Fibonacci numbers: practical application. Research paper "Fibonacci numbers"
Fibonacci Leonardo of Pisa (lat. Leonardo Pisano, Pisa, about 1170 - about 1250) was the first major mathematician of medieval Europe. He is better known by his nickname Fibonacci, which in Italian means “a good son is born” (Figlio Buono Nato Ci).
Little is known about the existence of Fibonacci. Even the exact date of his birth is unknown. Fibonacci is believed to have been born in 1170
Leonardo Fibonacci was a famous Italian mathematician, he was famous for his ability to do calculations. One day it dawned on him and he discovered a simple sequence of numbers, the relationships between which described the natural proportions of all bodies in the universe!
Leonardo Fibonacci was an outstanding mathematician of the Middle Ages. The fruits of his mathematical labors are used in many sciences, arts and everyday life to this day.
The merit of Leonardo Fibonacci is the series of Fibonacci numbers. It is believed that this series was known in the East, but it was Leonardo Fibonacci who published this series of numbers in the book “Liber Abaci” (he did this to demonstrate the reproduction of the rabbit population).
Elliott wrote: “The law of nature includes in consideration the most important element - rhythm. The law of nature is not a certain system, not a method of playing on the market, but a phenomenon that is characteristic, apparently, of the course of any human activity. Its application in forecasting is revolutionary.”
This chance to predict price movements keeps legions of analysts working day and night. We will focus on the ability to make predictions and try to find out whether it is possible or not. In introducing his approach, Elliott was very specific. He wrote: “Every human activity has three distinctive features: form, time and relation, and all of them are subject to the Fibonacci summation sequence.”
The Fibonacci sequence, known to everyone from the movie “The Da Vinci Code,” is a series of numbers described in the form of a riddle by the Italian mathematician Leonardo of Pisa, better known as 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.
Reflecting on this topic, Fibonacci built the following series of numbers.
A series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. known as the Fibonacci series. The peculiarity of the sequence of numbers is that each of its members, starting from the third, is equal to the sum of the previous two 2 + 3 = 5; 3 + 5 = 8; 5 + 8 = 13, 8 + 13 = 21; 13 + 21 = 34, etc., and the ratio of adjacent numbers in the series approaches the ratio of the golden division. So, 21: 34 = 0.617, and 34: 55 = 0.618. This ratio is denoted by the symbol F. Only this ratio - 0.618: 0.382 - gives a continuous division of a straight line segment in the golden proportion, increasing it or decreasing it to infinity, when the smaller segment is related to the larger one as the larger one is to everything.
Fibonacci also dealt with the practical needs of trade: what is the smallest number of weights that can be used to weigh a product? Fibonacci proves that the optimal system of weights is: 1, 2, 4, 8, 16...
This sequence has a number of mathematical features that definitely need to be touched upon. This sequence asymptotically (approaching more and more slowly) tends 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 is impossible to express it precisely.
Thus, the ratio of any member of the sequence to the one preceding it fluctuates around the number 1.618, sometimes exceeding it, sometimes not reaching it. The ratio to the next one similarly approaches the number 0.618, which is inversely proportional to 1.618. If we divide the elements of the sequence through one, we will get the numbers 2.618 and 0.382, which are also inversely proportional. These are the so-called Fibonacci ratios.
Nature, as it were, solves the problem from two sides at once and adds up the results obtained. As soon as it receives a total of 1, it moves to the next dimension, where it begins to build everything all over again. But then she must build this golden ratio according to a certain rule. Nature does not use the golden ratio right away. She obtains it through successive iterations and uses another series, the Fibonacci series, to generate the golden ratio.
The wonderful properties of the Fibonacci series are also manifested in the numbers themselves, which are members of this series. Let's arrange the terms of the Fibonacci series vertically, and then to the right, in descending order, write down the natural numbers.
21 20 19 18 17 16 15 14 13
34 33 32 31 30 29 28 27 26 25 24 23 22 21
55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34
Each line begins and ends with a Fibonacci number, i.e. there are only two such numbers in each line. "blue" numbers - 4, 7, 6, 11, 10, 18, 16, 29, 26, 47, 42 - have special properties (the second level of the Fibonacci series hierarchy):
(5-4)/(4-3) = 1/1
(8-7)/(7-5) = 1/2 and (8-6)/(6-5) = 2/1
(13-11)/(11-8) = 2/3 and (13-10)/(10-8) = 3/2
(21-18)/(18-13) = 3/5 and (21-16)/(1b-13) = 5/3
(34-29)/(29-21) = 5/8 and (34-26)/(26-21) = 8/5
(55-47)/(47-34) = 8/13 and (55-42)/(42-34) = 13/8
We have obtained the fractional Fibonacci series, which may be “professed” by the collective spins of elementary particles and atoms of chemical elements.
Let's imagine these numbers as a sequence of lever scales
What is all this for? This is how we approach one of the most mysterious natural phenomena. Fibonacci essentially did not discover anything new, he simply reminded the world of such a phenomenon as the 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. The golden ratio is 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 as 1, then segment a will be equal to 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). The ratio of c to a is 1.618, and c to b is 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.
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 sum of the 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 two previous ones. 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.
The Fibonacci series is not only a mathematical mystery, we encounter it every day in everyday life:
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.
A shell in the shape of a spiral - the shape of the shell interested Archimedes and he found out that the increase in the length of the curls of the shell is a constant value and is equal to 1.618.
Aloe multifolia.
Broccoli Romanesco.
Sunflower: The seeds in a sunflower are also arranged in a spiral.
Pine cone.
Plant growth also occurs in accordance with the Fibonacci number series - a branch leaves the trunk on which a leaf appears, then a long ejection occurs and a leaf appears again, but it is already shorter than the previous one. Then there is another surge, but it is also shorter than the previous one. In this picture, the first spike is 100%, the second is 62%, and the third is 38% (Fibonacci levels used in trading), etc. With the length of the petals everything looks exactly the same.
Lizard - if you divide a lizard into a tail and a body, then their ratio will be 0.62 to 0.38.
Pyramids - The edge length of the pyramid is 783.3 feet, and the height of the pyramid is 484.4 feet. The ratio of edge length/pyramid height is 1.618.
As you can see, the Fibonacci number series is widely represented in our lives: in the structure of living beings, structures, and even the structure of Galaxies is described with its help. All this testifies to the universality of the mathematical riddle of the Fibonacci number series.
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, the sequence itself is far from perfect, like everything in this world.
There is an assumption that the Fibonacci sequence is an attempt by nature to adapt to the 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 sequence, it is enough to know its three terms, following each other. But not for the golden sequence, two are enough for it, it is a geometric and arithmetic progression at the same time. 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; z0; z1; z2; z3; z4; z5 ... If we round the value of the Golden Ratio to three digits, 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 in a sequence 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 is 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 be 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...
The applied significance of the Fibonacci series and the Golden Ratio deserves a separate website. Now I’ll just say that, for example, the elements of the Fibonacci series are used to calculate moving averages (not to mention the growth of the rabbit population), and masterpieces of world art contain the Golden Ratio.
In the meantime, remember that Fibonacci is a legendary figure in mathematics, economics and finance; he promulgated Arabic numbers and introduced a magical series of numbers.
fibonacci number series
Ecology of life. Cognitive: Nature (including Man) develops according to the laws that are embedded in this numerical sequence...
Fibonacci numbers are a numerical sequence where each subsequent member of the series is equal to the sum of the two previous ones, 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,.. 4222970156496 25,.. 19581068021641812000,.. The complex and amazing properties of the Fibonacci series numbers were studied a wide variety of professional scientists and mathematics enthusiasts.
In 1997, several strange features of the series were described by researcher Vladimir Mikhailov, who was convinced that Nature (including Man) develops according to the laws that are embedded in this numerical sequence.
A remarkable property of the Fibonacci number series is that as the numbers of the series increase, the ratio of two neighboring members of this series asymptotically approaches the exact proportion of the Golden Ratio (1:1.618) - the basis of beauty and harmony in the nature around us, including in human relationships.
Note that Fibonacci himself opened his famous series while thinking about the problem of the number of rabbits that should be born from one pair within one year. It turned out that in each subsequent month after the second, the number of pairs of rabbits exactly follows the digital series that now bears his name. Therefore, it is no coincidence that man himself is structured according to the Fibonacci series. Each organ is arranged in accordance with internal or external duality.
Fibonacci numbers attracted mathematicians with their ability to appear in the most unexpected places. It has been noticed, for example, that the ratios of Fibonacci numbers, taken through one, correspond to the angle between adjacent leaves on a plant stem, more precisely, they say what fraction of a revolution this angle is: 1/2 - for elm and linden, 1/3 - for beech, 2/5 - for oak and apple trees, 3/8 - for poplar and roses, 5/13 - for willow and almonds, etc. You will find the same numbers when counting the seeds in the spirals of a sunflower, in the number of rays reflected from two mirrors, in the number of options for routes for a bee to crawl from one cell to another, in many mathematical games and tricks.
What is the difference between the golden ratio spirals and the Fibonacci spiral? The golden ratio spiral is ideal. It corresponds to the Primary Source of harmony. This spiral has neither beginning nor end. It is endless. The Fibonacci spiral has a beginning from which it begins to “unwind”. This is a very important property. It allows Nature, after the next closed cycle, to build a new spiral from scratch.
It should be said that the Fibonacci spiral can be double. There are numerous examples of these double helices found throughout the world. Thus, sunflower spirals always correlate with the Fibonacci series. Even in an ordinary pine cone you can see this Fibonacci double spiral. The first spiral goes in one direction, the second in the other. If you count the number of scales in a spiral rotating in one direction and the number of scales in another spiral, you can see that these are always two consecutive numbers of the Fibonacci series. The number of these spirals is 8 and 13. In sunflowers there are pairs of spirals: 13 and 21, 21 and 34, 34 and 55, 55 and 89. And there are no deviations from these pairs!..
In humans, 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...
But why does this particular series play a decisive role in Nature? This question can be answered comprehensively by the concept of trinity, which determines the conditions for its self-preservation. If the “balance of interests” of the triad is violated by one of its “partners,” the “opinions” of the other two “partners” must be adjusted. The concept of trinity is especially evident in physics, where “almost” all elementary particles are built from quarks. If we remember that the ratios of the fractional charges of quark particles form a series, and these are the first terms of the Fibonacci series, which are necessary for the formation of other elementary particles.
It is possible that the Fibonacci spiral can play a decisive role in the formation of the pattern of limited and closed hierarchical spaces. Indeed, let’s imagine that at some stage of evolution the Fibonacci spiral reached perfection (it became indistinguishable from the golden ratio spiral) and for this reason the particle should be transformed into the next “category”.
These facts once again confirm that the law of duality gives not only qualitative, but also quantitative results. They make us think that the Macroworld and Microworld around us evolve according to the same laws - the laws of hierarchy, and that these laws are the same for living and inanimate matter.
All this indicates that the Fibonacci number series represents a certain encrypted law of nature.
The digital code of the development of civilization can be determined using various methods in numerology. For example, by reducing complex numbers to single digits (for example, 15 is 1+5=6, etc.). Carrying out a similar addition procedure with all the complex numbers of the Fibonacci series, Mikhailov received the following series of these numbers: 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 8, 1, 9, then everything repeats 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 4, 8, 8,.. and repeats again and again... This series also has the properties of the Fibonacci series, each infinitely subsequent term is equal to the sum of the previous ones. For example, the sum of the 13th and 14th terms is 15, i.e. 8 and 8=16, 16=1+6=7. It turns out that this series is periodic, with a period of 24 terms, after which the entire order of numbers is repeated. Having received this period, Mikhailov put forward an interesting assumption - Isn't a set of 24 digits a kind of digital code for the development of civilization? published
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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, patterns described by Fibonacci ratios have been found in nature, architecture, fine arts, mathematics, physics, astronomy, biology and many other fields. 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 is no exaggeration to say that this is not just a game with numbers, 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 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 of the environment. The laws of golden symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and cosmic systems, in the gene structures of living organisms. 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 at the beginning of the 19th century.
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 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 pyramid at Giza. Unlike other 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.
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THE HIGHEST PURPOSE OF MATHEMATICS IS TO FIND THE HIDDEN ORDER IN THE CHAOS THAT SURROUND US.
Viner N.
A person strives for knowledge all his life, trying to study the world around him. And in the process of observation, questions arise that require answers. The answers are found, but new questions arise. In archaeological finds, in traces of civilization, distant from each other in time and space, one and the same element is found - a pattern in the form of a spiral. Some consider it a symbol of the sun and associate it with the legendary Atlantis, but its true meaning is unknown. What do the shapes of a galaxy and an atmospheric cyclone, the arrangement of leaves on a stem, and the arrangement of seeds in a sunflower have in common? These patterns come down to the so-called “golden” spiral, the amazing Fibonacci sequence discovered by the great Italian mathematician of the 13th century.
History of Fibonacci numbers
For the first time I heard about what Fibonacci numbers are from a mathematics teacher. But, besides, I didn’t know how the sequence of these numbers came together. This is what this sequence is actually famous for, how it affects a person, I want to tell you. Little is known about Leonardo Fibonacci. There is not even an exact date of his birth. It is known that he was born in 1170 into a merchant family in the city of Pisa in Italy. Fibonacci's father often visited Algeria on trade matters, and Leonardo studied mathematics there with Arab teachers. Subsequently, he wrote several mathematical works, the most famous of which is the “Book of Abacus,” which contains almost all the arithmetic and algebraic information of that time. 2
Fibonacci numbers are a sequence of numbers that have a number of properties. Fibonacci discovered this number sequence by accident when he was trying to solve a practical problem about rabbits in 1202. “Someone placed a pair of rabbits in a certain place, fenced on all sides by a wall, in order to find out how many pairs of rabbits would be born during the year, if the nature of rabbits is such that after a month a pair of rabbits gives birth to another pair, and rabbits give birth from the second months after your birth." When solving the problem, he took into account that each pair of rabbits gives birth to two more pairs throughout their lives, and then dies. This is how the sequence of numbers appeared: 1, 1, 2, 3, 5, 8, 13, 21, ... In this sequence, each next number is equal to the sum of the two previous ones. It was called the Fibonacci sequence. Mathematical properties of the sequence
I wanted to explore this sequence, and I discovered some of its properties. This pattern is of great importance. The sequence is slowly approaching a certain constant ratio of approximately 1.618, and the ratio of any number to the next one is approximately 0.618.
You can notice a number of interesting properties of Fibonacci numbers: two neighboring numbers are relatively prime; every third number is even; every fifteenth ends in zero; every fourth is a multiple of three. If you choose any 10 adjacent numbers from the Fibonacci sequence and add them together, you will always get a number that is a multiple of 11. But that's not all. Each sum is equal to the number 11 multiplied by the seventh term of the given sequence. Here's another interesting feature. For any n, the sum of the firstn terms of the sequence will always be equal to the difference between the (n+ 2)th and first terms of the sequence. This fact can be expressed by the formula: 1+1+2+3+5+…+an=a n+2 - 1. Now we have the following trick at our disposal: to find the sum of all terms
sequence between two given terms, it is enough to find the difference of the corresponding (n+2)-x terms. For example, a 26 +…+a 40 = a 42 - a 27. Now let's look for the connection between Fibonacci, Pythagoras and the “golden ratio”. The most famous evidence of the mathematical genius of mankind is the Pythagorean theorem: in any right triangle, the square of the hypotenuse is equal to the sum of the squares of its legs: c 2 =b 2 +a 2. From a geometric point of view, we can consider all the sides of a right triangle as the sides of three squares constructed on them. The Pythagorean theorem states that the total area of squares built on the sides of a right triangle is equal to the area of the square built on the hypotenuse. If the lengths of the sides of a right triangle are integers, then they form a group of three numbers called Pythagorean triplets. Using the Fibonacci sequence you can find such triplets. Let's take any four consecutive numbers from the sequence, for example, 2, 3, 5 and 8, and construct three more numbers as follows: 1) the product of the two extreme numbers: 2*8=16; 2) the double product of the two numbers in the middle: 2* (3*5)=30;3) the sum of the squares of two average numbers: 3 2 +5 2 =34; 34 2 =30 2 +16 2. This method works for any four consecutive Fibonacci numbers. Any three consecutive numbers in the Fibonacci series behave in a predictable way. If you multiply the two extreme ones and compare the result with the square of the average number, the result will always differ by one. For example, for the numbers 5, 8 and 13 we get: 5*13=8 2 +1. If you look at this property from a geometric point of view, you will notice something strange. Divide the square
8x8 in size (64 small squares in total) into four parts, the lengths of the sides being equal to the Fibonacci numbers. Now from these parts we will build a rectangle measuring 5x13. Its area is 65 small squares. Where does the extra square come from? The thing is that an ideal rectangle is not formed, but tiny gaps remain, which in total give this additional unit of area. Pascal's triangle also has a connection with the Fibonacci sequence. You just need to write the lines of Pascal's triangle one under the other, and then add the elements diagonally. The result is the Fibonacci sequence.
Now consider a golden rectangle, one side of which is 1.618 times longer than the other. At first glance, it may seem like an ordinary rectangle to us. However, let's do a simple experiment with two ordinary bank cards. Let's place one of them horizontally and the other vertically so that their lower sides are on the same line. If we draw a diagonal line in a horizontal map and extend it, we will see that it will pass exactly through the upper right corner of the vertical map - a pleasant surprise. Maybe this is an accident, or maybe these rectangles and other geometric shapes that use the “golden ratio” are especially pleasing to the eye. Did Leonardo da Vinci think about the golden ratio while working on his masterpiece? This seems unlikely. However, it can be argued that he attached great importance to the connection between aesthetics and mathematics.
Fibonacci numbers in nature
The connection of the golden ratio with beauty is not only a matter of human perception. It seems that nature itself has allocated a special role to F. If you inscribe squares sequentially into a “golden” rectangle, then draw an arc in each square, you will get an elegant curve called a logarithmic spiral. It is not a mathematical curiosity at all. 5
On the contrary, this remarkable line is often found in the physical world: from the shell of a nautilus to the arms of galaxies, and in the elegant spiral of petals of a blooming rose. The connections between the golden ratio and Fibonacci numbers are numerous and surprising. Let's consider a flower that looks very different from a rose - a sunflower with seeds. The first thing we see is that the seeds are arranged in two types of spirals: clockwise and counterclockwise. If we count the clockwise spirals, we get two seemingly ordinary numbers: 21 and 34. This is not the only example where Fibonacci numbers can be found in the structure of plants.
Nature gives us numerous examples of the arrangement of homogeneous objects described by Fibonacci numbers. In the various spiral arrangements of small plant parts, two families of spirals can usually be discerned. In one of these families the spirals curl clockwise, while in the other they curl counterclockwise. The numbers of spirals of one and another type often turn out to be adjacent Fibonacci numbers. So, taking a young pine twig, it is easy to notice that the needles form two spirals, going from bottom left to top right. On many cones, the seeds are arranged in three spirals, gently winding around the stem of the cone. They are located in five spirals, winding steeply in the opposite direction. In large cones it is possible to observe 5 and 8, and even 8 and 13 spirals. Fibonacci spirals are also clearly visible on a pineapple: there are usually 8 and 13 of them.
The chicory shoot makes a strong ejection into space, stops, releases a leaf, but this time is shorter than the first one, again makes an ejection into space, but with less force, releases a leaf of an even smaller size and is ejected again. The impulses of its growth gradually decrease in proportion to the “golden” section. To appreciate the enormous role of Fibonacci numbers, you just need to look at the beauty of the nature around us. Fibonacci numbers can be found in quantities
branches on the stem of each growing plant and in the number of petals.
Let's count the petals of some flowers - iris with its 3 petals, primrose with 5 petals, ragweed with 13 petals, cornflower with 34 petals, aster with 55 petals, etc. Is this a coincidence, or is it a law of nature? Look at the stems and flowers of yarrow. Thus, the total Fibonacci sequence can easily interpret the pattern of manifestations of “Golden” numbers found in nature. These laws operate regardless of our consciousness and desire to accept them or not. The patterns of “golden” symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and cosmic systems, in the gene structures of living organisms, in the structure of individual human organs and the body as a whole, and also manifest themselves in the biorhythms and functioning of the brain and visual perception.
Fibonacci numbers in architecture
The “Golden Ratio” is also evident in many remarkable architectural creations throughout human history. It turns out that ancient Greek and ancient Egyptian mathematicians knew these coefficients long before Fibonacci and called them the “golden ratio”. The Greeks used the principle of the “golden ratio” in the construction of the Parthenon, and the Egyptians used the Great Pyramid of Giza. Advances in construction technology and the development of new materials opened up new opportunities for twentieth-century architects. American Frank Lloyd Wright was one of the main proponents of organic architecture. Shortly before his death, he designed the Solomon Guggenheim Museum in New York, which is an inverted spiral, and the interior of the museum resembles a nautilus shell. Polish-Israeli architect Zvi Hecker also used spiral structures in his design for the Heinz Galinski School in Berlin, completed in 1995. Hecker started with the idea of a sunflower with a central circle, from where
All architectural elements are diverging. The building is a combination
orthogonal and concentric spirals, symbolizing the interaction of limited human knowledge and the controlled chaos of nature. Its architecture imitates a plant that follows the movement of the Sun, so classrooms are illuminated throughout the day.
In Quincy Park, located in Cambridge, Massachusetts (USA), the “golden” spiral can often be found. The park was designed in 1997 by artist David Phillips and is located near the Clay Mathematical Institute. This institution is a renowned center for mathematical research. In Quincy Park, you can stroll among “golden” spirals and metal curves, reliefs of two shells and a rock with a square root symbol. The sign contains information about the “golden” ratio. Even bicycle parking uses the F symbol.
Fibonacci numbers in psychology
In psychology, turning points, crises, and revolutions have been noted that mark transformations in the structure and functions of the soul in a person’s life path. If a person successfully overcomes these crises, then he becomes capable of solving problems of a new class that he had not even thought about before.
The presence of fundamental changes gives reason to consider life time as a decisive factor in the development of spiritual qualities. After all, nature does not measure out time generously for us, “no matter how much it will be, so much will be,” but just enough for the development process to materialize:
in body structures;
in feelings, thinking and psychomotor skills - until they acquire harmony necessary for the emergence and launch of the mechanism
creativity;
in the structure of human energy potential.
The development of the body cannot be stopped: the child becomes an adult. With the mechanism of creativity, everything is not so simple. Its development can be stopped and its direction changed.
Is there a chance to catch up with time? Undoubtedly. But for this you need to do a lot of work on yourself. What develops freely, naturally, does not require special efforts: the child develops freely and does not notice this enormous work, because the process of free development is created without violence against oneself.
How is the meaning of life's journey understood in everyday consciousness? The average person sees it this way: at the bottom there is birth, at the top there is the prime of life, and then everything goes downhill.
The sage will say: everything is much more complicated. He divides the ascent into stages: childhood, adolescence, youth... Why is this so? Few are able to answer, although everyone is sure that these are closed, integral stages of life.
To find out how the mechanism of creativity develops, V.V. Klimenko used mathematics, namely the laws of Fibonacci numbers and the proportion of the “golden section” - the laws of nature and human life.
Fibonacci numbers divide our lives into stages according to the number of years lived: 0 - the beginning of the countdown - the child is born. He still lacks not only psychomotor skills, thinking, feelings, imagination, but also operational energy potential. He is the beginning of a new life, a new harmony;
1 - the child has mastered walking and is mastering his immediate environment;
2 - understands speech and acts using verbal instructions;
3 - acts through words, asks questions;
5 - “age of grace” - harmony of psychomotor, memory, imagination and feelings, which already allow the child to embrace the world in all its integrity;
8 - feelings come to the fore. They are served by imagination, and thinking, through its criticality, is aimed at supporting the internal and external harmony of life;
13 - the mechanism of talent begins to work, aimed at transforming the material acquired in the process of inheritance, developing one’s own talent;
21 - the mechanism of creativity has approached a state of harmony and attempts are being made to perform talented work;
34—harmony of thinking, feelings, imagination and psychomotor skills: the ability to work ingeniously is born;
55 - at this age, provided the harmony of soul and body is preserved, a person is ready to become a creator. And so on…
What are the Fibonacci Numbers serifs? They can be compared to dams along the path of life. These dams await each of us. First of all, you need to overcome each of them, and then patiently raise your level of development until one fine day it falls apart, opening the way to the next one for free flow.
Now that we understand the meaning of these key points of age-related development, let’s try to decipher how it all happens.
B1 year the child masters walking. Before this, he experienced the world with the front of his head. Now he gets to know the world with his hands—an exceptional human privilege. The animal moves in space, and he, by learning, masters the space and masters the territory in which he lives.
2 years- understands the word and acts in accordance with it. It means that:
the child learns a minimum number of words - meanings and modes of action;
has not yet separated itself from the environment and is fused into integrity with the environment,
therefore he acts according to someone else's instructions. At this age he is the most obedient and pleasant to his parents. From a sensual person, a child turns into a cognitive person.
3 years- action using one's own word. The separation of this person from the environment has already occurred - and he learns to be an independently acting person. From here he:
consciously opposes the environment and parents, kindergarten teachers, etc.;
realizes its sovereignty and fights for independence;
tries to subjugate close and well-known people to his will.
Now for a child, a word is an action. This is where the active person begins.
5 years- “age of grace.” He is the personification of harmony. Games, dancing, deft movements - everything is saturated with harmony, which a person tries to master with his own strength. Harmonious psychomotor behavior helps bring about a new state. Therefore, the child is focused on psychomotor activity and strives for the most active actions.
Materialization of the products of sensitivity work is carried out through:
the ability to display the environment and ourselves as part of this world (we hear, see, touch, smell, etc. - all senses work for this process);
ability to design the external world, including oneself
(creation of second nature, hypotheses - do this and that tomorrow, build a new machine, solve a problem), by the forces of critical thinking, feelings and imagination;
the ability to create a second, man-made nature, products of activity (realization of plans, specific mental or psychomotor actions with specific objects and processes).
After 5 years, the imagination mechanism comes forward and begins to dominate the others. The child does a tremendous amount of work, creating fantastic images, and lives in the world of fairy tales and myths. The hypertrophied imagination of a child causes surprise in adults, because the imagination does not correspond to reality.
8 years— feelings come to the fore and one’s own standards of feelings (cognitive, moral, aesthetic) arise when the child unmistakably:
evaluates the known and the unknown;
distinguishes moral from immoral, moral from immoral;
beauty from what threatens life, harmony from chaos.
13 years— the mechanism of creativity begins to work. But this does not mean that it is working at full capacity. One of the elements of the mechanism comes to the fore, and all the others contribute to its work. If in this age period of development harmony is maintained, which almost constantly rebuilds its structure, then the youth will painlessly reach the next dam, unnoticed by himself will overcome it and will live at the age of a revolutionary. At the age of a revolutionary, a youth must take a new step forward: separate from the nearest society and live a harmonious life and activity in it. Not everyone can solve this problem that arises before each of us.
21 years old. If a revolutionary has successfully overcome the first harmonious peak of life, then his mechanism of talent is capable of performing talented
work. Feelings (cognitive, moral or aesthetic) sometimes overshadow thinking, but in general all elements work harmoniously: feelings are open to the world, and logical thinking is able to name and find measures of things from this peak.
The mechanism of creativity, developing normally, reaches a state that allows it to receive certain fruits. He starts working. At this age, the mechanism of feelings comes forward. As the imagination and its products are evaluated by the senses and the mind, antagonism arises between them. Feelings win. This ability gradually gains power, and the boy begins to use it.
34 years- balance and harmony, productive effectiveness of talent. The harmony of thinking, feelings and imagination, psychomotor skills, which are replenished with optimal energy potential, and the mechanism as a whole - the opportunity to perform brilliant work is born.
55 years- a person can become a creator. The third harmonious peak of life: thinking subjugates the power of feelings.
Fibonacci numbers refer to the stages of human development. Whether a person will go through this path without stopping depends on parents and teachers, the educational system, and then - on himself and on how a person will learn and overcome himself.
On the path of life, a person discovers 7 relationship objects:
From birthday to 2 years - discovery of the physical and objective world of the immediate environment.
From 2 to 3 years - self-discovery: “I am Myself.”
From 3 to 5 years - speech, the active world of words, harmony and the “I - You” system.
From 5 to 8 years - discovery of the world of other people's thoughts, feelings and images - the “I - We” system.
From 8 to 13 years - discovery of the world of tasks and problems solved by the geniuses and talents of humanity - the “I - Spirituality” system.
From 13 to 21 years - the discovery of the ability to independently solve well-known problems, when thoughts, feelings and imagination begin to work actively, the “I - Noosphere” system arises.
From 21 to 34 years old - discovery of the ability to create a new world or its fragments - awareness of the self-concept “I am the Creator”.
The life path has a spatiotemporal structure. It consists of age and individual phases, determined by many life parameters. A person masters, to a certain extent, the circumstances of his life, becomes the creator of his history and the creator of the history of society. A truly creative attitude to life, however, does not appear immediately and not even in every person. There are genetic connections between the phases of the life path, and this determines its natural character. It follows that, in principle, it is possible to predict future development on the basis of knowledge about its early phases.
Fibonacci numbers in astronomy
From the history of astronomy it is known that I. Titius, a German astronomer of the 18th century, using the Fibonacci series, found a pattern and order in the distances between the planets of the solar system. But one case seemed to contradict the law: there was no planet between Mars and Jupiter. But after the death of Titius at the beginning of the 19th century. concentrated observation of this part of the sky led to the discovery of the asteroid belt.
Conclusion
During the research, I found out that Fibonacci numbers are widely used in the technical analysis of stock prices. One of the simplest ways to use Fibonacci numbers in practice is to determine the time intervals after which a particular event will occur, for example, a price change. The analyst counts a certain number of Fibonacci days or weeks (13,21,34,55, etc.) from the previous similar event and makes a forecast. But this is still too difficult for me to figure out. Although Fibonacci was the greatest mathematician of the Middle Ages, the only monuments to Fibonacci are a statue in front of the Leaning Tower of Pisa and two streets that bear his name: one in Pisa and the other in Florence. And yet, in connection with everything I have seen and read, quite natural questions arise. Where did these numbers come from? Who is this architect of the universe who tried to make it ideal? What will be next? Having found the answer to one question, you will get the next one. If you solve it, you'll get two new ones. Once you deal with them, three more will appear. Having solved them too, you will have five unsolved ones. Then eight, thirteen, etc. Do not forget that two hands have five fingers, two of which consist of two phalanges, and eight of three.
Literature:
Voloshinov A.V. “Mathematics and Art”, M., Education, 1992.
Vorobyov N.N. “Fibonacci Numbers”, M., Nauka, 1984.
Stakhov A.P. “The Da Vinci Code and the Fibonacci Series”, St. Petersburg format, 2006
F. Corvalan “The Golden Ratio. Mathematical language of beauty", M., De Agostini, 2014.
Maksimenko S.D. "Sensitive periods of life and their codes."
"Fibonacci numbers". Wikipedia
Kanalieva Dana
In this work, we studied and analyzed the manifestation of the Fibonacci sequence numbers in the reality around us. We discovered an amazing mathematical relationship between the number of spirals in plants, the number of branches in any horizontal plane, and the Fibonacci sequence numbers. We also saw strict mathematics in the human structure. The human DNA molecule, in which the entire development program of a human being is encrypted, the respiratory system, the structure of the ear - everything obeys certain numerical relationships.
We are convinced that Nature has its own laws, expressed using mathematics.
And mathematics is very important tool of cognition secrets of Nature.
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MBOU "Pervomaiskaya Secondary School"
Orenburg district, Orenburg region
RESEARCH
"The Mystery of Numbers"
Fibonacci"
Completed by: Kanalieva Dana
6th grade student
Scientific adviser:
Gazizova Valeria Valerievna
Mathematics teacher of the highest category
n. Experimental
2012
Explanatory note………………………………………………………………………………........ 3.
Introduction. History of Fibonacci numbers.……………………………………………………...... 4.
Chapter 1. Fibonacci numbers in living nature.........……. …………………………………... 5.
Chapter 2. Fibonacci Spiral.................................................... ..........……………..... 9.
Chapter 3. Fibonacci numbers in human inventions.........…………………………….. 13
Chapter 4. Our research……………………………………………………………....... 16.
Chapter 5. Conclusion, conclusions………………………………………………………………………………...... 19.
List of used literature and Internet sites…………………………………........21.
Object of study:
Man, mathematical abstractions created by man, human inventions, the surrounding flora and fauna.
Subject of study:
form and structure of the objects and phenomena being studied.
Purpose of the study:
study the manifestation of Fibonacci numbers and the associated law of the golden ratio in the structure of living and non-living objects,
find examples of using Fibonacci numbers.
Job objectives:
Describe a method for constructing the Fibonacci series and Fibonacci spiral.
See mathematical patterns in the structure of humans, flora and inanimate nature from the point of view of the Golden Ratio phenomenon.
Novelty of the research:
Discovery of Fibonacci numbers in the reality around us.
Practical significance:
Using acquired knowledge and research skills when studying other school subjects.
Skills and abilities:
Organization and conduct of the experiment.
Use of specialized literature.
Acquiring the ability to review collected material (report, presentation)
Design of work with drawings, diagrams, photographs.
Active participation in discussions of your work.
Research methods:
empirical (observation, experiment, measurement).
theoretical (logical stage of cognition).
Explanatory note.
“Numbers rule the world! Number is the power that reigns over gods and mortals!” - this is what the ancient Pythagoreans said. Is this basis of Pythagoras’ teaching still relevant today? When studying the science of numbers at school, we want to make sure that, indeed, the phenomena of the entire Universe are subject to certain numerical relationships, to find this invisible connection between mathematics and life!
Is it really in every flower,
Both in the molecule and in the galaxy,
Numerical patterns
This strict “dry” mathematics?
We turned to a modern source of information - the Internet and read about Fibonacci numbers, about magic numbers that are fraught with a great mystery. It turns out that these numbers can be found in sunflowers and pine cones, in dragonfly wings and starfish, in the rhythms of the human heart and in musical rhythms...
Why is this sequence of numbers so common in our world?
We wanted to know about the secrets of Fibonacci numbers. This research work was the result of our activities.
Hypothesis:
in the reality around us, everything is built according to amazingly harmonious laws with mathematical precision.
Everything in the world is thought out and calculated by our most important designer - Nature!
Introduction. History of the Fibonacci series.
Amazing numbers were discovered by the Italian medieval mathematician Leonardo of Pisa, better known as Fibonacci. Traveling around the East, he became acquainted with the achievements of Arab mathematics and contributed to their transfer to the West. In one of his works, entitled “The Book of Calculations,” he introduced Europe to one of the greatest discoveries of all time - the decimal number system.
One day, he was racking his brains over solving a mathematical problem. He was trying to create a formula to describe the breeding sequence of rabbits.
The solution was a number series, each subsequent number of which is the sum of the two previous ones:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, ...
The numbers that form this sequence are called “Fibonacci numbers”, and the sequence itself is called the Fibonacci sequence.
"So what?" - you say, “Can we really come up with similar number series ourselves, increasing according to a given progression?” Indeed, when the Fibonacci series appeared, no one, including himself, had any idea how close he managed to come to solving one of the greatest mysteries of the universe!
Fibonacci led a reclusive lifestyle, spent a lot of time in nature, and while walking in the forest, he noticed that these numbers began to literally haunt him. Everywhere in nature he encountered these numbers again and again. For example, the petals and leaves of plants strictly fit into a given number series.
There is an interesting feature in Fibonacci numbers: the quotient of dividing the next Fibonacci number by the previous one, as the numbers themselves grow, tends to 1.618. It was this constant division number that was called the Divine proportion in the Middle Ages, and is now referred to as the golden section or golden proportion.
In algebra, this number is denoted by the Greek letter phi (Ф)
So, φ = 1.618
233 / 144 = 1,618
377 / 233 = 1,618
610 / 377 = 1,618
987 / 610 = 1,618
1597 / 987 = 1,618
2584 / 1597 = 1,618
No matter how many times we divide one by another, the number adjacent to it, we will always get 1.618. And if we do the opposite, that is, divide the smaller number by the larger one, we will get 0.618, this is the inverse of 1.618. also called the golden ratio.
The Fibonacci series could have remained only a mathematical incident, if not for the fact that all researchers of the golden division in the plant and animal world, not to mention art, invariably came to this series as an arithmetic expression of the law of the golden division.
Scientists, analyzing the further application of this number series to natural phenomena and processes, discovered that these numbers are contained in literally all objects of living nature, in plants, animals and humans.
The amazing mathematical toy turned out to be a unique code embedded in all natural objects by the Creator of the Universe himself.
Let's look at examples where Fibonacci numbers occur in living and inanimate nature.
Fibonacci numbers in living nature.
If you look at the plants and trees around us, you can see how many leaves there are on each of them. From a distance, it seems that the branches and leaves on the plants are located randomly, in no particular order. However, in all plants, in a miraculous, mathematically precise way, which branch will grow from where, how the branches and leaves will be located near the stem or trunk. From the first day of its appearance, the plant exactly follows these laws in its development, that is, not a single leaf, not a single flower appears by chance. Even before its appearance, the plant is already precisely programmed. How many branches will there be on the future tree, where will the branches grow, how many leaves will there be on each branch, and how and in what order the leaves will be arranged. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that the Fibonacci series manifests itself in the arrangement of leaves on a branch (phylotaxis), in the number of revolutions on the stem, in the number of leaves in a cycle, and therefore, the law of the golden ratio also manifests itself.
If you set out to find numerical patterns in living nature, you will notice that these numbers are often found in various spiral forms, which are so rich in the plant world. For example, leaf cuttings are adjacent to the stem in a spiral that runs betweentwo adjacent leaves:full rotation - at the hazel tree,- by the oak tree, - at the poplar and pear trees,- at the willow.
The seeds of sunflower, Echinacea purpurea and many other plants are arranged in spirals, and the number of spirals in each direction is the Fibonacci number.
Sunflower, 21 and 34 spirals. Echinacea, 34 and 55 spirals.
The clear, symmetrical shape of flowers is also subject to a strict law.
For many flowers, the number of petals is precisely the numbers from the Fibonacci series. For example:
iris, 3p. buttercup, 5 lep. golden flower, 8 lep. delphinium,
13 lep.
chicory, 21lep. aster, 34 lep. daisies, 55 lep.
The Fibonacci series characterizes the structural organization of many living systems.
We have already said that the ratio of neighboring numbers in the Fibonacci series is the number φ = 1.618. It turns out that man himself is simply a storehouse of phi numbers.
The proportions of the various parts of our body are a number very close to the golden ratio. If these proportions coincide with the golden ratio formula, then the person’s appearance or body is considered ideally proportioned. The principle of calculating the gold measure on the human body can be depicted in the form of a diagram.
M/m=1.618
The first example of the golden ratio in the structure of the human body:
If we take the navel point as the center of the human body, and the distance between a person’s foot and the navel point as a unit of measurement, then a person’s height is equivalent to the number 1.618.
Human hand
It is enough just to bring your palm closer to you and look carefully at your index finger, and you will immediately find the formula of the golden ratio in it. Each finger of our hand consists of three phalanges.
The sum of the first two phalanges of the finger in relation to the entire length of the finger gives the number of the golden ratio (with the exception of the thumb).
In addition, the ratio between the middle finger and little finger is also equal to the golden ratio.
A person has 2 hands, the fingers on each hand consist of 3 phalanges (except for the thumb). There are 5 fingers on each hand, that is, 10 in total, but with the exception of two two-phalanx thumbs, only 8 fingers are created according to the principle of the golden ratio. Whereas all these numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence.
The golden ratio in the structure of the human lungs
American physicist B.D. West and Dr. A.L. Goldberger, during physical and anatomical studies, established that the golden ratio also exists in the structure of the human lungs.
The peculiarity of the bronchi that make up the human lungs lies in their asymmetry. The bronchi consist of two main airways, one of which (the left) is longer and the other (the right) is shorter.
It was found that this asymmetry continues in the branches of the bronchi, in all the smaller respiratory tracts. Moreover, the ratio of the lengths of short and long bronchi is also the golden ratio and is equal to 1:1.618.
Artists, scientists, fashion designers, designers make their calculations, drawings or sketches based on the ratio of the golden ratio. They use measurements from the human body, which was also created according to the principle of the golden ratio. Before creating their masterpieces, Leonardo Da Vinci and Le Corbusier took the parameters of the human body, created according to the law of the Golden Proportion.
There is another, more prosaic application of the proportions of the human body. For example, using these relationships, crime analysts and archaeologists use fragments of parts of the human body to reconstruct the appearance of the whole.
Golden proportions in the structure of the DNA molecule.
All information about the physiological characteristics of living beings, be it a plant, an animal or a person, is stored in a microscopic DNA molecule, the structure of which also contains the law of the golden proportion. The DNA molecule consists of two vertically intertwined helices. The length of each of these spirals is 34 angstroms and the width is 21 angstroms. (1 angstrom is one hundred millionth of a centimeter).
So, 21 and 34 are numbers following each other in the sequence of Fibonacci numbers, that is, the ratio of the length and width of the logarithmic spiral of the DNA molecule carries the formula of the golden ratio 1:1.618.
Not only erect walkers, but also all swimming, crawling, flying and jumping creatures did not escape the fate of being subject to the number phi. The human heart muscle contracts to 0.618 of its volume. The structure of a snail shell corresponds to the Fibonacci proportions. And such examples can be found in abundance - if there was a desire to explore natural objects and processes. The world is so permeated with Fibonacci numbers that sometimes it seems that the Universe can only be explained by them.
Fibonacci spiral.
There is no other form in mathematics that has the same unique properties as the spiral, because
The structure of the spiral is based on the Golden Ratio rule!
To understand the mathematical construction of a spiral, let us repeat what the Golden Ratio is.
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 related to the larger one as the larger one is to the whole.
That is (a+b) /a = a / b
A rectangle with exactly this aspect ratio came to be called the golden rectangle. Its long sides are in relation to its short sides in a ratio of 1.168:1.
The golden rectangle has many unusual properties. Cutting a square from a golden rectangle whose side is equal to the smaller side of the rectangle,
we will again get a smaller golden rectangle.
This process can be continued indefinitely. As we continue to cut off squares, we will end up with smaller and smaller golden rectangles. Moreover, they will be located in a logarithmic spiral, which is important in mathematical models of natural objects.
For example, the spiral shape can be seen in the arrangement of sunflower seeds, in pineapples, cacti, the structure of rose petals, and so on.
We are surprised and delighted by the spiral structure of shells.
In most snails that have shells, the shell grows in a spiral shape. However, there is no doubt that these unreasonable creatures not only have no idea about the spiral, but do not even have the simplest mathematical knowledge to create a spiral-shaped shell for themselves.
But then how were these unreasonable creatures able to determine and choose for themselves the ideal form of growth and existence in the form of a spiral shell? Could these living creatures, which the scientific world calls primitive life forms, calculate that the spiral shape of a shell would be ideal for their existence?
Trying to explain the origin of such even the most primitive form of life by a random combination of certain natural circumstances is absurd, to say the least. It is clear that this project is a conscious creation.
Spirals also exist in humans. With the help of spirals we hear:
Also, in the human inner ear there is an organ called Cochlea (“Snail”), which performs the function of transmitting sound vibration. This bony structure is filled with fluid and created in the shape of a snail with golden proportions.
There are spirals on our palms and fingers:
In the animal kingdom we can also find many examples of spirals.
The horns and tusks of animals develop in a spiral shape; the claws of lions and the beaks of parrots are logarithmic shapes and resemble the shape of an axis that tends to turn into a spiral.
It’s interesting that a hurricane and a cyclone’s clouds are twisting like a spiral, and this is clearly visible from space:
In ocean and sea waves, the spiral can be mathematically represented on a graph with points 1,1,2,3,5,8,13,21,34 and 55.
Everyone will also recognize such an “everyday” and “prosaic” spiral.
After all, the water escapes from the bathroom in a spiral:
Yes, and we live in a spiral, because the galaxy is a spiral corresponding to the formula of the Golden Ratio!
So, we found out that if we take the Golden Rectangle and break it into smaller rectanglesin the exact Fibonacci sequence, and then divide each of them in such proportions again and again, you get a system called the Fibonacci spiral.
We discovered this spiral in the most unexpected objects and phenomena. Now it’s clear why the spiral is also called the “curve of life.”
The spiral has become a symbol of evolution, because everything develops in a spiral.
Fibonacci numbers in human inventions.
Having observed a law in nature expressed by the sequence of Fibonacci numbers, scientists and artists try to imitate it and embody this law in their creations.
The phi proportion allows you to create masterpieces of painting and correctly fit architectural structures into space.
Not only scientists, but also architects, designers and artists are amazed by this perfect spiral of the nautilus shell,
occupying the least space and providing the least heat loss. American and Thai architects, inspired by the example of the “chambered nautilus” in the matter of placing the maximum in the minimum space, are busy developing corresponding projects.
Since time immemorial, the Golden Ratio proportion has been considered the highest proportion of perfection, harmony and even divinity. The golden ratio can be found in sculptures and even in music. An example is the musical works of Mozart. Even stock exchange rates and the Hebrew alphabet contain a golden ratio.
But we want to focus on a unique example of creating an efficient solar installation. An American schoolboy from New York, Aidan Dwyer, put together his knowledge of trees and discovered that the efficiency of solar power plants can be increased by using mathematics. While on a winter walk, Dwyer wondered why trees needed such a “pattern” of branches and leaves. He knew that branches on trees are arranged according to the Fibonacci sequence, and leaves carry out photosynthesis.
At some point, the smart boy decided to check whether this position of the branches helps to collect more sunlight. Aidan built a pilot plant in his backyard using small solar panels instead of leaves and tested it in action. It turned out that compared to a conventional flat solar panel, its “tree” collects 20% more energy and operates efficiently for 2.5 hours longer.
Dwyer solar tree model and graphs made by a student.
“This installation also takes up less space than a flat panel, collects 50% more sun in winter even where it does not face south, and it does not accumulate as much snow. In addition, a tree-shaped design is much more suitable for the urban landscape,” notes the young inventor.
Aidan was recognized one of the best young naturalists of 2011. The 2011 Young Naturalist competition was hosted by the New York Museum of Natural History. Aidan has filed a provisional patent application for his invention.
Scientists continue to actively develop the theory of Fibonacci numbers and the golden ratio.
Yu. Matiyasevich solves Hilbert's 10th problem using Fibonacci numbers.
Elegant methods are emerging for solving a number of cybernetic problems (search theory, games, programming) using Fibonacci numbers and the golden ratio.
In the USA, even the Mathematical Fibonacci Association is being created, which has been publishing a special journal since 1963.
So, we see that the scope of the Fibonacci sequence of numbers is very multifaceted:
Observing the phenomena occurring in nature, scientists have made striking conclusions that the entire sequence of events occurring in life, revolutions, crashes, bankruptcies, periods of prosperity, laws and waves of development in the stock and foreign exchange markets, cycles of family life, and so on , are organized on a time scale in the form of cycles and waves. These cycles and waves are also distributed according to the Fibonacci number series!
Based on this knowledge, a person will learn to predict and manage various events in the future.
4. Our research.
We continued our observations and studied the structure
pine cone
yarrow
mosquito
person
And we became convinced that in these objects, so different at first glance, the same numbers of the Fibonacci sequence were invisibly present.
So, step 1.
Let's take a pine cone:
Let's take a closer look at it:
We notice two series of Fibonacci spirals: one - clockwise, the other - counterclockwise, their number 8 and 13.
Step 2.
Let's take yarrow:
Let's carefully consider the structure of the stems and flowers:
Note that each new branch of the yarrow grows from the axil, and new branches grow from the new branch. By adding up the old and new branches, we found the Fibonacci number in each horizontal plane.
Step 3.
Do Fibonacci numbers appear in the morphology of various organisms? Consider the well-known mosquito:
We see: 3 pairs of legs, head 5 antennae, the abdomen is divided into 8 segments.
Conclusion:
In our research, we saw that in the plants around us, living organisms and even in the human structure, numbers from the Fibonacci sequence manifest themselves, which reflects the harmony of their structure.
The pine cone, the yarrow, the mosquito, and the human being are arranged with mathematical precision.
We were looking for an answer to the question: how does the Fibonacci series manifest itself in the reality around us? But, answering it, we received more and more questions.
Where did these numbers come from? Who is this architect of the universe who tried to make it ideal? Is the spiral curling or unwinding?
How amazing it is for a person to experience this world!!!
Having found the answer to one question, he gets the next one. If he solves it, he gets two new ones. Once he deals with them, three more will appear. Having solved them too, he will have five unsolved ones. Then eight, then thirteen, 21, 34, 55...
Do you recognize?
Conclusion.
by the creator himself into all objects
A unique code is provided
And the one who is friendly with mathematics,
He will know and understand!
We have studied and analyzed the manifestation of the Fibonacci sequence numbers in the reality around us. We also learned that the patterns of this number series, including the patterns of “Golden” symmetry, are manifested in the energy transitions of elementary particles, in planetary and cosmic systems, in the gene structures of living organisms.
We discovered a surprising mathematical relationship between the number of spirals in plants, the number of branches in any horizontal plane, and the numbers in the Fibonacci sequence. We saw how the morphology of various organisms also obeys this mysterious law. We also saw strict mathematics in the human structure. The human DNA molecule, in which the entire development program of a human being is encrypted, the respiratory system, the structure of the ear - everything obeys certain numerical relationships.
We learned that pine cones, snail shells, ocean waves, animal horns, cyclone clouds and galaxies all form logarithmic spirals. Even the human finger, which is composed of three phalanges in the Golden Ratio relative to each other, takes on a spiral shape when squeezed.
An eternity of time and light years of space separate the pine cone and the spiral galaxy, but the structure remains the same: coefficient 1,618 ! Perhaps this is the primary law governing natural phenomena.
Thus, our hypothesis about the existence of special numerical patterns that are responsible for harmony is confirmed.
Indeed, everything in the world is thought out and calculated by our most important designer - Nature!
We are convinced that Nature has its own laws, expressed using mathematics. And mathematics is a very important tool
to learn the secrets of nature.
List of literature and Internet sites:
1.
Vorobyov N. N. Fibonacci numbers. - M., Nauka, 1984.
2. Ghika M. Aesthetics of proportions in nature and art. - M., 1936.
3. Dmitriev A. Chaos, fractals and information. // Science and Life, No. 5, 2001.
4. Kashnitsky S. E. Harmony woven from paradoxes // Culture and
Life. - 1982.- No. 10.
5. Malay G. Harmony - the identity of paradoxes // MN. - 1982.- No. 19.
6. Sokolov A. Secrets of the golden section // Youth technology. - 1978.- No. 5.
7. Stakhov A.P. Codes of the golden proportion. - M., 1984.
8. Urmantsev Yu. A. Symmetry of nature and the nature of symmetry. - M., 1974.
9. Urmantsev Yu. A. Golden section // Nature. - 1968.- No. 11.
10. Shevelev I.Sh., Marutaev M.A., Shmelev I.P. Golden Ratio/Three
A look at the nature of harmony.-M., 1990.
11. Shubnikov A. V., Koptsik V. A. Symmetry in science and art. -M.:
