If you look at the plants and trees around us, you can see how many leaves each of them has. From afar, it seems that the branches and leaves on the plants are arranged randomly, in an arbitrary order. However, in all plants it is miraculously, mathematically precisely planned which branch will grow from where, how 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 the appearance of the plant is already precisely programmed. How many branches will be on the future tree, where the branches will grow, how many leaves will be on each branch, and how, in what order the leaves will be arranged. Collaboration botanists and mathematicians shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch (phylotaxis), in the number of turns on the stem, in the number of leaves in the cycle, the Fibonacci series manifests itself, and therefore, the law of the golden section also manifests itself.
If you set out to find numerical patterns in wildlife, you will notice that these numbers are often found in various spiral forms, which the plant world is so rich in. For example, leaf cuttings adjoin the stem in a spiral that runs between two adjacent leaves: a full turn - in hazel, - in oak, - in poplar and pear, - in 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.
A clear, symmetrical form of flowers is also subject to a strict law.
Many flowers have the number of petals - exactly the numbers from the Fibonacci series. For example:
iris, 3 lep. buttercup, 5 lep. golden flower, 8 lep. delphinium,
chicory, 21 lep. 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 the man himself is just a storehouse of the number phi.
The proportions of the various parts of our body make up a number very close to the golden ratio. If these proportions coincide with the formula of the golden ratio, then the appearance or body of a person is considered to be ideally built. The principle of calculating the golden measure on the human body can be depicted in the form of a diagram.
M/m=1.618
The first example of the golden section in the structure of the human body:
If we take the navel point as the center of the human body, and the distance between the human foot and the navel point as a unit of measurement, then the height of a person is equivalent to the number 1.618.
Human hand
It is enough just to bring your palm closer to you now and carefully look at forefinger, and you will immediately find the golden section formula 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 golden ratio (with the exception of the thumb).
In addition, the ratio between the middle finger and the little finger is also equal to the golden ratio.
A person has 2 hands, the fingers on each hand consist of 3 phalanges (with the exception of the thumb). Each hand has 5 fingers, that is, 10 in total, but with the exception of two two-phalangeal thumbs only 8 fingers are created according to the principle of the golden section. Whereas all these numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence.
golden ratio in the structure of the human lungs
American physicist B.D. West and Dr. A.L. Goldberger during physical and anatomical studies found that in the structure of the human lungs 
also exists golden ratio.
The peculiarity of the bronchi that make up the lungs of a person lies in their asymmetry. The bronchi are made up of two main airways, one (left) is longer and the other (right) is shorter.
It was found that this asymmetry continues in the branches of the bronchi, in all smaller airways. Moreover, the ratio of the length 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, also created according to the principle of the golden ratio. Leonardo Da Vinci and Le Corbusier, before creating their masterpieces, took the parameters of the human body, created according to the law of the Golden Ratio.
There is another, more prosaic application of the proportions of the human body. For example, using these ratios, criminal analysts and archaeologists restore the appearance of the whole from fragments of parts of the human body.
Leonardo Fibonacci is one of the greatest mathematicians of the Middle Ages. In one of his works, The Book of Calculations, Fibonacci described the Indo-Arabic calculus and the advantages of using it over the Roman one.
Definition
Fibonacci numbers or Fibonacci Sequence is a numerical sequence that has a number of properties. For example, the sum of two neighboring numbers in the sequence gives the value of the next one (for example, 1+1=2; 2+3=5, etc.), which confirms the existence of the so-called Fibonacci coefficients, i.e. constant ratios.
The Fibonacci sequence starts like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233…
2.
Complete definition of Fibonacci numbers
3.
Properties of the Fibonacci Sequence
4.
1. The ratio of each number to the next more and more tends to 0.618 as the serial number increases. The ratio of each number to the previous one tends to 1.618 (reverse to 0.618). The number 0.618 is called (FI).
2. When dividing each number by the next one, the number 0.382 is obtained through one; vice versa - respectively 2.618.
3. Selecting ratios in this way, we obtain the main set of Fibonacci coefficients: … 4.235, 2.618, 1.618, 0.618, 0.382, 0.236.
5.
Relationship between the Fibonacci sequence and the "golden section"
6.
The Fibonacci sequence asymptotically (approaching more and more slowly) tends to some constant ratio. However, this ratio is irrational, that is, it is a number with an infinite, unpredictable sequence of decimal digits in the fractional part. It cannot be expressed exactly.
If any member of the Fibonacci sequence is divided by the one preceding it (for example, 13:8), the result will be a value that fluctuates around the irrational value of 1.61803398875 ... and after a time either exceeding it or not reaching it. But even having spent Eternity on it, it is impossible to know the ratio exactly, to the last decimal digit. For the sake of brevity, we will give it in the form of 1.618. Special names for this ratio began to be given even before Luca Pacioli (a medieval mathematician) called it the Divine Proportion. Among its modern names are such as the Golden Ratio, the Golden Mean and the ratio of rotating squares. Kepler called this relation one of the "treasures of geometry". In algebra, it is commonly denoted by the Greek letter phi
Let's imagine the golden section on the example of a segment.
Consider a segment with ends A and B. Let point C divide segment AB so that,
AC/CB = CB/AB or
AB/CB = CB/AC.
You can imagine it like this: A-–C--–B
7.
The golden section is such a proportional division of a segment into unequal parts, in which the entire segment relates to the larger part in the same way as the larger part itself relates to the smaller one; or in other words, the smaller section is related to the larger one as the larger one is to everything.
8.
Segments of the golden ratio are expressed as an infinite irrational fraction 0.618 ..., if AB is taken as one, AC = 0.382 .. As we already know, the numbers 0.618 and 0.382 are the coefficients of the Fibonacci sequence.
9.
Fibonacci proportions and the golden ratio in nature and history
10.
It is important to note that Fibonacci, as it were, reminded humanity of his sequence. It was known to the ancient Greeks and Egyptians. Indeed, since then, patterns described by Fibonacci coefficients have been found in nature, architecture, fine arts, mathematics, physics, astronomy, biology and many other areas. It is simply amazing how many constants can be calculated using the Fibonacci sequence, and how its terms appear in a huge number of combinations. However, it would not be an exaggeration to say that this is not just a number game, but the most important mathematical expression. natural phenomena of all ever discovered.
11.
The examples below show some interesting applications of this mathematical sequence.
12.
1. The shell is twisted in a spiral. If you unfold it, you get a length slightly inferior to the length of the snake. A small ten-centimeter shell has a spiral 35 cm long. The shape of the spirally curled shell attracted the attention of Archimedes. The fact is that the ratio of measurements of the volutes of the shell is constant and equal to 1.618. Archimedes studied the spiral of shells and derived the equation for the spiral. The spiral drawn by this equation is called by his name. The increase in her step is always uniform. At present, the Archimedes spiral is widely used in engineering.
2. Plants and animals. Even Goethe emphasized the tendency of nature to spirality. The spiral and spiral arrangement of leaves on tree branches was noticed long ago. The spiral was seen in the arrangement of sunflower seeds, in pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch of sunflower seeds, pine cones, the Fibonacci series manifests itself, and therefore, the law of the golden section manifests itself. The spider spins its web in a spiral pattern. A hurricane is spiraling. A frightened herd of reindeer scatter in a spiral. The DNA molecule is twisted into a double helix. Goethe called the spiral "the curve of life."
Among the roadside grasses, an unremarkable plant grows - chicory. Let's take a closer look at it. A branch was formed from the main stem. Here is the first leaf. The process makes a strong ejection into space, stops, releases a leaf, but is shorter than the first one, again makes an ejection into space, but of less force, releases an even smaller leaf and ejection again. If the first outlier is taken as 100 units, then the second is equal to 62 units, the third is 38, the fourth is 24, and so on. The length of the petals is also subject to the golden ratio. In growth, the conquest of space, the plant retained certain proportions. Its growth impulses gradually decreased in proportion to the golden ratio.
The lizard is viviparous. In the lizard, at first glance, proportions that are pleasing to our eyes are caught - the length of its tail relates to the length of the rest of the body as 62 to 38.
Both in the plant and animal worlds, the shaping tendency of nature persistently breaks through - symmetry with respect to the direction of growth and movement. Here the golden ratio appears in the proportions of parts perpendicular to the direction of growth. Nature has carried out the division into symmetrical parts and golden proportions. In parts, a repetition of the structure of the whole is manifested.
Pierre Curie at the beginning of our century formulated a number of profound ideas of symmetry. He argued that one cannot consider the symmetry of any body without taking into account the symmetry 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 space systems, in the gene structures of living organisms. These patterns, as indicated above, are in the structure of individual human organs and the body as a whole, and are also manifested in biorhythms and the functioning of the brain and visual perception.
3. Space. It is known from the history of astronomy that I. Titius, a German astronomer of the 18th century, using this series (Fibonacci) found regularity and order in the distances between the planets of the solar system
However, one case that seemed to be against the law: there was no planet between Mars and Jupiter. Focused observation of this area of the sky led to the discovery of the asteroid belt. This happened after the death of Titius in early XIX in.
The Fibonacci series is widely used: with its help, they represent the architectonics of living beings, and man-made structures, and the structure of the Galaxies. These facts are evidence of the independence of the number series from the conditions of its manifestation, which is one of the signs of its universality.
4. Pyramids. Many have tried to unravel the secrets of the Giza pyramid. Unlike other Egyptian pyramids, this is not a tomb, but rather an unsolvable puzzle of numerical combinations. The remarkable ingenuity, skill, time and labor of the architects of the pyramid, which they used in the construction of the eternal symbol, indicate the extreme importance of the message that they wanted to convey to future generations. Their era was pre-literate, pre-hieroglyphic, and symbols were the only means of recording discoveries. The key to the geometrical-mathematical secret of the Giza pyramid, which had been a mystery to mankind for so long, was actually given to Herodotus by the temple priests, who informed him that the pyramid was built so that the area of each of its faces was equal to the square of its height.
Triangle area
356 x 440 / 2 = 78320
square area
280 x 280 = 78400
The length of the edge of the base of the pyramid at Giza is 783.3 feet (238.7 m), the height of the pyramid is 484.4 feet (147.6 m). The length of the edge of the base, divided by the height, leads to the ratio Ф=1.618. The height of 484.4 feet corresponds to 5813 inches (5-8-13) - these are numbers from the Fibonacci sequence. These interesting observations suggest that the construction of the pyramid is based on the proportion Ф=1.618. Some modern scholars tend to interpret that the ancient Egyptians built it for the sole purpose of passing on the knowledge they wanted to preserve for future generations. Intensive studies of the pyramid at Giza showed how extensive knowledge in mathematics and astrology was at that time. In all internal and external proportions of the pyramid, the number 1.618 plays a central role.
Pyramids in Mexico. Not only the Egyptian pyramids were built in accordance with the perfect proportions of the golden ratio, the same phenomenon was found in the Mexican pyramids. The idea arises that both Egyptian and Mexican pyramids were erected at approximately the same time by people of a common origin.
Kanalieva Dana
In this paper, we have studied and analyzed the manifestation of the numbers of the Fibonacci sequence in the reality around us. We have discovered an amazing mathematical relationship between the number of spirals in plants, the number of branches in any horizontal plane and numbers in the Fibonacci sequence. We also saw strict mathematics in the structure of man. The human DNA molecule, in which the entire program for the development of a human being is encrypted, respiratory system, the structure of the ear - everything obeys certain numerical ratios.
We have seen that Nature has its own laws, expressed with the help of mathematics.
And mathematics is very important tool knowledge secrets of nature.
Download:
Preview:
MBOU "Pervomaiskaya secondary school"
Orenburgsky district of the Orenburg region
RESEARCH
"The riddle of numbers
Fibonacci"
Completed by: Kanalieva Dana
6th grade student
Supervisor:
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 wildlife.......……. …………………………………... 5.
Chapter 2. Fibonacci Spiral............................................... ..........……………..... nine.
Chapter 3. Fibonacci numbers in human inventions .........…………………………….
Chapter 4. Our Research………………………………………………………………………………………………….
Chapter 5. Conclusion, conclusions……………………………………………………………….....
List of used literature and Internet sites……………………………………........21.
Object of study:
Man, mathematical abstractions, created by man, inventions of man, the surrounding flora and fauna.
Subject of study:
the form and structure of the studied objects and phenomena.
Purpose of the study:
to study the manifestation of Fibonacci numbers and the law of the golden section associated with it in the structure of living and inanimate objects,
find examples of using Fibonacci numbers.
Work tasks:
Describe how to construct a Fibonacci series and a Fibonacci spiral.
See mathematical patterns in the structure of man, flora and inanimate nature from the point of view of the Golden Section phenomenon.
Research novelty:
The discovery of Fibonacci numbers in the reality around us.
Practical significance:
Use of the acquired knowledge and research skills in the study of 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)
Registration of work with drawings, diagrams, photographs.
Active participation in the discussion of their work.
Research methods:
empirical (observation, experiment, measurement).
theoretical (logical stage of knowledge).
Explanatory note.
“Numbers rule the world! Number is the power that reigns over gods and mortals!” - so said the ancient Pythagoreans. Is this basis of the Pythagorean teaching relevant today? 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 ratios, 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 the Fibonacci numbers, about the magic numbers that are fraught with great riddle. 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 learn about the secrets of Fibonacci numbers. This research work is the result of our work.
Hypothesis:
in the reality around us, everything is built according to surprisingly harmonious laws with mathematical precision.
Everything in the world is thought out and calculated by our most important designer - Nature!
Introduction. The history of the Fibonacci series.
Amazing numbers were discovered by the Italian mathematician of the Middle Ages, Leonardo of Pisa, better known as Fibonacci. Traveling in the East, he became acquainted with the achievements of Arabic mathematics and contributed to their transfer to the West. In one of his works entitled "The Book of Calculations" he presented to Europe one of greatest discoveries of all times and peoples - the decimal number system.
One day, he puzzled over the solution of one mathematical problem. He was trying to create a formula describing the breeding sequence of rabbits.
The solution was 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 will say, - “Can we ourselves come up with similar numerical series, growing according to a given progression?” Indeed, when the Fibonacci series appeared, no one, including himself, suspected how close he managed to get closer to unraveling one of the greatest mysteries of the universe!
Fibonacci led a reclusive life, spent a lot of time in nature, and while walking in the forest, he noticed that these numbers literally began to haunt him. Everywhere in nature, he met these numbers again and again. For example, the petals and leaves of plants strictly fit into a given number series.
In Fibonacci numbers there is interesting feature: quotient from dividing the next Fibonacci number by the previous one, as the numbers themselves grow, tend 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 Ratio.
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 the other, the number adjacent to it, we will always get 1.618. And if we do the opposite, that is, we divide the smaller number by the larger one, we 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 it were 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 golden division law.
Scientists analyzing further application of this numerical series to natural phenomena and processes, they found that these numbers are contained in literally all objects of wildlife, in plants, in animals and in humans.
An amazing mathematical toy turned out to be a unique code embedded in all natural objects by the Creator of the Universe himself.
Consider examples where Fibonacci numbers are found in animate and inanimate nature.
Fibonacci numbers in wildlife.
If you look at the plants and trees around us, you can see how many leaves each of them has. From afar, it seems that the branches and leaves on the plants are arranged randomly, in an arbitrary order. However, in all plants it is miraculously, mathematically precisely planned which branch will grow from where, how 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 the appearance of the plant is already precisely programmed. How many branches will be on the future tree, where the branches will grow, how many leaves will be on each branch, and how, 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 in the arrangement of leaves on a branch (phylotaxis), in the number of revolutions on the stem, in the number of leaves in the cycle, the Fibonacci series manifests itself, and therefore, the law of the golden section also manifests itself.
If you set out to find numerical patterns in wildlife, you will notice that these numbers are often found in various spiral forms, which the plant world is so rich in. For example, leaf cuttings adjoin the stem in a spiral that runs betweentwo adjacent leaves:full turn - at the hazel,- at the oak - at the poplar and pear,- 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.
A clear, symmetrical shape of flowers is also subject to a strict law.
Many flowers have the number of petals - exactly the numbers from the Fibonacci series. For example:
iris, 3 lep. buttercup, 5 lep. golden flower, 8 lep. delphinium,
13 lep.
chicory, 21 lep. 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 the man himself is just a storehouse of the number phi.
The proportions of the various parts of our body make up a number very close to the golden ratio. If these proportions coincide with the formula of the golden ratio, then the appearance or body of a person is considered to be ideally built. The principle of calculating the golden measure on the human body can be depicted in the form of a diagram.
M/m=1.618
The first example of the golden section in the structure of the human body:
If we take the navel point as the center of the human body, and the distance between the human foot and the navel point as a unit of measurement, then the height of a person is equivalent to the number 1.618.
Human hand
It is enough just to bring your palm closer to you now and carefully look at your index finger, and you will immediately find the golden section formula 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 golden ratio (with the exception of the thumb).
In addition, the ratio between the middle finger and the little finger is also equal to the golden ratio.
A person has 2 hands, the fingers on each hand consist of 3 phalanges (with the exception of the thumb). Each hand has 5 fingers, that is, 10 in total, but with the exception of two two-phalangeal 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 found that the golden section also exists in the structure of the human lungs.
The peculiarity of the bronchi that make up the lungs of a person lies in their asymmetry. The bronchi are made up of two main airways, one (left) is longer and the other (right) is shorter.
It was found that this asymmetry continues in the branches of the bronchi, in all smaller airways. Moreover, the ratio of the length 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, also created according to the principle of the golden ratio. Leonardo Da Vinci and Le Corbusier, before creating their masterpieces, took the parameters of the human body, created according to the law of the Golden Ratio.
There is another, more prosaic application of the proportions of the human body. For example, using these ratios, criminal analysts and archaeologists restore the appearance of the whole from fragments of parts of the human body.
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 ratio. The DNA molecule consists of two vertically intertwined helices. Each of these spirals is 34 angstroms long and 21 angstroms wide. (1 angstrom is one hundred millionth of a centimeter).
So 21 and 34 are numbers, following friend one after the other in the sequence of Fibonacci numbers, that is, the ratio of the length and width of the logarithmic helix of the DNA molecule carries the formula of the golden section 1:1.618.
Not only upright walkers, but also all those who swim, crawl, fly and jump, did not escape the fate of obeying the number phi. The human heart muscle contracts to 0.618 of its volume. The structure of the snail shell corresponds to the Fibonacci proportions. And there are plenty of such examples - there would be a desire to explore natural objects and processes. The world is so permeated with Fibonacci numbers that sometimes it seems that the Universe can be explained only by them.
Fibonacci spiral.
There is no other form in mathematics that has the same unique properties like a spiral because
The structure of the spiral is based on the rule of the Golden Section!
To understand the mathematical construction of the spiral, let's 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 in the same way 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 everything.
That is, (a + b) / a = a / b
A rectangle with exactly this ratio of sides was called the golden rectangle. Its long sides are related to the short sides in a ratio of 1.168:1.
The golden rectangle has many unusual properties. Cutting off from the golden rectangle a square whose side is equal to the smaller side of the rectangle,
we again get a smaller golden rectangle.
This process can be continued ad infinitum. As we keep cutting off the squares, we'll get smaller and smaller golden rectangles. Moreover, they will be located in a logarithmic spiral having importance in mathematical models of natural objects.
For example, a spiral shape can also 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 beings not only have no idea about the spiral, but do not even have the simplest mathematical knowledge to create a spiral shell for themselves.
But then how could these unintelligent beings determine and choose for themselves the ideal form of growth and existence in the form of a spiral shell? Could these living beings whom scientists world calls primitive life forms, to calculate that the spiral shape of the shell will be ideal for their existence?
Trying to explain the origin of such even the most primitive form of life by a random coincidence of some natural circumstances is at least absurd. It is clear that this project is a conscious creation.
Spirals are also in man. With the help of spirals we hear:
Also, in the human inner ear there is an organ Cochlea ("Snail"), which performs the function of transmitting sound vibration. This bone-like structure is filled with liquid and created in the form of a snail with golden proportions.
Spirals are 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 the form of a spiral, the claws of lions and the beaks of parrots are logarithmic forms and resemble the shape of an axis that tends to turn into a spiral.
It is interesting that a hurricane, cyclone clouds are spiraling, and this is clearly visible from space:
in ocean and sea waves the spiral can be shown mathematically on a graph with points 1,1,2,3,5,8,13,21,34 and 55.
Everyone will also recognize such a “everyday” and “prosaic” spiral.
After all, water runs away from the bathroom in a spiral:
Yes, and we live in a spiral, because the galaxy is a spiral that corresponds to the formula of the Golden Section!
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 found 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 peeped from nature the law expressed by the sequence of Fibonacci numbers, scientists and people of art try to imitate it, to embody this law in their creations.
The proportion of phi allows you to create masterpieces of painting, competently fit architectural structures into space.
Not only scientists, but also architects, designers and artists are amazed at this flawless spiral at the nautilus shell,
occupying smallest space and providing least loss heat. American and Thai architects, inspired by the “camera nautilus” example of putting the maximum in the minimum of space, are busy developing their designs.
Since time immemorial, the proportion of the Golden Ratio 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 prices and the Hebrew alphabet contain a golden ratio.
But we want to dwell on a unique example of creating an effective solar installation. Aidan Dwyer, an American high school student from New York City, brought 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 the trees needed such a “pattern” of branches and leaves. He knew that the branches on the trees are arranged according to the Fibonacci sequence, and the leaves carry out photosynthesis.
At some point, a quick-witted boy decided to check if this position of the branches helps to collect more sunlight. Aidan built a pilot plant in his backyard with small solar panels instead of leaves and tested it in action. It turned out that in comparison with the usual flat solar panel its "tree" collects 20% more energy and works 2.5 hours more efficiently.
Model solar tree Dwyer and graphs built by a schoolboy.
"And such an installation takes less space, than a flat panel, collects 50% more sun in winter even where it does not face south, and it does not accumulate snow in that amount. In addition, the tree design is much more suitable for the urban landscape," notes the young inventor.
Aidan recognized one of the best young natural scientists of 2011. The 2011 Young Naturalist competition was hosted by the New York Museum of Natural History. Aidan filed a provisional patent application for his invention.
Scientists continue to actively develop the theory of Fibonacci numbers and the golden section.
Yu. Matiyasevich solves Hilbert's 10th problem using Fibonacci numbers.
There are elegant methods for solving a number of cybernetic problems (search theory, games, programming) using Fibonacci numbers and the golden section.
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 is very multifaceted:
Observing the phenomena occurring in nature, scientists have made amazing conclusions that the whole sequence of events occurring in life, revolutions, collapses, bankruptcies, periods of prosperity, laws and waves of development in the stock and currency markets, cycles family life, and so on, are organized on the timeline in the form of cycles, waves. These cycles and waves are also distributed according to the Fibonacci number series!
Based on this knowledge, a person will learn to predict various events in the future and manage them.
4. Our research.
We continued our observations and studied the structure
Pine cone
yarrow
mosquito
human
And we made sure that in these objects, so different at first glance, the very numbers of the Fibonacci sequence are invisibly present.
So step 1.
Let's take pine cone:
Let's take a closer look at it:
We notice two series of Fibonacci spirals: one - clockwise, the other - against, their number 8 and 13.
Step 2
Let's take a yarrow:
Let's take a closer look at the structure of stems and flowers:
Note that each new branch of the yarrow grows from the sinus, and new branches grow from the new branch. Adding old and new branches, we found the Fibonacci number in each horizontal plane.
Step 3
Do Fibonacci numbers show up in morphology various organisms? Consider the well-known mosquito:
We see: 3 pair of legs, head 5 antennae - 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.
Pine cone, yarrow, mosquito, man 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, received new and new questions.
Where did these numbers come from? Who is this architect of the universe who tried to make it perfect? Is the coil twisting or untwisting?
How amazingly man knows this world!!!
Having found the answer to one question, he receives the next one. Solve it, get two new ones. Deal with them, three more will appear. Having solved them, he will acquire five unresolved ones. Then eight, then thirteen, 21, 34, 55...
Do you recognize?
Conclusion.
By the creator himself in all objects
A unique code has been assigned
And the one who is friendly with mathematics,
He will know and understand!
We have studied and analyzed the manifestation of the numbers of the Fibonacci sequence in the reality around us. We also learned that the patterns of this number series, including the patterns of the "Golden" symmetry, are manifested in the energy transitions of elementary particles, in planetary and cosmic systems, in the gene structures of living organisms.
We have 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 have seen how the morphology of various organisms also obeys this mysterious law. We also saw strict mathematics in the structure of man. The human DNA molecule, in which the whole program of the development of a human being is encrypted, the respiratory system, the structure of the ear - everything obeys certain numerical ratios.
We have learned that pine cones, snail shells, ocean waves, animal horns, cyclone clouds, and galaxies all form logarithmic spirals. Even the human finger, which is made up of three phalanges in relation to each other in the Golden ratio, takes on a spiral shape when compressed.
eternity of time and light years space divide a pinecone and a spiral galaxy, but the structure remains the same: the coefficient 1,618 ! Perhaps this is the supreme law that governs 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 with the help of mathematics. And math is a very important tool
to discover the mysteries of nature.
List of literature and Internet sites:
1.
Vorobyov N. N. Fibonacci numbers. - M., Nauka, 1984.
2. Gika 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
A life. - 1982.- No. 10.
5. Malay G. Harmony - the identity of paradoxes // MN. - 1982.- No. 19.
6. Sokolov A. Secrets of the golden section // Technique of youth. - 1978.- No. 5.
7. Stakhov A. P. Codes of the golden ratio. - M., 1984.
8. Urmantsev Yu. A. Symmetry of nature and the nature of symmetry. - M., 1974.
9. Urmantsev Yu. A. Golden section // Priroda. - 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.:
The surrounding world, starting with the smallest invisible particles, and ending with distant galaxies of boundless space, is fraught with many unsolved mysteries. However, the veil of mystery has already been lifted over some of them thanks to the inquisitive minds of a number of scientists.
One such example is golden ratio and Fibonacci numbers that form its basis. This pattern has been displayed in mathematical form and is often found in human environment nature, once again excluding the possibility that it arose by chance.
Fibonacci numbers and their sequence
Fibonacci number sequence called a series of numbers, each of which is the sum of the previous two:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 …
A feature of this sequence is the numerical values that are obtained by dividing the numbers of this series by each other.
A series of Fibonacci numbers has its own interesting patterns:
- In the Fibonacci series, each number divided by the next will show a value tending towards 0,618 . The farther the numbers are from the beginning of the series, the more accurate the ratio will be. For example, the numbers taken at the beginning of the row 5 and 8 will show 0,625 (5/8=0,625 ). If we take the numbers 144 and 233 , then they will show the ratio 0.618 .
- In turn, if in a series of Fibonacci numbers we divide the number by the previous one, then the result of the division will tend to 1,618 . For example, the same numbers were used as mentioned above: 8/5=1,6 and 233/144=1,618 .
- The number divided by the next one after it will show a value approaching 0,382 . And the farther from the beginning of the series the numbers are taken, the more precisely meaning ratios: 5/13=0,385 and 144/377=0,382 . Division of digits in reverse order will give a result 2,618 : 13/5=2,6 and 377/144=2,618 .
Using the above calculation methods and increasing the gaps between the numbers, you can display the following range of values: 4.235, 2.618, 1.618, 0.618, 0.382, 0.236, which is widely used in Fibonacci tools in the forex market.
Golden Ratio or Divine Proportion
The “golden section” and Fibonacci numbers are very clearly represented by the analogy with a segment. If segment AB is divided by point C in such a ratio that the condition is met:
AC / BC \u003d BC / AB, then it will be the "golden section"
READ ALSO THE FOLLOWING ARTICLES:
Surprisingly, it is this ratio that can be traced in the series of Fibonacci numbers. Taking a few numbers from the series, you can check by calculation that this is so. For example, such a sequence of Fibonacci numbers ... 55, 89, 144 ... Let the number 144 be the whole segment AB, which was mentioned above. Since 144 is the sum of the two previous numbers, then 55+89=AC+BC=144.
Dividing the segments will show the following results:
AC/BC=55/89=0.618
BC/AB=89/144=0.618
If we take the segment AB as a whole, or as a unit, then AC \u003d 55 will be 0.382 of this whole, and BC \u003d 89 will be equal to 0.618.
Where are Fibonacci numbers found?
The regular sequence of Fibonacci numbers was known to the Greeks and Egyptians long before Leonardo Fibonacci himself. This number series acquired such a name after the famous mathematician ensured the wide distribution of this mathematical phenomenon in scientific ranks.
It is important to note that the golden Fibonacci numbers are not just science, but a mathematical representation of the world around them. Many natural phenomena, representatives of the flora and fauna have the "golden section" in their proportions. These are spiral curls of the shell, and the arrangement of sunflower seeds, cacti, pineapples.
The spiral, the proportions of the branches of which are subject to the laws of the "golden section", underlies the formation of a hurricane, the weaving of a web by a spider, the shape of many galaxies, the interweaving of DNA molecules and many other phenomena.
The length of the lizard's tail to its body has a ratio of 62 to 38. The chicory shoot, before releasing a leaf, makes a release. After the first sheet is released, a second ejection occurs before the release of the second sheet, with a force equal to 0.62 of the conditionally accepted unit of force of the first ejection. The third outlier is 0.38 and the fourth is 0.24.
Also for the trader great importance has the fact that the price movement in the forex market is often subject to the patterns of golden Fibonacci numbers. Based on this sequence, a number of tools have been created that a trader can use in his arsenal.
Often used by traders, the tool "" can high precision show the price movement targets, as well as the levels of its correction.
Among the many inventions made by great scientists in past centuries, the discovery of the patterns of development of our universe in the form of a system of numbers is the most interesting and useful. This fact was described in his work by the Italian mathematician Leonardo Fibonacci. A number series is a sequence of digits in which each member value is the sum of the two previous ones. This system expresses the information embedded in the structure of all living things in accordance with the harmonious development.
The great scientist FibonacciThe Italian scientist lived and worked in the XIII century in the city of Pisa. He was born into a merchant family and at first worked with his father in trade. Leonardo Fibonacci came to mathematical discoveries when he was trying to establish contacts at that time with business partners.
The scientist made his discovery when calculating the planning of the offspring of rabbits at the request of one of his distant relatives. He opened the number series, according to which the reproduction of animals will be carried out. He described this pattern in his work "The Book of Calculations", where he also presented information about the decimal for European countries.
"Golden" discovery
The number series can be expressed graphically as an expanding spiral. It can be noted that in nature there are many examples that are based on this figure, for example, rolling waves, the structure of galaxies, microcapillaries in the human body, and
Interestingly, the numbers in this system (Fibonacci coefficients) are considered “live” numbers, since all living things evolve according to this progression. This pattern was known even to people of ancient civilizations. There is a version that already at that time it was known how to investigate the convergence of a number series - the most important issue in the sequence of numbers.
Application of Fibonacci Theory
Having examined his number series, the Italian scientist discovered that the ratio of a digit from a given sequence to the next member is 0.618. This value is called the proportionality factor, or the "golden section". It is known that this number was used by the Egyptians in the construction of the famous pyramid, as well as the ancient Greeks and Russian architects in the construction of classical structures - temples, churches, etc.
But an interesting fact is that the Fibonacci number series is also used in evaluating the movement of prices for The use of this sequence in technical analysis was proposed by engineer Ralph Elliot at the beginning of the last century. In the 30s, the American financier was engaged in forecasting stock prices, in particular, the study of the Dow Jones index, which is one of the main components in the stock market. After a series of successful predictions, he published several of his articles in which he described methods for using the Fibonacci series.
On the this moment Almost all traders use the Fibonacci theory when predicting price movements. This dependence is also used for many scientific research in various fields. Thanks to the discovery of a great scientist, many useful inventions can be created even after many centuries.
