sequence are far from the Golden Ratio. But the further we move along it, the more these deviations are smoothed out. To define any series, it is enough to know its three terms, coming one after another. But not for the golden sequence, two are enough for it, it is geometric and arithmetic progression simultaneously. One might think that it is the basis for all other sequences.

Each term of the golden logarithmic sequence is a power of the Golden Ratio ( z). Part of the series looks something like this: ... z -5 ; z -4 ; z -3 ; z -2 ; z -1 ; z 0 ; z 1 ; z 2 ; z 3 ; z 4 ; z 5... If we round the value of the Golden Ratio to three decimal places, we get z=1.618, then the series looks like this: ... 0,090 0,146; 0,236; 0,382; 0,618; 1; 1,618; 2,618; 4,236; 6,854; 11,090 ... Each next term can be obtained not only by multiplying the previous one by 1,618 , but also by adding the two previous ones. Thus, exponential growth is achieved by simply adding two adjacent elements. It's a series without beginning or end, and that's what the Fibonacci sequence tries to be like. Having a very definite beginning, she strives for the ideal, never achieving it. That's life.

And yet, in connection with everything we have seen and read, quite logical questions arise:
Where did these numbers come from? Who is this architect of the universe who tried to make it ideal? Was everything ever the way he wanted? And if so, why did it go wrong? Mutations? Free choice? What will happen next? Is the spiral curling or unwinding?

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, then thirteen, 21, 34, 55...

Sources: ; ; ;

Fibonacci numbers... in nature and life

Leonardo Fibonacci is one of the greatest mathematicians of the Middle Ages. In one of his works, “The Book of Calculations,” Fibonacci described the Indo-Arabic system of calculation and the advantages of its use over the Roman one.

Definition
Fibonacci numbers or Fibonacci Sequence is a number sequence that has a number of properties. For example, the sum of two adjacent numbers in a sequence gives the value of the next one (for example, 1+1=2; 2+3=5, etc.), which confirms the existence of the so-called Fibonacci coefficients, i.e. constant ratios.

The Fibonacci sequence starts like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233…

2.

Complete definition of Fibonacci numbers

3.


Properties of the Fibonacci sequence

4.

1. The ratio of each number to the next one tends more and more to 0.618 as the serial number increases. The ratio of each number to the previous one tends to 1.618 (the reverse of 0.618). The number 0.618 is called (FI).

2. When dividing each number by the one following it, the number after one is 0.382; on the contrary – respectively 2.618.

3. Selecting the ratios in this way, we obtain the main set of Fibonacci ratios: ... 4.235, 2.618, 1.618, 0.618, 0.382, 0.236.

5.


The connection between the Fibonacci sequence and the “golden ratio”

6.

The Fibonacci sequence asymptotically (approaching slower and slower) tends to some constant relationship. However, this ratio is irrational, that is, it represents a number with an infinite, unpredictable sequence of decimal digits in the fractional part. It is impossible to express it precisely.

If any member of the Fibonacci sequence is divided by its predecessor (for example, 13:8), the result will be a value that fluctuates around the irrational value 1.61803398875... and sometimes exceeds it, sometimes does not reach it. But even after spending Eternity on this, it is impossible to find out the ratio exactly, down to the last decimal digit. For the sake of brevity, we will present it in the form of 1.618. Special names began to be given to this ratio even before Luca Pacioli (a medieval mathematician) called it the Divine proportion. Among its modern names are the Golden Ratio, the Golden Average and the ratio of rotating squares. Kepler called this relationship one of the “treasures of geometry.” In algebra, it is generally accepted to be denoted by the Greek letter phi

Let's imagine the golden ratio using the example of a segment.

Consider a segment with ends A and B. Let point C divide the segment AB so that,

AC/CB = CB/AB or

AB/CB = CB/AC.

You can imagine it something like this: A-–C--–B

7.

The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part as the larger part itself is related to the smaller one; or in other words, the smaller segment is to the larger as the larger is to the whole.

8.

Segments of the golden proportion are expressed as an infinite irrational fraction 0.618..., if AB is taken as one, AC = 0.382.. As we already know, the numbers 0.618 and 0.382 are the coefficients of the Fibonacci sequence.

9.

Fibonacci proportions and the golden ratio in nature and history

10.


It is important to note that Fibonacci seemed to remind humanity of his sequence. It was known to the ancient Greeks and Egyptians. And indeed, since then in nature, architecture, fine arts, mathematics, physics, astronomy, biology and many other fields, patterns described by Fibonacci coefficients were found. It's amazing how many constants can be calculated using the Fibonacci sequence, and how its terms appear in a huge number of combinations. However, it would not be an exaggeration to say that this is not just a game with numbers, but the most important mathematical expression natural phenomena of all ever opened.

11.

The examples below show some interesting applications of this mathematical sequence.

12.

1. The sink is twisted in a spiral. If you unfold it, you get a length slightly shorter than the length of the snake. The small ten-centimeter shell has a spiral 35 cm long. The shape of the spirally curled shell attracted the attention of Archimedes. The fact is that the ratio of the dimensions of the shell curls is constant and equal to 1.618. Archimedes studied the spiral of shells and derived the equation of the spiral. The spiral drawn according to this equation is called by his name. The increase in her step is always uniform. Currently, the Archimedes spiral is widely used in technology.

2. Plants and animals. Goethe also emphasized the tendency of nature towards spirality. The helical and spiral arrangement of leaves on tree branches was noticed a long time ago. The spiral was seen in the arrangement of sunflower seeds, pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch of sunflower seeds and pine cones, the Fibonacci series manifests itself, and therefore, the law of the golden ratio manifests itself. The spider weaves its web in a spiral pattern. A hurricane is spinning like a spiral. A frightened herd of reindeer scatters in a spiral. The DNA molecule is twisted in a double helix. Goethe called the spiral the “curve of life.”

Among the roadside herbs grows an unremarkable plant - chicory. Let's take a closer look at it. A shoot has formed from the main stem. The first leaf was located right there. The shoot makes a strong ejection into space, stops, releases a leaf, but this time it 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. If the first emission is taken as 100 units, then the second is equal to 62 units, the third – 38, the fourth – 24, etc. The length of the petals is also subject to the golden proportion. In growing and conquering space, the plant maintained certain proportions. The impulses of its growth gradually decreased in proportion to the golden ratio.

The lizard is viviparous. At first glance, the lizard has proportions that are pleasant to our eyes - the length of its tail is related to the length of the rest of the body, as 62 to 38.

In both the plant and animal worlds, the formative tendency of nature persistently breaks through - symmetry regarding the direction of growth and movement. Here the golden ratio appears in the proportions of parts perpendicular to the direction of growth. Nature has carried out division into symmetrical parts and golden proportions. The parts reveal a repetition of the structure of the whole.

Pierre Curie at the beginning of this century formulated a number of profound ideas about symmetry. He argued that one cannot consider the symmetry of any body without taking into account the symmetry environment. Patterns of golden symmetry manifest themselves in energy transitions elementary particles, in the structure of some chemical compounds, in planetary and space systems, in the gene structures of living organisms. These patterns, as indicated above, exist 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.

3. Space. From the history of astronomy it is known that I. Titius, a German astronomer of the 18th century, with the help of this series (Fibonacci) found a pattern and order in the distances between the planets of the solar system

However, one case that seemed to contradict the law: there was no planet between Mars and Jupiter. Focused observation of this part of the sky led to the discovery of the asteroid belt. This happened after the death of Titius in early XIX V.

The Fibonacci series is widely used: it is used to represent the architectonics of living beings, man-made structures, and the structure of Galaxies. These facts are evidence of independence number series on the conditions of its manifestation, which is one of the signs of its universality.

4. Pyramids. Many have tried to unravel the secrets of the pyramid at Giza. Unlike others Egyptian pyramids This is not a tomb, but rather an unsolvable puzzle of number combinations. The remarkable ingenuity, skill, time and labor that the pyramid's architects employed in constructing the eternal symbol indicate the extreme importance of the message they wished to convey to future generations. Their era was preliterate, prehieroglyphic, and symbols were the only means of recording discoveries. The key to the geometric-mathematical secret of the Pyramid of Giza, which had been a mystery to mankind for so long, was actually given to Herodotus by the temple priests, who informed him that the pyramid was built so that the area of ​​each of its faces was equal to the square of its height.

Area of ​​a triangle

356 x 440 / 2 = 78320

Square area

280 x 280 = 78400

The length of the edge of the base of the pyramid at Giza is 783.3 feet (238.7 m), the height of the pyramid is 484.4 feet (147.6 m). The length of the base edge divided by the height leads to the ratio Ф=1.618. The height of 484.4 feet corresponds to 5813 inches (5-8-13) - these are the numbers from the Fibonacci sequence. These interesting observations suggest that the design of the pyramid is based on the proportion Ф=1.618. Some modern scholars are inclined to interpret that the ancient Egyptians built it for the sole purpose of passing on knowledge that they wanted to preserve for future generations. Intensive studies of the pyramid at Giza showed how extensive the knowledge of mathematics and astrology was at that time. In all internal and external proportions of the pyramid, the number 1.618 plays a central role.

Pyramids in Mexico. Not only were the Egyptian pyramids built in accordance with the perfect proportions of the golden ratio, the same phenomenon was found in the Mexican pyramids. The idea arises that both the Egyptian and Mexican pyramids were erected at approximately the same time by people of common origin.

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 at that time 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 remains large number 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 just 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 + 1 = 2
2. 2 + 1 = 3
3. 3 + 2 = 5
4. 5 + 3 = 8
5. 8 + 5 = 13
6. 13 + 8 = 21
7. 21 + 13 = 34
8. 34 + 21 = 55
9. 55 + 34 = 89
10. 89 + 55 = 144
11. 144 + 89 = 233
12. 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 we take two consecutive pairs from a row and divide larger number to less, 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

1. 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.
2. Find the square number. It is known that if you add 5 to it or subtract 5, you will 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 numerically The golden ratio represents a 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. Well, 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 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. Required 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 side 1. To which, guided by the principle stated above, figures with sides are completed, equal numbers Fibonacci. 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 doors amazing worlds What can mathematics reveal to 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 a 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, which also do not contain double letter"b" and begin with the letter "a", this 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 that 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 it is unusual phenomenon, like 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 American film 2009 "Mr. Nobody" in one of the episodes the address of the house is part of the Fibonacci sequence - 12358. In addition, in another episode main character must call the phone number, which is essentially the same, but slightly distorted (extra digit after the 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.
• Java game developers for mobile phones Doom RPG placed on one of the levels secret door. 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<…>.

• limerick ( short poem a certain form - usually five lines, with a certain rhyme scheme, humorous in content, in which the first and last lines are repeated or partially duplicate each other) 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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