Discovery of Leonardo Fibonacci: number series. The Golden Ratio and Fibonacci Numbers

MOU Talovskaya secondary school

Completed by students of grade 9

Head Dankova Valentina Anatolievna

2015

Fibonacci number sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55…

FIBONACCCI (Leonardo of Pisa)
Fibonacci (Leonardo of Pisa), c. 1175–1250

Italian mathematician. Born in Pisa, became the first great mathematician of Europe in the late Middle Ages. It was the practical need to establish business contacts that led him to mathematics. He published his books on arithmetic, algebra and other mathematical disciplines. From Muslim mathematicians, he learned about the system of numbers invented in India and already adopted in the Arab world, and was convinced of its superiority (these numbers were the forerunners of modern Arabic numerals).

The Italian merchant Leonardo of Pisa (1180-1240), better known as Fibonacci, was by far the most important mathematician of the Middle Ages. The role of his books in the development of mathematics and the dissemination of mathematical knowledge in Europe can hardly be overestimated.

In the age of Fibonacci, the renaissance was still far away, but history gave Italy a brief period of time that could well be called a rehearsal for the impending Renaissance. This rehearsal was led by Frederick II, Emperor (since 1220) of the Holy Roman Empire. Brought up in the traditions of southern Italy, Frederick II was internally deeply far from European Christian chivalry.

Frederick II did not recognize the jousting tournaments so beloved by his grandfather. Instead, he cultivated much less bloody math competitions, in which opponents exchanged not blows, but problems.

At such tournaments, the talent of Leonardo Fibonacci shone. This was facilitated a good education, which was given to his son by the merchant Bonacci, who took him with him to the East and assigned Arab teachers to him.

Frederick's patronage stimulated the release of Fibonacci's scientific treatises:

The book of the abacus (Liber Abaci), written in 1202, but which has come down to us in its second version, which dates back to 1228.

Geometry Practices" (1220)

Book of Squares (1225)

According to these books, superior in their level to Arabic and medieval European works, mathematics was taught almost until the time of Descartes (XVII century).

According to the documents of 1240, the admiring citizens of Pisa said that he was a "reasonable and erudite man", and not so long ago Joseph Gies (Joseph Gies), Chief Editor The Encyclopædia Britannica stated that future scientists of all time "will pay their debt to Leonardo of Pisa as one of the world's greatest intellectual pioneers." His work after years just now being translated from Latin to English. For those who are interested, the book entitled Lenardo of Pisa and the New Mathematics of the Middle Ages by Joseph and Frances Gies is an excellent treatise on the Fibonacci age and his works.

Of greatest interest to us is the work "The Book of the Abacus" ("Liber Abaci"). This book is a voluminous work containing almost all the arithmetic and algebraic information of that time and played a significant role in the development of mathematics in Western Europe over the next few centuries. In particular, it was from this book that Europeans got acquainted with Hindu (Arabic) numerals.

In "Liber Abaci" Fibonacci gives his sequence of numbers as a solution mathematical problem- Finding the breeding formula for rabbits. The numerical sequence is as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 (then ad infinitum).

On pages 123-124 of this manuscript, Fibonacci placed the following problem: "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 the rabbits is such that in a month a pair of rabbits gives birth to another pair, and rabbits give birth from the second months after his birth.

In the figure, segment AB is divided by point C so that AC: AB = CB: AC.

which is approximately 1.618 ... Thus, the ratio of the greater part of the segment to the smaller and the entire length of the segment to its greater part (Ф) is approximately 1.618 ... The reciprocal value - the ratio of the smaller part of the segment to the larger and the greater part to the entire segment - is approximately 0.618 ... This fact is embedded in the equation for the number Ф (**).

If we divide any segment into two parts so that the ratio of the larger part of the segment to the whole is equal to the ratio of the smaller part to the larger one, we get a section that is called golden.

One of the most beautiful works of ancient Greek architecture is the Parthenon (V century BC). The figures show a number of patterns associated with the golden ratio. The proportions of the building can be expressed through various degrees of the number Ф = 0.618 ...

On the floor plan of the Parthenon, you can also see the "golden rectangles":

We can see the golden ratio in the building of Notre Dame Cathedral (Notre Dame de Paris)

The proportions of the pyramid of Cheops, temples, bas-reliefs, household items and decorations from the tomb of Tutankhamun indicate that the Egyptian craftsmen used the ratios of the golden division when creating them. The French architect Le Corbusier found that in the relief from the temple of Pharaoh Seti I in Abydos and in the relief depicting Pharaoh Ramses, the proportions of the figures correspond to the values of the golden division. The architect Khesira depicted on a relief wooden board from the tomb of his name, holds in his hands measuring tools, in which the proportions of the golden division are fixed.

Turning to examples of the "golden section" in painting, one cannot but stop one's attention on the work of Leonardo da Vinci. Let's look closely at the painting "La Gioconda". The composition of the portrait is based on "golden triangles".

FIBONACCCI NUMBERS - a numerical sequence, where each subsequent term

row is equal to the sum of the previous two, that is: 1, 1, 2, 3, 5, 8, 13, 21, 34,

55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711,

28657, 46368,.. 75025,.. 3478759200, 5628750625,.. 260993908980000,..

422297015649625,.. 19581068021641812000,.. A variety of professional scientists and amateurs of mathematics have been studying the complex and amazing properties of the Fibonacci series numbers.

In 1997, several strange features of the series were described by the researcher

Vladimir MIKHAILOV. [Computer bulletin of RIA-Novosti "Terra-Incognita"]

32(209) dated 08.08.1997]. Mikhailov is convinced that Nature (including

Man) develops according to the laws that are embedded in this numerical

sequences. AT pine cone if you look at it from the side

handle, you can find two spirals, one twisted against the other along

hour hand. The number of these spirals is 8 and 13.

There are pairs of spirals in sunflowers: 13 and 21, 21 and 34, 34 and 55, 55 and 89. And there are no deviations from these pairs!..

Let's take a closer look at the chicory shoot. Its growth impulses gradually decreased in proportion to the golden ratio.

In a lizard, proportions that are pleasing to our eyes are captured at first glance - the length of its tail relates to the length of the rest of the body as 62 to 38. You can notice the golden proportions if you look closely at the bird's egg.

In Man, in the set of chromosomes of a somatic cell (there are 23 pairs of them), the source of hereditary diseases are 8, 13 and 21 pairs of chromosomes ... Perhaps all this indicates that the series of Fibonacci numbers is a kind of encrypted law of nature.

It is known from the history of astronomy that I. Titius, a German astronomer of the 18th century, using this series, found regularity and order in the distances between the planets of the solar system.
However, one case that seemed to be against the law: there was no planet between Mars and Jupiter. Focused observation of this area of the sky led to the discovery of the asteroid belt. This happened after the death of Titius in early XIX in. The Fibonacci series is widely used: with its help, they represent the architectonics of living beings, and man-made structures, and the structure of the Galaxies. These facts are evidence of independence number series on the conditions of its manifestation, which is one of the signs of its universality.

H directing all his attention to the study of the behavior of the stock market. It interested and interests many. Exploring the features of price patterns, After a series of successful predictions, he came to the conclusion thatthat "Any human activity three distinctive features: form, time and relation, all of which are subject to the total Fibonacci sequence."

Ralph Nelson Elliott

Properties research

MOU Talovskaya secondary school

Abstract of the integrated lesson

in computer science and mathematics

Prepared by the teacher

computer science and mathematics

Dankova Valentina Anatolievna

year 2009

During the classes:

1. Organizing moment.

Greetings. Definition of absent. Checking students' readiness for the lesson.

2. Results of the research work

Teacher: Let's write the topic of the lesson in a notebook: "Sequence of Fibonacci numbers."

And who was this man? Scientist? Writer? Mathematician? Why does the sequence of numbers called "Fibonacci numbers" still haunt scientists, philosophers, and even you and me?

In preparing for today's lesson, in addition to solving problems, you spent research work. And I think that it will not be difficult for you to answer the question: What is special about Fibonacci numbers and why are they associated with the golden ratio, and what do these numbers have in common with nature? How does this sequence relate to our history?

I ask you to state the essence of your research and briefly write down the features of Fibonacci numbers in your notebook. …

A presentation is shown that accompanies the story of the students.

History reference Fibonacci life.

Fibonacci numbers in nature

Fibonacci numbers in painting, architecture.

Mathematical basis of Fibonacci numbers

Summing up what has been said, answer where this sequence manifested itself?

What are the sciences associated with it?

In what areas of human knowledge did it manifest itself?

What does this indicate?

These facts are evidence of the independence of the number series from the conditions of its manifestation, which is one of the signs of its universality.

After researching this topic, what features of this sequence did you notice?

Are all the numbers on the board even? where are they located?

But can it be argued that the 27th place will also be an even number, and the 28th odd number?

What can be said about the numbers 5 and 8, what are they? What about 13 and 21? And if you take the numbers standing on the 37th and 38th place?

Every fifteenth number ends in zero

So, today in the lesson we have to study some properties of numbers.

every third Fibonacci number even,

every fifteenth ends zero,

two adjacent Fibonacci numbers coprime and etc.

Only the first and third properties for the first 12 Fibonacci numbers are obvious to us, the second property we need to find out experimentally. Now in your notebooks you will make programs that approve these properties or, on the contrary, deny them. That is, we will conduct a study of these properties of Fibonacci numbers using the PASCAL programming language. (The first group works at computers, the second group works in notebooks, one student at the teacher's computer is typing this program.). At the end of the work, a self-check is carried out.

Task for the first group

№1 . Fill the array A(N) with elements of the Fibonacci sequence. Let's check the parity of each number standing in the places of multiples of 3.

Task for the second group

№1. Fill the array A(N) with elements of the Fibonacci sequence. Check if adjacent Fibonacci numbers are prime

Homework

№1. Fill the array A(N) with elements of the Fibonacci sequence. Check if every fifteenth number from the sequence will end with zero,

According to the research of historians, it can be argued: chronology and periodization, historical development with the help of the Fibonacci series, it is divided into 18 time steps that have a planetary character. Events, the chronology of which is outside the series, have a regional character, i.e. local, moving boundaries. The chronological boundaries of archaeological epochs and periods found using the Fibonacci series are rigid. There is no agreement in them: they are either acceptable or they are not. This is because such a choice is based on a scientific worldview, which is always strictly defined.

Ralph Helson Elliott being a simple engineer. After a serious illness in the early 1930s. engaged in the analysis of stock prices. H directing all his attention to the study of the behavior of the stock market. It interested and interests many. Exploring the features of price patterns, After a series of successful predictions, he came to the conclusion that "Any human activity has three distinctive features: form, time and attitude, and they all obey the total Fibonacci sequence."

Lesson Analysis

Lesson type: integrated (mathematics and computer science)

Type of lesson: Research.

Lesson Objectives.

Educational:

Create conditions for understanding the term “Fibonacci sequence”;

To promote the use of the sequence of these numbers in solving problems of filling and processing one-dimensional arrays;

Help in developing existing knowledge on the topics “Array”, “Filling array elements using formulas” and skills in working in the PASCAL environment;

Contribute to the implementation of interdisciplinary connections in the informatics lesson.

Develop research work in the informatics class.

Educational:

To promote the development of cognitive interest and creative activity of students;

To promote the development of logical thinking and the ability to model a problem.

Educational:

Contribute to the formation of cognitive interest as a component of educational motivation;

Encourage students to become interested in historical events, associated with the numbers of the Fibonacci sequence;

Contribute to the development of the skills of conscious and rational use computers in their educational and then professional activities.

Teaching methods and techniques: explanatory and illustrative; partial search; verbal (frontal conversation); visual (demonstration of a computer presentation); practical, research method.

Means of education: author's multimedia presentation integrated with the PASCAL program; technical (computer, multimedia projector with screen), board, marker. Computer software security: PowerPoint and PASCAL programs.

1. Every third even

program n1;

var i,w,f,k: longint;

begin

a:=1; a:=1;

for i:=3 to 40 do

a[i]:=a+a;

for i:=1 to 40 do

write(a[i]," ");

for i:=1 to 40 do begin

if (a[i] mod 2<>0)and (i mod 3=0) then begin w:=1; k:=i; end;

if (a[i] mod 2=0) and (i mod 3<>0) then f:=1;

end; writeln;

if w=0 then writeln ("every third even")else writeln (k);

if f=0 then writeln ("if the index is not a multiple of 3 then the number is odd");

readln;

end.

2. Every fifteenth ends with a zero

program no. 2;

var i,w,f,k: longint;

a:array of integer;

begin

a:=1; a:=1;

for i:=3 to 40 do

a[i]:=a+a;

for i:=1 to 40 do

write(a[i]," ");

for i:=1 to 40 do begin

if (a[i] mod 10<>0)and (i mod 15=0) then begin w:=1; k:=i; end;

if (a[i] mod 10=0) and (i mod 15<>0) then f:=1;

end; writeln;

if w=0 then writeln ("only the fifteenth ends in zero") else writeln (k);

if f=0 then writeln (" every fifteenth ends with a zero");

readln;

end.

3. Neighboring elements are coprime.

program n3;

var x,y,i,w,f,k: longint;

a:array of integer;

begin

a:=1; a:=1;

for i:=3 to 40 do

a[i]:=a+a;

for i:=1 to 40 do

write(a[i]," ");

for i:=2 to 40 do begin

x:=a[i]; y:=a;

repeat

if x>y then x:=x mod y else y:=y mod x;

until (x=0) or (y=0);

if x+y<>1 then f:=1;

end; writeln;

if f=0 then writeln ("adjacent elements are coprime");

readln;

end.

4. Display all Fibonacci numbers not exceeding 50.

program n 4;

var i,w,f,k,l: longint;

a:array of longint;

begin

a:=1; a:=1; i:=3;

While a[i]<50 do begin

a[i]:=a+a;

i:=i+1;

end;

l:= i-1;

for i:=1 to l do

write(a[i]," ");

readln;

end.

Tasks

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 calculus and the advantages of using it over the Roman one.

Definition
Fibonacci numbers or Fibonacci Sequence is a numerical sequence that has a number of properties. For example, the sum of two neighboring numbers in the 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…

Complete definition of Fibonacci numbers

3.

Properties of the Fibonacci Sequence

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

2. When dividing each number by the next one, the number 0.382 is obtained through one; vice versa - respectively 2.618.

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

5.

Relationship between the Fibonacci sequence and the "golden section"

The Fibonacci sequence 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 cannot be expressed exactly.

If any member of the Fibonacci sequence is divided by the one preceding it (for example, 13:8), the result will be a value that fluctuates around the irrational value of 1.61803398875 ... and after a time either exceeding it or not reaching it. But even having spent Eternity on it, it is impossible to know the ratio exactly, to the last decimal digit. For the sake of brevity, we will give it in the form of 1.618. Special names for this ratio began to be given even before Luca Pacioli (a medieval mathematician) called it the Divine Proportion. Among its modern names are such as the Golden Ratio, the Golden Mean and the ratio of rotating squares. Kepler called this relation one of the "treasures of geometry". In algebra, it is commonly denoted by the Greek letter phi

Let's imagine the golden section on the example of a segment.

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

AC/CB = CB/AB or

AB/CB = CB/AC.

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

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

Segments of the golden ratio 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.

Fibonacci proportions and the golden ratio in nature and history

10.

It is important to note that Fibonacci, as it were, reminded humanity of his sequence. It was known to the ancient Greeks and Egyptians. Indeed, since then, patterns described by Fibonacci coefficients have been found in nature, architecture, fine arts, mathematics, physics, astronomy, biology and many other areas. It is simply 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 number game, but the most important mathematical expression of natural phenomena ever discovered.

11.

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

12.

1. The shell is twisted in a spiral. If you unfold it, you get a length slightly inferior to the length of the snake. A 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 measurements of the volutes of the shell is constant and equal to 1.618. Archimedes studied the spiral of shells and derived the equation for the spiral. The spiral drawn by this equation is called by his name. The increase in her step is always uniform. At present, the Archimedes spiral is widely used in engineering.

2. Plants and animals. Even Goethe emphasized the tendency of nature to spirality. The spiral and spiral arrangement of leaves on tree branches was noticed long ago. The spiral was seen in the arrangement of sunflower seeds, in 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, pine cones, the Fibonacci series manifests itself, and therefore, the law of the golden section manifests itself. The spider spins its web in a spiral pattern. A hurricane is spiraling. A frightened herd of reindeer scatter in a spiral. The DNA molecule is twisted into a double helix. Goethe called the spiral "the curve of life."

Among the roadside grasses, an unremarkable plant grows - chicory. Let's take a closer look at it. A branch was formed from the main stem. Here is the first leaf. The process makes a strong ejection into space, stops, releases a leaf, but is shorter than the first one, again makes an ejection into space, but of less force, releases an even smaller leaf and ejection again. If the first outlier is taken as 100 units, then the second is equal to 62 units, the third is 38, the fourth is 24, and so on. The length of the petals is also subject to the golden ratio. In growth, the conquest of space, the plant retained certain proportions. Its growth impulses gradually decreased in proportion to the golden ratio.

The lizard is viviparous. In the lizard, at first glance, proportions that are pleasing to our eyes are caught - the length of its tail relates to the length of the rest of the body as 62 to 38.

Both in the plant and animal worlds, the shaping tendency of nature persistently breaks through - symmetry with respect to 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 the division into symmetrical parts and golden proportions. In parts, a repetition of the structure of the whole is manifested.

Pierre Curie at the beginning of our century formulated a number of profound ideas of symmetry. He argued that one cannot consider the symmetry of any body without taking into account the symmetry of the environment. The patterns of golden symmetry are manifested in the energy transitions of 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, are in the structure of individual human organs and the body as a whole, and are also manifested in biorhythms and the functioning of the brain and visual perception.

3. Space. It is known from the history of astronomy that I. Titius, a German astronomer of the 18th century, using this series (Fibonacci) found regularity and order in the distances between the planets of the solar system

However, one case that seemed to be against the law: there was no planet between Mars and Jupiter. Focused observation of this area of the sky led to the discovery of the asteroid belt. This happened after the death of Titius at the beginning of the 19th century.

The Fibonacci series is widely used: with its help, they represent the architectonics of living beings, and man-made structures, and the structure of the Galaxies. These facts are evidence of the independence of the number series from 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 Giza pyramid. Unlike other Egyptian pyramids, this is not a tomb, but rather an unsolvable puzzle of numerical combinations. The remarkable ingenuity, skill, time and labor of the architects of the pyramid, which they used in the construction of the eternal symbol, indicate the extreme importance of the message that they wanted to convey to future generations. Their era was pre-literate, pre-hieroglyphic, and symbols were the only means of recording discoveries. The key to the geometrical-mathematical secret of the Giza pyramid, 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.

Triangle area

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 edge of the base, 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 numbers from the Fibonacci sequence. These interesting observations suggest that the construction of the pyramid is based on the proportion Ф=1.618. Some modern scholars tend to interpret that the ancient Egyptians built it for the sole purpose of passing on the knowledge they wanted to preserve for future generations. Intensive studies of the pyramid at Giza showed how extensive knowledge in 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 the Egyptian pyramids were 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 Egyptian and Mexican pyramids were erected at approximately the same time by people of a common origin.

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Introduction

THE HIGHEST PURPOSE OF MATHEMATICS IS TO FIND THE HIDDEN ORDER IN THE CHAOS THAT SURROUNDS US.

Viner N.

A person strives for knowledge all his life, tries to study the world around him. And in the process of observation, he has questions that need to be answered. Answers are found, but new questions appear. In archaeological finds, in the 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 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 about what Fibonacci numbers are, I heard from a mathematics teacher. But, besides, how the sequence of these numbers is formed, I did not know. This is what this sequence is actually famous for, how it affects a person, and 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 in the family of a merchant, in the city of Pisa in Italy. Fibonacci's father was often in Algiers on business, and Leonardo studied mathematics there with Arab teachers. Subsequently, he wrote several mathematical works, the most famous of which is the "Book of the abacus", which contains almost all the arithmetic and algebraic information of that time. 2

Fibonacci numbers are a sequence of numbers with a number of properties. Fibonacci discovered this numerical sequence by accident when he tried to solve a practical problem about rabbits in 1202. “Someone placed a pair of rabbits in a certain place, enclosed on all sides by a wall, in order to find out how many pairs of rabbits will be born during the year, if the nature of rabbits is such that in a month a pair of rabbits gives birth to another pair, and rabbits give birth from the second months after his birth. When solving the problem, he took into account that each pair of rabbits gives birth to two more pairs during their life, 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's called the Fibonacci sequence. Mathematical properties of a sequence

I wanted to explore this sequence, and I identified some of its properties. This rule is of great importance. The sequence slowly approaches some constant ratio of about 1.618, and the ratio of any number to the next is about 0.618.

One can notice a number of curious properties of Fibonacci numbers: two neighboring numbers are coprime; every third number is even; every fifteenth ends in zero; every fourth is a multiple of three. If you choose any 10 neighboring 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 member of the given sequence. And here is another interesting feature. For any n, the sum of the first n members of the sequence will always be equal to the difference of the (n + 2) -th and first member 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: to find the sum of all terms

sequence between two given members, it suffices to find the difference of the corresponding (n+2)-x members. For example, a 26 + ... + a 40 \u003d a 42 - a 27. Now let's look for a connection between Fibonacci, Pythagoras and the "golden section". 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 \u003d 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 built on them. The Pythagorean theorem says that the total area of the squares built on the legs 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 triples. Using the Fibonacci sequence, you can find such triples. 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) \u003d 30; 3) the sum of the squares of two average numbers: 3 2 +5 2 \u003d 34; 34 2 =30 2 +16 2 . This method works for any four consecutive Fibonacci numbers. Predictably, any three consecutive numbers of the Fibonacci series behave in a predictable way. If you multiply the two extremes of them and compare the result with the square of the average number, then the result will always differ by one. For example, for numbers 5, 8 and 13 we get: 5*13=8 2 +1. If we consider this property from the point of view of geometry, we can notice something strange. Divide the square

size 8x8 (total 64 small squares) into four parts, the lengths of the sides of which are 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 a perfect 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. Get 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 put 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 such rectangles and other geometric shapes using 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 section with beauty is not only a matter of human perception. It seems that nature itself has allocated a special role to F. If squares are sequentially entered into the "golden" rectangle, then an arc is drawn in each square, then an elegant curve is obtained, which is called a logarithmic spiral. It is not a mathematical curiosity at all. 5

On the contrary, this wonderful 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 the petals of a full-blown rose. The connections between the golden ratio and Fibonacci numbers are numerous and unexpected. 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 kinds 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 when you can find Fibonacci numbers 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 seen. In one of these families, the spirals curl clockwise, and in the other - counterclockwise. Spiral numbers of one type and another often turn out to be neighboring Fibonacci numbers. So, taking a young pine twig, it is easy to notice that the needles form two spirals, going from bottom left to right up. On many cones, the seeds are arranged in three spirals, gently winding around the stem of the cone. They are arranged 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. The Fibonacci spirals are also clearly visible on the pineapple: there are usually 8 and 13 of them.

The chicory shoot makes a strong ejection into space, stops, releases a leaf, but already shorter than the first one, again makes an ejection into space, but of lesser force, releases an even smaller leaf and ejection again. Its growth impulses gradually decrease in proportion to the "golden" section. To appreciate the huge role of Fibonacci numbers, just look at the beauty of the nature around us. Fibonacci numbers can be found in quantity

branches on the stem of each growing plant and in the number of petals.

Let's count the petals of some flowers - the iris with its 3 petals, the primrose with 5 petals, the ragweed with 13 petals, the daisy with 34 petals, the aster with 55 petals, and so on. Is this a coincidence, or is it the law of nature? Look at the stems and flowers of the yarrow. Thus, the total Fibonacci sequence can easily interpret the pattern of manifestations of the "Golden" numbers found in nature. These laws operate regardless of our consciousness and the 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 space 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 biorhythms and the functioning of the brain and visual perception.

Fibonacci numbers in architecture

The Golden Ratio also manifests itself in many remarkable architectural creations throughout the history of mankind. It turns out that even ancient Greek and Egyptian mathematicians knew these coefficients long before Fibonacci and called them the "golden section". The principle of the "golden section" was used by the Greeks in the construction of the Parthenon, the Egyptians - the Great Pyramid of Giza. Advances in building technology and the development of new materials opened up new possibilities for 20th-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 the design of 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 diverge. The building is a combination

orthogonal and concentric spirals, symbolizing the interaction of limited human knowledge and controlled chaos of nature. Its architecture mimics a plant that follows the movement of the sun, so the classrooms are lit up 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 well-known center for mathematical research. In Quincy Park, you can walk among the "golden" spirals and metal curves, reliefs of two shells and a rock with a square root symbol. On the plate is written information about the "golden" proportion. Even bike parking uses the F symbol.

Fibonacci numbers in psychology

In psychology, there are turning points, crises, upheavals that mark the transformation of the structure and functions of the soul on a person's life path. If a person has successfully overcome these crises, then he becomes able to solve problems of a new class, which he had not even thought about before.

The presence of fundamental changes gives reason to consider the time of life as a decisive factor in the development of spiritual qualities. After all, nature measures time for us not generously, “no matter how much it will be, so much will be,” but just enough so that the development process materializes:

in the structures of the body;

in feelings, thinking and psychomotor - 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 the life path understood in everyday consciousness? The inhabitant sees it like this: at the foot - the birth, at the top - the prime of life, and then - everything goes downhill.

The wise man will say: everything is much more complicated. He divides the ascent into stages: childhood, adolescence, youth ... Why is that? Few people 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 life into stages according to the number of years lived: 0 - the beginning of the countdown - the child was 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 masters the immediate environment;

2 - understands speech and acts using verbal instructions;

3 - acts through the word, asks questions;

5 - "age of grace" - the 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, by the forces of 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 brilliant work is born;

55 - at this age, subject to the preserved harmony of soul and body, a person is ready to become a creator. Etc…

What are Fibonacci serifs? They can be compared to dams on the path of life. These dams await each of us. First of all, it is necessary to overcome each of them, and then patiently raise your level of development, until one day it falls apart, opening the way to the next free flow.

Now that we understand the meaning of these nodal points of age development, let's try to decipher how it all happens.

At 1 year the child learns to walk. Before that, he knew the world with the front of his head. Now he knows the world with his hands - the exclusive privilege of man. The animal moves in space, and he, cognizing, masters the space and masters the territory on which he lives.

2 years understands the word and acts in accordance with it. It means that:

the child learns the minimum number of words - meanings and patterns of action;

yet does not separate itself from the environment and is merged into integrity with the environment,

Therefore, he acts on someone else's instructions. At this age, he is the most obedient and pleasant for parents. From a man of the senses, the child turns into a man of knowledge.

3 years- action with the help of one's own word. The separation of this person from the environment has already taken place - and he is learning to be an independently acting person. Hence he:

consciously opposes the environment and parents, kindergarten teachers, etc.;

is aware of 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 acting person begins.

5 years- Age of Grace. He is the personification of harmony. Games, dances, dexterous movements - everything is saturated with harmony, which a person tries to master with his own strength. Harmonious psychomotor contributes to bringing to a new state. Therefore, the child is directed to psychomotor activity and strives for the most active actions.

Materialization of the products of the work of sensitivity is carried out through:

the ability to display the environment and ourselves as part of this world (we hear, see, touch, smell, etc. - all sense organs work for this process);

ability to design the outside world, including yourself

(creation of a second nature, hypotheses - to do both 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 (implementation of the plan, specific mental or psychomotor actions with specific objects and processes).

After 5 years, the imagination mechanism comes forward and begins to dominate the rest. The child does a gigantic job, creating fantastic images, and lives in the world of fairy tales and myths. The hypertrophy of the child's imagination causes surprise in adults, because the imagination does not correspond to reality in any way.

8 years- feelings come to the fore and their own measurements of feelings (cognitive, moral, aesthetic) arise when the child unmistakably:

evaluates the known and the unknown;

distinguishes the moral from the immoral, the moral from the immoral;

beauty from what threatens life, harmony from chaos.

13 years old- the mechanism of creativity begins to work. But that doesn't mean it's working at full capacity. One of the elements of the mechanism comes to the fore, and all the others contribute to its work. If even in this age period of development harmony is preserved, which almost all the time rebuilds its structure, then the child will painlessly get to the next dam, overcome it imperceptibly and will live at the age of a revolutionary. At the age of a revolutionary, the youth must take a new step forward: to separate from the nearest society and live in it a harmonious life and activity. 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 fulfilling a talented

work. Feelings (cognitive, moral, or aesthetic) sometimes overshadow thinking, but in general, all elements work in harmony: 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 to work. At this age, the mechanism of feelings comes forward. As the imagination and its products are evaluated by feelings and thinking, antagonism arises between them. Feelings win. This ability is gradually gaining power, and the boy begins to use it.

34 years- balance and harmony, productive effectiveness of talent. Harmony of thinking, feelings and imagination, psychomotor skills, which is replenished with optimal energy potential, and the mechanism as a whole - an opportunity is born to perform brilliant work.

55 years- a person can become a creator. The third harmonious peak of life: thinking subdues the power of feelings.

Fibonacci numbers name the stages of human development. Whether a person goes 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 objects of relationships:

From birthday to 2 years - the discovery of the physical and objective world of the immediate environment.

From 2 to 3 years - the discovery of oneself: "I am Myself."

From 3 to 5 years - speech, the effective world of words, harmony and the "I - You" system.

From 5 to 8 years old - the discovery of the world of other people's thoughts, feelings and images - the "I - We" system.

From 8 to 13 years old - the discovery of the world of tasks and problems solved by the geniuses and talents of mankind - the system "I - Spirituality".

From 13 to 21 years old - the discovery of the ability to independently solve well-known tasks, when thoughts, feelings and imagination begin to work actively, the "I - Noosphere" system arises.

From 21 to 34 years old - the discovery of the ability to create a new world or its fragments - the realization of the self-concept "I am the Creator".

The life path has a space-time structure. It consists of age and individual phases, determined by many parameters of life. 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 links 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 of its early phases.

Fibonacci numbers in astronomy

It is known from the history of astronomy that I. Titius, a German astronomer of the 18th century, using the Fibonacci series, found regularity and order in the distances between the planets of the solar system. But one case seemed to be against the law: there was no planet between Mars and Jupiter. But after the death of Titius at the beginning of the XIX century. concentrated observation of this part of the sky led to the discovery of the asteroid belt.

Conclusion

In the process of 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 length of time after which an 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 too hard for me to figure out. Although Fibonacci was the greatest mathematician of the Middle Ages, the only monuments to Fibonacci are the 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 perfect? What will be next? Finding the answer to one question, you get the next. If you solve it, you get two new ones. Deal with them, three more will appear. Having solved them, you will acquire five unresolved ones. Then eight, thirteen, and so on. Do not forget that there are five fingers on two hands, two of which consist of two phalanges, and eight of which consist of three.

Literature:

Voloshinov A.V. "Mathematics and Art", M., Enlightenment, 1992

Vorobyov N.N. "Fibonacci numbers", M., Nauka, 1984

Stakhov A.P. "The Da Vinci Code and the Fibonacci Series", Peter 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

Fibonacci numbers... in nature and life

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

Complete definition of Fibonacci numbers

3.

Properties of the Fibonacci Sequence

2. When dividing each number by the next one, the number 0.382 is obtained through one; vice versa - respectively 2.618.

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

5.

Relationship between the Fibonacci sequence and the "golden section"

Let's imagine the golden section on the example of a segment.

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

AC/CB = CB/AB or

AB/CB = CB/AC.

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

Fibonacci proportions and the golden ratio in nature and history

11.

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

12.

The lizard is viviparous. In the lizard, at first glance, proportions that are pleasing to our eyes are caught - the length of its tail relates to the length of the rest of the body as 62 to 38.

Triangle area

356 x 440 / 2 = 78320

square area

280 x 280 = 78400

Among the many inventions made by great scientists in past centuries, the discovery of the patterns of development of our universe in the form of a system of numbers is the most interesting and useful. This fact was described in his work by the Italian mathematician Leonardo Fibonacci. A number series is a sequence of digits in which each member value is the sum of the two previous ones. This system expresses the information embedded in the structure of all living things in accordance with the harmonious development.

The great scientist Fibonacci

The Italian scientist lived and worked in the XIII century in the city of Pisa. He was born into a merchant family and at first worked with his father in trade. Leonardo Fibonacci came to mathematical discoveries when he was trying to establish contacts at that time with business partners.

The scientist made his discovery when calculating the planning of the offspring of rabbits at the request of one of his distant relatives. He opened the number series, according to which the reproduction of animals will be carried out. He described this pattern in his work "The Book of Calculations", where he also presented information about the decimal for European countries.

"Golden" discovery

The number series can be expressed graphically as an expanding spiral. It can be noted that in nature there are many examples that are based on this figure, for example, rolling waves, the structure of galaxies, microcapillaries in the human body, and

Interestingly, the numbers in this system (Fibonacci coefficients) are considered “live” numbers, since all living things evolve according to this progression. This pattern was known even to people of ancient civilizations. There is a version that already at that time it was known how to investigate the convergence of a number series - the most important issue in the sequence of numbers.

Application of Fibonacci Theory

Having examined his number series, the Italian scientist discovered that the ratio of a digit from a given sequence to the next member is 0.618. This value is called the proportionality factor, or the "golden section". It is known that this number was used by the Egyptians in the construction of the famous pyramid, as well as the ancient Greeks and Russian architects in the construction of classical structures - temples, churches, etc.

But an interesting fact is that the Fibonacci number series is also used in evaluating the movement of prices for The use of this sequence in technical analysis was proposed by engineer Ralph Elliot at the beginning of the last century. In the 30s, the American financier was engaged in forecasting stock prices, in particular, the study of the Dow Jones index, which is one of the main components in the stock market. After a series of successful predictions, he published several of his articles in which he described methods for using the Fibonacci series.

At the moment, almost all traders use the Fibonacci theory when predicting price movements. Also, this dependence is used in many scientific studies in various fields. Thanks to the discovery of a great scientist, many useful inventions can be created even after many centuries.