Number 13 On the energies of the Earth channel, the number 13, like a four, has a green color - the level of sound and information. This is the fourth digit of the new reality, its double sign. The number 13 adds up to the number 4, the fourth point of reality. In Nature's understanding, this is a flower waiting for pollination.

Number 14 On the energies of the Earth channel, the number 14 manifests itself in representatives of the new, not yet mastered by our civilization, the first intellectual level of the Sky-blue color. Under the code number 14, people born on the last day of the year come. These people are not

Number 11 On the energies of the Cosmic Channel, the number 11 personifies the energy of two worlds: manifested and unmanifested. Symbolically, this is the Sun reflected in water, two Suns: in the sky and in water, two units. This is a sign of play, a sign of creativity. The person of this sign is a mirror that

Number 12 On the energies of the Cosmic Channel, the number 12 personifies the harmony and completeness of space at a new level of reality, which includes three basic concepts of life: past, present and future. The number 12 contains one - the sign of the leader and two - the sign of the owner

Number 13 On the energies of the Cosmic Channel, the number 13 personifies the wind energy of all four cardinal points, mobility, sociability at a new level of development. Symbolically, the energy of the number 13 looks like the same Wind Rose as the number 4, but without space restrictions.

Number 14 On the energies of the Cosmic Channel, the number 14 is the messenger of the Cosmos. Royal number 13 is not the last in the levels of development of our civilization. There is one more day in the year when missionaries come from the Cosmos itself, these people do not have a clear body code (Earth channel), they do not have

Step one. We calculate the number of birth, or the number of personality The number of birth reveals the natural characteristic of a person, it, as we have already said, remains unchanged for life. Unless we are talking about the numbers 11 and 22, which can “simplify” to 2 and 4

5th number. "Bor" Bor is often lucky at birth, and he inherits certain capitals, "factories" and "steamboats". Perhaps he will not squander the inheritance, and will pass it on to his heirs. His personal preferences are vague - whether he loves harmony and feels, or loves power and

Leonardo Fibonacci is one of the greatest mathematicians of the Middle Ages. In one of his own works, The Book of Calculations, Fibonacci outlined the Indo-Arabic calculus and the benefits of using it over the Roman one.

Definition

Fibonacci numbers or Fibonacci sequence - a numerical sequence that has a number of parameters. For example, the sum of 2 adjoining numbers of 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

Characteristics 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 comes out through one; on the contrary - respectively 2.618.

3. Therefore, choosing the ratios, we obtain the main set of Fibonacci coefficients: … 4.235, 2.618, 1.618, 0.618, 0.382, 0.236.

Relationship between the Fibonacci sequence and the "golden section"

The Fibonacci sequence asymptotically (approaching slower and slower) tends to some constant ratio. But, this ratio is irrational, in other words it is a number with an endless, unpredictable sequence of decimal digits in the fractional part. It is impossible to express it exactly.

In that case, 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 1.61803398875... and through time it sometimes surpasses it, sometimes it does not reach it. But even having spent an eternity on this, it is unrealistic to find out the ratio exactly, to the last decimal digit. For the sake of brevity, we will present it in the form 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 titles there are such as golden ratio, Golden mean and 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

Ф=1.618

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

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

AC/CB = CB/AB or

It is possible to represent it approximately 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 largest part in the same way as the largest part itself relates to the smallest; or in other words, the smallest segment is related to the larger one as the larger one is to everything.

Segments of the golden ratio are expressed by an endless irrational fraction 0.618..., in that case, take AB as a unit, 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

It is important to note that Fibonacci seems to have reminded the population of the earth of his sequence. It was known to the ancient Greeks and Egyptians. Indeed, since that time in nature, architecture, fine arts, arithmetic, physics, astronomy, biology and many other areas, patterns described by Fibonacci coefficients have been found. It is simply mind-boggling how many constants it is possible to calculate using the Fibonacci sequence, and how its members appear in an unlimited number of combinations. But it would not be an exaggeration to say that this is not just a game with numbers, but the most fundamental mathematical expression of natural phenomena ever discovered.

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

1. The shell is wrapped in a spiral . In that case, unfold it, then the length comes out, 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 interested 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. A spiral drawn according to this equation is called by its name. Raising her stride is always moderate. At present, the Archimedes spiral is widely used in engineering.

2. Plants and animals . Even Goethe emphasized the laws of nature towards helicity. The helical and spiral arrangement of leaves on tree branches has been noticed for a long time. The spiral was seen in the arrangement of sunflower seeds, in pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch of sunflower seeds, pine cones manifests itself Fibonacci series, and therefore the law manifests itself golden section. The spider weaves a web in a spiral pattern. A hurricane is spiraling. A frightened herd of reindeer scatter in a spiral. The DNA molecule is wrapped in a double helix. Goethe called the spiral "the curve of life."

Among the roadside herbs, an inconspicuous plant grows - chicory . Let's take a close look at him. A branch was formed from the main stem. Here is the 1st sheet. The process makes a strong ejection into place, stops, releases a leaf, however, it is already shorter than the first one, again makes an ejection into place, but already of the smallest force, releases a leaf of an even smaller size and again ejection. In that case, the 1st outlier is taken as 100 units, then the 2nd is equal to 62 units, the 3rd - 38, the 4th - 24, etc. The length of the petals is also subject to the golden ratio. In growth, the conquest of a place, the plant retained certain proportions. Its growth impulses decreased uniformly in proportion to the golden section.

The lizard is viviparous. In the lizard, at first glance, proportions that are pleasant to our eyes are captured - 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 world, the form-forming regularity of nature is aggressively breaking 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 made a 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 identified a number of the deepest thoughts of symmetry. He argued that one cannot consider the symmetry of any body without taking into account the symmetry of the medium. Patterns of golden symmetry appear in the energy transitions of simple particles, in the structure of certain chemical compounds, in planetary and galactic 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, also appear in biorhythms and the functioning of the brain and visual perception.

3.Space. From the history of astronomy, it is clear that I. Titius, a German astrologer of the 18th century, using this series (Fibonacci) found a pattern and order in the distances between the planets of the galaxy

But one case that seemed to be contrary to 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. It came out after the death of Titius in the early 19th century.

The Fibonacci series is widely used: with its help, they represent the architectonics of living creatures, man-made structures, and the structure of the Galaxies. These facts are evidence independence of the number series from the criterion of its manifestation , which is one of the hallmarks of its versatility.

4.Pyramids. Many tried to unravel the secrets pyramids at Giza. Unlike other Egyptian pyramids, this is not a tomb, but rather an unsolvable puzzle of numerical compositions. The remarkable ingenuity, skill, time and labor of the architects of the pyramid, which they used in the construction of the never-ending sign, indicate the extreme significance of the message they wished to convey to future generations. Their era was pre-literate, pre-hieroglyphic, and signs were the only means of recording discoveries. The key to the geometrical-mathematical secret of the Giza pyramid, which had been a mystery for the population of the earth for so long, was in reality handed over 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 noteworthy observations give a hint that the construction 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 transmitting the knowledge they wished to preserve for future generations. The intense study of the pyramid at Giza showed how vast the knowledge of arithmetic and astrology was in those periods. 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. There is an idea that both the Egyptian and Mexican pyramids were built at about the same time by people of a common origin.

In preparing the answer, the following material was used:

The Phi number is recognized as the most beautiful in the universe... Despite its mystical origin, the Phi number has played a unique role - the role of the basic block in the construction of all living things. All plants, animals, and human beings correspond to physical proportions approximately equal to the root of the ratio of Phi to 1... Phi is 1.618. The Phi number is derived from the Fibonacci sequence, a mathematical progression known not only because the sum of two neighboring numbers in it is equal to the next number, but also because the quotient of two neighboring numbers has a unique property - proximity to the number 1.618, that is, to the number Phi! This omnipresence of Phi in nature indicates the connection of all living beings. Sunflower seeds are arranged in spirals, counterclockwise and the ratio of the diameter of each of the spirals to the diameter of the next one is Phi. Spiral-shaped corn cob leaves, arrangement of leaves on plant stems, segmentation parts of insect bodies. And all of them in their structure obediently follow the law of "divine proportion". Drawing by Leonardo da Vinci depicting a naked man in a circle. No one better than da Vinci understood the divine structure of the human body, its structure. He was the first to show that the human body consists of "building blocks", the ratio of the proportions of which is always equal to our cherished number. If you measure the distance from the top of your head to the floor, then divide by your height, then we will see what the number will be. It is Phi - 1.618. The mathematician Fibonacci lived in the twelfth century (1175). He was one of the most famous scientists of his time. Among his greatest achievements is the introduction of Arabic numerals to replace Roman numerals. He discovered the Fibonacci summation sequence. This mathematical sequence occurs when, starting from 1, 1, the next number is obtained by adding the previous two. This sequence tends asymptotically to some constant relation. 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 1.61803398875... and sometimes exceeds it, sometimes not reaching it. But, even after spending Eternity on this, it is impossible to know the ratio exactly, to the last decimal digit. When dividing any member of the Fibonacci sequence by the next one, the result is simply the reciprocal of 1.618 (1:1.618). But this is also a very unusual, even remarkable phenomenon. Since the original ratio is an infinite fraction, this ratio must also have no end. 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-written, pre-hieroglyphic, and symbols were the only means of recording discoveries. The key to the geometric and 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. The area of ​​a triangle is 356 * 440 / 2 = 78320. The area of ​​a square is 280 * 280 = 78400. The length of the face of the Giza pyramid is 783.3 feet (238.7 m), the height of the pyramid is 484.4 feet (147.6 m). The length of the 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 numbers from the Fibonacci sequence. These interesting observations suggest that the construction of the pyramid is based on the proportion Ф = 1.618. Modern scholars lean towards the interpretation 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. Not only are the Egyptian pyramids built according to the perfect proportions of the golden ratio, the same phenomenon is 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 a common origin.

Camposanto (Camposanto monumentale). Pisa

Today I already told you about it, but I wanted to continue this topic in this way ...

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

The life and scientific career of Leonardo is closely connected with the development of European culture and science.

The Renaissance was still far away, but history gave Italy a short period of time that could well be called a rehearsal for the impending Renaissance. This rehearsal was led by Frederick II, Holy Roman Emperor. Brought up in the traditions of southern Italy, Frederick II was internally deeply far from European Christian chivalry. Frederick II did not recognize knightly tournaments at all. Instead, he cultivated mathematical competitions, in which opponents exchanged not blows, but problems.

At such tournaments, the talent of Leonardo Fibonacci shone. This was facilitated by 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. The meeting between Fibonacci and Frederick II took place in 1225 and was an event of great importance for the city of Pisa. The emperor rode at the head of a long procession of trumpeters, courtiers, knights, officials, and a wandering menagerie of animals. Some of the problems that the Emperor posed to the famous mathematician are detailed in the Book of the Abacus. Fibonacci, apparently, solved the problems posed by the Emperor, and forever became a welcome guest at the Royal Court.

When Fibonacci revised the Book of the Abacus in 1228, he dedicated the revised edition to Frederick II. In total, he wrote three significant mathematical works: the Book of the Abacus, published in 1202 and reprinted in 1228, Practical Geometry, published in 1220, and the Book of Quadratures. These books, surpassing in their level Arabic and medieval European writings, taught mathematics almost until the time of Descartes. As stated in documents from 1240, the admiring citizens of Pisa said that he was "a sensible and erudite man", and not so long ago, Joseph of Guise, editor-in-chief of the Encyclopædia Britannica, declared that future scientists at all times "will pay their debt to Leonardo of Pisa, as one of the world's greatest intellectual pioneers."

Rabbit problem.

Of greatest interest to us is the essay "The Book of the Abacus". 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.

The material is explained by examples of tasks that make up a significant part of this path.

In 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.

It is clear that if we consider the first pair of rabbits as newborns, then in the second month we will still have one pair; for the 3rd month — 1+1=2; on the 4th - 2 + 1 = 3 pairs (because of the two available pairs, only one pair gives offspring); on the 5th month - 3 + 2 = 5 pairs (only 2 couples born on the 3rd month will give offspring on the 5th month); on the 6th month - 5 + 3 = 8 pairs (because only those pairs that were born on the 4th month will give offspring), etc.

Thus, if we denote the number of pairs of rabbits available in the nth month as Fk, then F1=1, F2=1, F3=2, F4=3, F5=5, F6=8, F7=13, F8=21 etc., and the formation of these numbers is regulated by the general law: Fn=Fn-1+Fn-2 for all n>2, because the number of pairs of rabbits in the nth month is equal to the number Fn-1 of pairs of rabbits in the previous month plus the number newly born pairs, which coincides with the number of Fn-2 pairs of rabbits born in the (n-2)th month (because only these pairs of rabbits give birth).

The numbers Fn that form the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ... are called "Fibonacci numbers", and the sequence itself is called the Fibonacci sequence.

Special names for this ratio began to be given even before Luca Pacioli (a medieval mathematician) called it the Divine Proportion. Kepler called this relation one of the treasures of geometry. In algebra, its designation by the Greek letter "phi" (Ф=1.618033989...) is generally accepted.

The following are the ratios of the second term to the first, the third to the second, the fourth to the third, and so on:

1:1 = 1.0000, which is less than phi by 0.6180

2:1 = 2.0000, which is 0.3820 more phi

3:2 = 1.5000, which is less than phi by 0.1180

5:3 = 1.6667, which is 0.0486 more phi

8:5 = 1.6000, which is less than phi by 0.0180

As we move along the Fibonacci summation sequence, each new term will divide the next with more and more approximation to the unattainable "phi". Fluctuations of ratios around the value of 1.618 by a larger or smaller value, we will find in the Elliott Wave Theory, where they are described by the Rule of Alternation. It should be noted that in nature it is precisely the approximation to the number "phi" that occurs, while mathematics operates with a "pure" value. It was introduced by Leonardo da Vinci and called the "golden section" (golden proportion). Among its modern names there are also such as "golden mean" and "rotating squares ratio". The golden ratio is the division of the segment AC into two parts in such a way that its greater part AB relates to the smaller part BC in the same way that the entire segment AC relates to AB, that is: AB: BC \u003d AC: AB \u003d F (exact irrational number " phi").

When dividing any member of the Fibonacci sequence by the next one, the value inverse to 1.618 is obtained (1: 1.618=0.618). This is also a very unusual, even remarkable phenomenon. Since the original ratio is an infinite fraction, this ratio must also have no end.

When dividing each number by the next one after it, we get the number 0.382.

Selecting ratios in this way, we obtain the main set of Fibonacci coefficients: 4.235, 2.618, 1.618, 0.618, 0.382, 0.236. All of them play a special role in nature and in particular in technical analysis.

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.

These numbers are undoubtedly part of a mystical natural harmony that feels good, looks good, and even sounds good. Music, for example, is based on an 8-note octave. On a piano this is represented by 8 white keys and 5 black keys for a total of 13.

A more visual representation can be obtained by studying spirals in nature and works of art. Sacred geometry explores two types of spirals: the golden section spiral and the Fibonacci spiral. Comparison of these spirals allows us to draw the following conclusion. The golden ratio spiral is perfect: it has no beginning and no end, it continues indefinitely. Unlike it, the Fibonacci spiral has a beginning. All natural spirals are Fibonacci spirals, and works of art use both spirals, sometimes at the same time.

Mathematics.

The pentagram (pentacle, five-pointed star) is one of the commonly used symbols. The pentagram is a symbol of a perfect person standing on two legs with outstretched arms. We can say that a person is a living pentagram. This is true both physically and spiritually - a person possesses five virtues and manifests them: love, wisdom, truth, justice and kindness. These are the virtues of Christ, which can be represented by a pentagram. These five virtues, necessary for human development, are directly related to the human body: kindness is associated with the feet, justice with the hands, love with the mouth, wisdom with the ears, eyes with the truth.

Truth belongs to the spirit, love to the soul, wisdom to the intellect, kindness to the heart, justice to the water. There is also a correspondence between the human body and the five elements (earth, water, air, fire and ether): will corresponds to earth, heart to water, intellect to air, soul to fire, spirit to ether. Thus, by his will, intellect, heart, soul, spirit, man is connected with the five elements working in the cosmos, and he can consciously work in harmony with it. This is the meaning of another symbol - a double pentagram, a person (microcosm) lives and acts inside the universe (microcosm).

The inverted pentagram pours energy into the earth and is therefore a symbol of materialistic tendencies, while the normal pentagram directs energy upward, thus being spiritual. On one point everyone agrees: the pentagram certainly represents the "spiritual form" of the human figure.

Note CF:FH=CH:CF=AC:CH=1.618. The actual proportions of this symbol are based on a sacred proportion called the golden ratio: this is the position of a point on any line drawn when it divides the line so that the smaller part is in the same ratio to the larger part as the larger part to the whole. In addition, the regular pentagon in the center suggests that the proportions are preserved for infinitesimal pentagons. This "divine proportion" is manifested in each individual ray of the pentagram and helps to explain the awe with which mathematicians have looked at this symbol at all times. Moreover, if the side of the pentagon is equal to one, then the diagonal is equal to 1.618.

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.

Scientists have discovered that the three pyramids at Giza are arranged in a spiral. In the 1980s, it was found that both the golden spiral and the Fibonacci spiral were present there.

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 face 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 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. Modern scholars lean toward the interpretation 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.

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 built at approximately the same time by people of common origin.

Biology.

In the 19th century, scientists noticed that the flowers and seeds of sunflowers, chamomile, scales in pineapple fruits, coniferous cones, etc. are “packed” in double spirals, curling towards each other. At the same time, the numbers of "right" and "left" spirals always refer to each other as neighboring Fibonacci numbers (13:8, 21:13, 34:21, 55:34). Numerous examples of double helixes found throughout nature always follow this rule.

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 work of botanists and mathematicians has 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."

Any good book will show the nautilus shell as an example. Moreover, in many publications it is said that this is a golden ratio spiral, but this is not true - this is a Fibonacci spiral. You can see the perfection of the arms of the spiral, but if you look at the beginning, it doesn't look so perfect. Its two innermost bends are actually equal. The second and third bends are a little closer to phi. Then, finally, this elegant smooth spiral is obtained. Remember the relationship of the second term to the first, the third to the second, the fourth to the third, and so on. It will be clear that the mollusk follows the mathematics of the Fibonacci series exactly.

Fibonacci numbers show up in the morphology of various organisms. For example, starfish. The number of rays they have corresponds to a series of Fibonacci numbers and is equal to 5, 8, 13, 21, 34, 55. The well-known mosquito has three pairs of legs, the abdomen is divided into eight segments, and there are five antennae on the head. The mosquito larva is divided into 12 segments. The number of vertebrae in many domestic animals is 55. The proportion of "phi" is also manifested in the human body.

Drunvalo Melchizedek in The Ancient Secret of the Flower of Life writes: “Da Vinci calculated that if you draw a square around the body, then draw a diagonal from the feet to the tips of the outstretched fingers, and then draw a parallel horizontal line (the second of these parallel lines) from the navel to the side of the square, then this horizontal line will intersect the diagonal exactly in phi proportion, as well as the vertical line from the head to the feet. If we consider that the navel is at that perfect point, and not slightly higher for women or slightly lower for men, then this means that the human body is divided in the proportion of phi from the top of the head to the feet ... If these lines were the only ones where in the human body there is a phi proportion, that would probably be just an interesting fact. In fact, the proportion of phi is found in thousands of places throughout the body, and this is not just a coincidence.

Here are some distinct places in the human body where the proportion of phi is found. The length of each phalanx of the finger is in the proportion of phi to the next phalanx ... The same proportion is noted for all fingers and toes. If you correlate the length of the forearm with the length of the palm, then you get the proportion of phi, just as the length of the shoulder refers to the length of the forearm. Or take the length of the leg to the length of the foot and the length of the thigh to the length of the leg. The proportion of phi is found throughout the skeletal system. It is usually marked in places where something bends or changes direction. It is also found in the ratio of the sizes of some parts of the body to others. Studying this, you are always surprised.”

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.

Conclusion.

Although he was the greatest mathematician of the Middle Ages, the only monuments to Fibonacci are a statue opposite the Leaning Tower of Pisa across the Arno River and two streets that bear his name, one in Pisa and the other in Florence.

If you put your open palm vertically in front of you, pointing your thumb to your face, and, starting with the little finger, successively clench your fingers into a fist, you get a movement that is a Fibonacci spiral.