However, this is not all that can be done with the golden ratio. If we divide the unit by 0.618, then we get 1.618, if we square it, then we get 2.618, if we raise it into a cube, we get the number 4.236. These are the Fibonacci expansion coefficients. The only thing missing here is the number 3.236, which was proposed by John Murphy.
What do experts think about sequence?
Some will say that these numbers are already familiar because they are used in technical analysis programs to determine the amount of correction and expansion. In addition, these same series play an important role in the Eliot wave theory. They are its numerical basis.
Our expert Nikolay Proven portfolio manager of Vostok investment company.
- — Nikolai, what do you think, is the appearance of Fibonacci numbers and its derivatives on the charts of various instruments by chance? And is it possible to say: "Fibonacci series practical application" takes place?
- - I have a bad attitude towards mysticism. And even more so on the stock exchange charts. Everything has its reasons. in the book "Fibonacci Levels" he beautifully told where the golden ratio appears, that he was not surprised that it appeared on the stock exchange charts. But in vain! Pi often appears in many of the examples he gave. But for some reason it is not in the price ratio.
- - So you do not believe in the effectiveness of the Elliot wave principle?
- “No, no, that’s not the point. The wave principle is one thing. The numerical ratio is different. And the reasons for their appearance on price charts are the third
- What do you think are the reasons for the appearance of the golden section on stock charts?
- - The correct answer to this question may be able to earn the Nobel Prize in Economics. While we can guess the true reasons. They are clearly out of harmony with nature. There are many models of exchange pricing. They do not explain the indicated phenomenon. But not understanding the nature of the phenomenon should not deny the phenomenon as such.
- - And if this law is ever open, will it be able to destroy the exchange process?
- - As the same theory of waves shows, the law of change in stock prices is pure psychology. It seems to me that knowledge of this law will not change anything and will not be able to destroy the stock exchange.
The material is provided by webmaster Maxim's blog.
The coincidence of the foundations of the principles of mathematics in a variety of theories seems incredible. Maybe it's fantasy or an adjustment to the end result. Wait and see. Much of what was previously considered unusual or impossible: space exploration, for example, has become commonplace and does not surprise anyone. Also, the wave theory, which may be incomprehensible, will become more accessible and understandable with time. What was previously unnecessary, in the hands of an experienced analyst, will become a powerful tool for predicting future behavior.
Fibonacci numbers in nature.
Look
And now, let's talk about how you can refute the fact that the Fibonacci digital series is involved in any patterns in nature.
Let's take any other two numbers and build a sequence with the same logic as the Fibonacci numbers. That is, the next member of the sequence is equal to the sum of the two previous ones. For example, let's take two numbers: 6 and 51. Now we will build a sequence that we will complete with two numbers 1860 and 3009. Note that when dividing these numbers, we get a number close to the golden ratio.
At the same time, the numbers that were obtained by dividing other pairs decreased from the first to the last, which allows us to assert that if this series is continued indefinitely, then we will get a number equal to the golden ratio.
Thus, the Fibonacci numbers themselves are not distinguished by anything. There are other sequences of numbers, of which there are an infinite number, which give the golden number phi as a result of the same operations.
Fibonacci was not an esotericist. He didn't want to put any mysticism into the numbers, he was just solving an ordinary rabbit problem. And he wrote a sequence of numbers that followed from his task, in the first, second and other months, how many rabbits there would be after breeding. Within a year, he received that same sequence. And didn't make a relationship. There was no golden ratio, no Divine relation. All this was invented after him in the Renaissance.
Before mathematics, Fibonacci's virtues are enormous. He adopted the number system from the Arabs and proved its validity. It was a hard and long struggle. From the Roman number system: heavy and inconvenient for counting. She disappeared after the French Revolution. It has nothing to do with the golden section of Fibonacci.
There are infinitely many spirals, the most popular are: natural logarithm spiral, Archimedes spiral, hyperbolic spiral.
Now let's take a look at the Fibonacci spiral. This piecewise-composite aggregate consists of several quarters of circles. And it is not a spiral, as such.
Conclusion
No matter how long we look for confirmation or refutation of the applicability of the Fibonacci series on the stock exchange, this practice exists.
Huge masses of people act according to the Fibonacci ruler, which is found in many user terminals. Therefore, whether we like it or not: Fibonacci numbers have an impact on, and we can take advantage of this influence.
Without fail we read the article -.
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 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 also saw strict mathematics in the structure of man. The human DNA molecule, in which the entire 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 seen that Nature has its own laws, expressed with the help of mathematics.
And mathematics is very important learning tool secrets of nature.
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MBOU "Pervomaiskaya secondary school"
Orenburgsky district of the Orenburg region
RESEARCH
"The riddle 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 wildlife.......……. …………………………………... 5.
Chapter 2. Fibonacci Spiral............................................... ..........……………..... 9.
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.
To see mathematical patterns in the structure of man, the plant world and inanimate nature from the point of view of the phenomenon of the Golden Section.
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.
Acquisition of the ability to review the 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 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 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 introduced Europe to one of the greatest discoveries of all times and peoples - the decimal number system.
Once, he puzzled over the solution of a mathematical problem. He was trying to create a formula describing the breeding sequence of rabbits.
The answer 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 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.
There is an interesting feature in the Fibonacci numbers: the quotient of dividing the next Fibonacci number by the previous one tends to 1.618 as the numbers themselves grow. 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 the further application of this number series to natural phenomena and processes, 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 section. 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 one after another 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 as 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, which is important 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 creatures, which the scientific world calls primitive life forms, have calculated that the spiral shape of the shell would 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 a spiral form, 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 mathematically plotted 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 the smallest space and providing the least heat loss. Inspired by the “camera nautilus” example of putting the maximum in the minimum of space, American and Thai architects are busy developing designs to match.
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 efficient solar installation. American schoolboy from New York Aidan Dwyer 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 smart little 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 a conventional flat solar panel, his “tree” collects 20% more energy and works effectively for 2.5 hours longer.
Dwyer's solar tree model and student plots.
“It 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 snow as much. In addition, the design in the form of a tree 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 of family life, and so on , are organized on a time scale 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 cones
yarrow
mosquito
human
And they made sure that in these objects, so different at first glance, the same numbers of the Fibonacci sequence are 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 - 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 the morphology of 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 of space separate 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
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 Italian mathematician Leonardo Fibonacci lived in the 13th century and was one of the first in Europe to use Arabic (Indian) numerals. He came up with a somewhat artificial problem about rabbits that are raised on a farm, with all of them being considered females, males are ignored. Rabbits start breeding after they are two months old and then give birth to a rabbit every month. Rabbits never die.
It is necessary to determine how many rabbits will be on the farm in n months, if at the initial moment of time there was only one newborn rabbit.
Obviously, the farmer has one rabbit in the first month and one rabbit in the second month. In the third month there will be two rabbits, in the fourth month there will be three, and so on. Let us denote the number of rabbits in n month like . Thus, 
,
,
,
,
,
…
We can construct an algorithm to find 
for any n.
According to the condition of the problem, the total number of rabbits 
V n+1 month is decomposed into three components:
one-month-old rabbits, not capable of reproduction, in the amount
;
Thus, we get
. (8.1)
Formula (8.1) allows you to calculate a series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ...
The numbers in this sequence are called Fibonacci numbers .
If accept 
And 
, then with the help of formula (8.1) one can determine all other Fibonacci numbers. Formula (8.1) is called recurrent
formula ( recurrence
- "return" in Latin).
Example 8.1. Suppose there is a staircase in n steps. We can climb it with a step of one step, or with a step of two steps. How many combinations of different lifting methods are there?
If n= 1, there is only one solution to the problem. For n= 2 there are 2 options: two single steps or one double step. For n= 3 there are 3 options: three single steps, or one single and one double, or one double and one single.
In the next case n= 4, we have 5 possibilities (1+1+1+1, 2+1+1, 1+2+1, 1+1+2, 2+2).
In order to answer a given question with an arbitrary n, denote the number of options as 
, and try to determine 
according to famous 
And 
. If we start from a single step, then we have 
combinations for the remaining n steps. If we start with a double step, then we have 
combinations for the remaining n-1 steps. The total number of options for n+1 steps equals
. (8.2)
The resulting formula, like a twin, resembles formula (8.1). However, this does not allow one to identify the number of combinations 
with Fibonacci numbers 
. We see, for example, that 
, But 
. However, there is the following relationship:
.
This is true for n= 1, 2, and is also valid for each n. Fibonacci numbers and number of combinations 
are calculated using the same formula, but the initial values 
,
And 
,
they differ.
Example 8.2. This example is of practical importance for problems of error-correcting coding. Find the number of all binary words of length n, not containing multiple zeros in a row. Let's denote this number by 
. Obviously, 
, and the words of length 2 that satisfy our constraint are: 10, 01, 11, i.e. 
. Let 
- a word from n characters. If the symbol 
, That 
can be arbitrary ( 
)-literal word that does not contain multiple zeros in a row. So the number of words with a unit at the end is 
.
If the symbol 
, then necessarily 
, and the first 
symbol 
can be arbitrary, taking into account the considered restrictions. Therefore, there is 
word length
n with zero at the end. Thus, the total number of words of interest to us is
.
Taking into account the fact that 
And 
, the resulting sequence of numbers is the Fibonacci numbers.
Example 8.3. In Example 7.6 we found that the number of binary words of constant weight t(and length k) equals 
. Now let's find the number of binary words of constant weight t, not containing multiple zeros in a row.
You can reason like this. Let 
the number of zeros in the words under consideration. Every word has 
gaps between the nearest zeros, each of which contains one or more ones. It is assumed that 
. Otherwise, there is not a single word without adjacent zeros.
If we remove exactly one unit from each interval, then we get a word of length 
containing 
zeros. Any such word can be obtained in the specified way from some (and only one) k-literal word containing 
zeros, no two of which are adjacent. Hence, the required number coincides with the number of all words of length 
containing exactly 
zeros, i.e. equals 
.
Example 8.4. Let us prove that the sum 
equals Fibonacci numbers for any integer 
. Symbol 
stands for smallest integer greater than or equal to
. For example, if 
, That 
; and if 
, That 
ceil("ceiling"). There is also a symbol 
, which stands for largest integer less than or equal to
. In English, this operation is called floor
("floor").
If 
, That 
. If 
, That 
. If 
, That 
.
Thus, for the considered cases, the sum is indeed equal to the Fibonacci numbers. We now give a proof for the general case. Since the Fibonacci numbers can be obtained using the recursive equation (8.1), the equality must hold:
.
And it actually does:
Here we used the previously obtained formula (4.4): 
.
Sum of Fibonacci Numbers
Let us determine the sum of the first n Fibonacci numbers.
0+1+1+2+3+5 = 12,
0+1+1+2+3+5+8 = 20,
0+1+1+2+3+5+8+13 = 33.
It is easy to see that by adding one to the right side of each equation, we again get the Fibonacci number. The general formula for determining the sum of the first n Fibonacci numbers has the form:
We will prove this using the method of mathematical induction. To do this, we write:
This amount must be equal to 
.
Reducing the left and right sides of the equation by –1, we obtain equation (6.1).
Formula for Fibonacci numbers
Theorem 8.1. Fibonacci numbers can be calculated using the formula
.
Proof. Let us verify the validity of this formula for n= 0, 1, and then we prove the validity of this formula for an arbitrary n by induction. Let's calculate the ratio of the two closest Fibonacci numbers:
We see that the ratio of these numbers fluctuates around the value of 1.618 (if we ignore the first few values). This property of Fibonacci numbers resemble members of a geometric progression. Accept 
,
(
). Then the expression
converted to
which after simplification looks like this
.
We have obtained a quadratic equation whose roots are equal to:
Now we can write:
(Where c is a constant). Both members 
And 
do not give Fibonacci numbers, for example 
, while 
. However, the difference 
satisfies the recursive equation:
For n=0 this difference gives 
, that is: 
. However, when n=1 we have 
. To obtain 
must be accepted: 
.
Now we have two sequences: 
And 
, which start with the same two numbers and satisfy the same recursive formula. They must be equal: 
. The theorem has been proven.
With increasing n member 
becomes very large while 
, and the role of the member 
is reduced in difference. Therefore, at large n we can write approximately
.
We are ignoring 1/2 (because the Fibonacci numbers increase to infinity as n to infinity).
Attitude 
called golden ratio, it is used outside of mathematics (for example, in sculpture and architecture). The golden ratio is the ratio between the diagonal and the side regular pentagon(Fig. 8.1).
Rice. 8.1. Regular pentagon and its diagonals
To denote the golden section, it is customary to use the letter 
in honor of the famous Athenian sculptor Phidias.
prime numbers
All natural numbers, large ones, fall into two classes. The first includes numbers that have exactly two natural divisors, one and itself, the second includes all the rest. Numbers of the first class are called simple, and the second constituent. Prime numbers within the first three tens: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
The properties of prime numbers and their connection with all natural numbers were studied by Euclid (3rd century BC). If you write out prime numbers in a row, you can see that their relative density decreases. The first ten of them account for 4, i.e. 40%, for a hundred - 25, i.e. 25%, per thousand - 168, i.e. less than 17%, per million - 78498, i.e. less than 8%, etc. However, their total number is infinite.
Among prime numbers, there are pairs of such, the difference between which is equal to two (the so-called simple twins), but the finiteness or infinity of such pairs has not been proved.
Euclid considered it obvious that by multiplying only prime numbers, one can obtain all natural numbers, and each natural number can be represented as a product of prime numbers in a unique way (up to the order of factors). Thus, the prime numbers form a multiplicative basis of the natural series.
The study of the distribution of primes led to the creation of an algorithm that allows one to obtain tables of primes. Such an algorithm is sieve of Eratosthenes(3rd century BC). This method consists in sifting (for example, by crossing out) those integers of a given sequence 
, which are divisible by at least one of the prime numbers less than 
.
Theorem 8 . 2 . (Euclid's theorem). The number of prime numbers is infinite.
Proof. Euclid's theorem on the infinity of the number of primes will be proved by the method proposed by Leonhard Euler (1707–1783). Euler considered the product over all prime numbers p:
at 
. This product converges, and if it is expanded, then, due to the uniqueness of the decomposition of natural numbers into prime factors, it turns out that it is equal to the sum of the series 
, whence the Euler identity follows:
.
Since at 
series on the right diverges (harmonic series), then the Euler identity implies Euclid's theorem.
Russian mathematician P.L. Chebyshev (1821–1894) derived a formula that determines the limits within which the number of primes is contained 
, not exceeding X:
,
Where 
,
.
Have you ever heard that mathematics is called "the queen of all sciences"? Do you agree with this statement? As long as mathematics remains a boring textbook puzzle for you, you can hardly feel the beauty, versatility and even humor of this science.
But there are topics in mathematics that help to make curious observations on things and phenomena that are common to us. And even try to penetrate the veil of the mystery of the creation of our universe. There are curious patterns in the world that can be described with the help of mathematics.
Introducing Fibonacci Numbers
Fibonacci numbers name the elements of a sequence. In it, each next number in the series is obtained by summing the two previous numbers.
Sample sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…
You can write it like this:
F 0 = 0, F 1 = 1, F n = F n-1 + F n-2, n ≥ 2
You can start a series of Fibonacci numbers with negative values n. Moreover, the sequence in this case is two-sided (that is, it covers negative and positive numbers) and tends to infinity in both directions.
An example of such a sequence: -55, -34, -21, -13, -8, 5, 3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
The formula in this case looks like this:
F n = F n+1 - F n+2 or otherwise you can do it like this: F-n = (-1) n+1 Fn.
What we now know as "Fibonacci numbers" was known to ancient Indian mathematicians long before they were used in Europe. And with this name, in general, one continuous historical anecdote. Let's start with the fact that Fibonacci himself never called himself Fibonacci during his lifetime - this name began to be applied to Leonardo of Pisa only several centuries after his death. But let's talk about everything in order.
Leonardo of Pisa aka Fibonacci
The son of a merchant who became a mathematician, and subsequently received the recognition of his descendants as the first major mathematician of Europe during the Middle Ages. Not least thanks to the Fibonacci numbers (which then, we recall, were not yet called that). Which he described at the beginning of the 13th century in his work “Liber abaci” (“The Book of the Abacus”, 1202).
Traveling with his father to the East, Leonardo studied mathematics with Arab teachers (and in those days they were one of the best specialists in this matter, and in many other sciences). He read the works of mathematicians of Antiquity and Ancient India in Arabic translations.
Having properly comprehended everything he read and connected his own inquisitive mind, Fibonacci wrote several scientific treatises on mathematics, including the “Book of the Abacus” already mentioned above. In addition to her, he created:
- "Practica geometriae" ("Practice of Geometry", 1220);
- "Flos" ("Flower", 1225 - a study on cubic equations);
- "Liber quadratorum" ("The Book of Squares", 1225 - problems on indefinite quadratic equations).
He was a great lover of mathematical tournaments, so in his treatises he paid much attention to the analysis of various mathematical problems.
Very little biographical information remains about Leonardo's life. As for the name Fibonacci, under which he entered the history of mathematics, it was fixed to him only in the 19th century.
Fibonacci and his problems
After Fibonacci, a large number of problems remained, which were very popular among mathematicians in the following centuries. We will consider the problem of rabbits, in the solution of which the Fibonacci numbers are used.
Rabbits are not only valuable fur
Fibonacci set the following conditions: there is a pair of newborn rabbits (male and female) of such an interesting breed that they regularly (starting from the second month) produce offspring - always one new pair of rabbits. Also, as you might guess, male and female.
These conditional rabbits are placed in a closed space and breed enthusiastically. It is also stipulated that no rabbit dies from some mysterious rabbit disease.
We need to calculate how many rabbits we will get in a year.
- At the beginning of 1 month we have 1 pair of rabbits. At the end of the month they mate.
- The second month - we already have 2 pairs of rabbits (a pair has parents + 1 pair - their offspring).
- Third month: The first pair gives birth to a new pair, the second pair mates. Total - 3 pairs of rabbits.
- Fourth month: The first couple gives birth to a new couple, the second couple does not lose time and also gives birth to a new couple, the third couple is just mating. Total - 5 pairs of rabbits.
Number of rabbits in n-th month = number of pairs of rabbits from the previous month + number of newborn pairs (there are the same number of pairs of rabbits 2 months before now). And all this is described by the formula that we have already given above: F n \u003d F n-1 + F n-2.
Thus, we obtain a recurrent (explanation of recursion- below) numerical sequence. In which each next number is equal to the sum of the previous two:
- 1 + 1 = 2
- 2 + 1 = 3
- 3 + 2 = 5
- 5 + 3 = 8
- 8 + 5 = 13
- 13 + 8 = 21
- 21 + 13 = 34
- 34 + 21 = 55
- 55 + 34 = 89
- 89 + 55 = 144
- 144 + 89 = 233
- 233+ 144 = 377 <…>
You can continue the sequence for a long time: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987<…>. But since we have set a specific period - a year, we are interested in the result obtained on the 12th "move". Those. 13th member of the sequence: 377.
The answer is in the problem: 377 rabbits will be obtained if all the stated conditions are met.
One of the properties of the Fibonacci sequence is very curious. If you take two consecutive pairs from a row and divide the larger number by the smaller one, the result will gradually approach golden ratio(You can read more about it later in the article).
In the language of mathematics, "relationship limit a n+1 To a n equal to the golden ratio.
More problems in number theory
- Find a number that can be divided by 7. Also, if you divide it by 2, 3, 4, 5, 6, the remainder will be one.
- Find a square number. It is known about him that if you add 5 to it or subtract 5, you again get a square number.
We invite you to find answers to these questions on your own. You can leave us your options in the comments to this article. And then we will tell you if your calculations were correct.
An explanation about recursion
recursion- definition, description, image of an object or process, which contains this object or process itself. That is, in fact, an object or process is a part of itself.
Recursion finds wide application in mathematics and computer science, and even in art and popular culture.
Fibonacci numbers are defined using a recursive relation. For number n>2 n- e number is (n - 1) + (n - 2).
Explanation of the golden ratio
golden ratio- the division of a whole (for example, a segment) into such parts that are related according to the following principle: a large part belongs to a smaller one in the same way as the entire value (for example, the sum of two segments) to a larger part.
The first mention of the golden ratio can be found in Euclid's treatise "Beginnings" (about 300 BC). In the context of building a regular rectangle.
The term familiar to us in 1835 was introduced by the German mathematician Martin Ohm.
If you describe the golden ratio approximately, it is a proportional division into two unequal parts: approximately 62% and 38%. Numerically, the golden ratio is the number 1,6180339887 .
The golden ratio finds practical application in the visual arts (paintings by Leonardo da Vinci and other Renaissance painters), architecture, cinema (S. Ezenstein's Battleship Potemkin) and other areas. For a long time it was believed that the golden ratio is the most aesthetic proportion. This view is still popular today. Although, according to the results of research, visually, most people do not perceive such a proportion as the most successful option and consider it too elongated (disproportionate).
- Cut length With = 1, A = 0,618, b = 0,382.
- Attitude With To A = 1, 618.
- Attitude With To b = 2,618
Now back to the Fibonacci numbers. Take two successive terms from its sequence. Divide the larger number by the smaller and get approximately 1.618. And now let's use the same larger number and the next member of the series (i.e., an even larger number) - their ratio is early 0.618.
Here is an example: 144, 233, 377.
233/144 = 1.618 and 233/377 = 0.618
By the way, if you try to do the same experiment with numbers from the beginning of the sequence (for example, 2, 3, 5), nothing will work. Almost. The golden ratio rule is almost not respected for the beginning of the sequence. But on the other hand, as you move along the row and the numbers increase, it works fine.
And in order to calculate the entire series of Fibonacci numbers, it is enough to know three members of the sequence, following each other. You can see for yourself!
Golden Rectangle and Fibonacci Spiral
Another curious parallel between the Fibonacci numbers and the golden ratio allows us to draw the so-called "golden rectangle": its sides are related in the proportion of 1.618 to 1. But we already know what the number 1.618 is, right?
For example, let's take two consecutive terms of the Fibonacci series - 8 and 13 - and build a rectangle with the following parameters: width = 8, length = 13.
And then we break the large rectangle into smaller ones. Mandatory condition: the lengths of the sides of the rectangles must correspond to the Fibonacci numbers. Those. the side length of the larger rectangle must be equal to the sum of the sides of the two smaller rectangles.
The way it is done in this figure (for convenience, the figures are signed in Latin letters).
By the way, you can build rectangles in the reverse order. Those. start building from squares with a side of 1. To which, guided by the principle voiced above, figures with sides equal to the Fibonacci numbers are completed. Theoretically, this can be continued indefinitely - after all, the Fibonacci series is formally infinite.
If we connect the corners of the rectangles obtained in the figure with a smooth line, we get a logarithmic spiral. Rather, its special case is the Fibonacci spiral. It is characterized, in particular, by the fact that it has no boundaries and does not change shape.
Such a spiral is often found in nature. Mollusk shells are one of the most striking examples. Moreover, some galaxies that can be seen from Earth have a spiral shape. If you pay attention to weather forecasts on TV, you may have noticed that cyclones have a similar spiral shape when shooting them from satellites.
It is curious that the DNA helix also obeys the golden section rule - the corresponding pattern can be seen in the intervals of its bends.
Such amazing “coincidences” cannot but excite the minds and give rise to talk about a certain single algorithm that all phenomena in the life of the Universe obey. Now do you understand why this article is called that way? And the doors to what amazing worlds can mathematics open for you?
Fibonacci numbers in nature
The connection between Fibonacci numbers and the golden ratio suggests curious patterns. So curious that it is tempting to try to find sequences like Fibonacci numbers in nature and even in the course of historical events. And nature indeed gives rise to such assumptions. But can everything in our life be explained and described using mathematics?
Examples of wildlife that can be described using the Fibonacci sequence:
- the order of arrangement of leaves (and branches) in plants - the distances between them are correlated with Fibonacci numbers (phyllotaxis);
- the location of sunflower seeds (the seeds are arranged in two rows of spirals twisted in different directions: one row is clockwise, the other is counterclockwise);
- location of scales of pine cones;
- flower petals;
- pineapple cells;
- the ratio of the lengths of the phalanges of the fingers on the human hand (approximately), etc.
Problems in combinatorics
Fibonacci numbers are widely used in solving problems in combinatorics.
Combinatorics- this is a branch of mathematics that deals with the study of a selection of a given number of elements from a designated set, enumeration, etc.
Let's look at examples of combinatorics tasks designed for the high school level (source - http://www.problems.ru/).
Task #1:
Lesha climbs a ladder of 10 steps. He jumps up either one step or two steps at a time. In how many ways can Lesha climb the stairs?
The number of ways that Lesha can climb the stairs from n steps, denote and n. Hence it follows that a 1 = 1, a 2= 2 (after all, Lesha jumps either one or two steps).
It is also agreed that Lesha jumps up the stairs from n > 2 steps. Suppose he jumped two steps the first time. So, according to the condition of the problem, he needs to jump another n - 2 steps. Then the number of ways to complete the climb is described as a n-2. And if we assume that for the first time Lesha jumped only one step, then we will describe the number of ways to finish the climb as a n–1.
From here we get the following equality: a n = a n–1 + a n–2(looks familiar, doesn't it?).
Since we know a 1 And a 2 and remember that there are 10 steps according to the condition of the problem, calculate in order all a n: a 3 = 3, a 4 = 5, a 5 = 8, a 6 = 13, a 7 = 21, a 8 = 34, a 9 = 55, a 10 = 89.
Answer: 89 ways.
Task #2:
It is required to find the number of words with a length of 10 letters, which consist only of the letters "a" and "b" and should not contain two letters "b" in a row.
Denote by a n number of words long n letters that consist only of the letters "a" and "b" and do not contain two letters "b" in a row. Means, a 1= 2, a 2= 3.
In sequence a 1, a 2, <…>, a n we will express each next term in terms of the previous ones. Therefore, the number of words of length n letters that also do not contain a doubled letter "b" and begin with the letter "a", this a n–1. And if the word is long n letters begins with the letter "b", it is logical that the next letter in such a word is "a" (after all, there cannot be two "b" according to the condition of the problem). Therefore, the number of words of length n letters in this case, denoted as a n-2. In both the first and second cases, any word (of length n - 1 And n - 2 letters respectively) without doubled "b".
We were able to explain why a n = a n–1 + a n–2.
Let's calculate now a 3= a 2+ a 1= 3 + 2 = 5, a 4= a 3+ a 2= 5 + 3 = 8, <…>, a 10= a 9+ a 8= 144. And we get the familiar Fibonacci sequence.
Answer: 144.
Task #3:
Imagine that there is a tape divided into cells. It goes to the right and lasts indefinitely. Place a grasshopper on the first cell of the ribbon. On whichever of the cells of the tape he is, he can only move to the right: either one cell, or two. How many ways are there for a grasshopper to jump from the beginning of the ribbon to n th cell?
Let us denote the number of ways the grasshopper moves along the tape up to n th cell as a n. In this case a 1 = a 2= 1. Also in n + 1-th cell the grasshopper can get either from n th cell, or by jumping over it. From here n + 1 = a n – 1 + a n. Where a n = F n – 1.
Answer: F n – 1.
You can create similar problems yourself and try to solve them in math lessons with your classmates.
Fibonacci numbers in popular culture
Of course, such an unusual phenomenon as the Fibonacci numbers cannot but attract attention. There is still something attractive and even mysterious in this strictly verified pattern. It is not surprising that the Fibonacci sequence somehow “lit up” in many works of modern mass culture of various genres.
We will tell you about some of them. And you try to look for yourself more. If you find it, share it with us in the comments - we are also curious!
- Fibonacci numbers are mentioned in Dan Brown's bestseller The Da Vinci Code: the Fibonacci sequence serves as the code by which the main characters of the book open the safe.
- In the 2009 American film Mr. Nobody, in one of the episodes, the address of the house is part of the Fibonacci sequence - 12358. In addition, in another episode, the main character must call the phone number, which is essentially the same, but slightly distorted (an extra number after the number 5) sequence: 123-581-1321.
- In the 2012 TV series The Connection, the main character, an autistic boy, is able to discern patterns in the events taking place in the world. Including through the Fibonacci numbers. And manage these events also through numbers.
- The developers of the java-game for Doom RPG mobile phones placed a secret door on one of the levels. The code that opens it is the Fibonacci sequence.
- In 2012, the Russian rock band Splin released a concept album called Illusion. The eighth track is called "Fibonacci". In the verses of the leader of the group Alexander Vasiliev, the sequence of Fibonacci numbers is beaten. For each of the nine consecutive members, there is a corresponding number of rows (0, 1, 1, 2, 3, 5, 8, 13, 21):
0 Set off on the road
1 Clicked one joint
1 One sleeve trembled
2 Everything, get the staff
Everything, get the staff
3 Request for boiling water
The train goes to the river
The train goes to the taiga<…>.
- limerick (a short poem of a certain form - usually five lines, with a certain rhyming scheme, comic in content, in which the first and last lines are repeated or partially duplicate each other) by James Lyndon also uses a reference to the Fibonacci sequence as a humorous motif:
Dense food of the Fibonacci wives
It was only for their benefit, not otherwise.
The wives weighed, according to rumor,
Each is like the previous two.
Summing up
We hope that we were able to tell you a lot of interesting and useful things today. For example, you can now look for the Fibonacci spiral in the nature around you. Suddenly, it is you who will be able to unravel the "secret of life, the universe and in general."
Use the formula for Fibonacci numbers when solving problems in combinatorics. You can build on the examples described in this article.
site, with full or partial copying of the material, a link to the source is required.
There are still many unsolved mysteries in the universe, some of which scientists have already been able to identify and describe. The Fibonacci numbers and the golden ratio form the basis for unraveling the world around us, building its shape and optimal visual perception by a person, with the help of which he can feel beauty and harmony.
golden ratio
The principle of determining the size of the golden section underlies the perfection of the whole world and its parts in its structure and functions, its manifestation can be seen in nature, art and technology. The doctrine of the golden ratio was founded as a result of research by ancient scientists on the nature of numbers.
It is based on the theory of the proportions and ratios of segment divisions, which was made by the ancient philosopher and mathematician Pythagoras. He proved that when dividing a segment into two parts: X (smaller) and Y (larger), the ratio of the larger to the smaller will be equal to the ratio of their sum (of the entire segment):
The result is an equation: x 2 - x - 1=0, which is solved as x=(1±√5)/2.
If we consider the ratio 1/x, then it is equal to 1,618…
Evidence of the use of the golden ratio by ancient thinkers is given in the book of Euclid's "Beginnings", written back in the 3rd century. BC, who used this rule to construct regular 5-gons. Among the Pythagoreans, this figure is considered sacred, since it is both symmetrical and asymmetrical. The pentagram symbolized life and health.
Fibonacci numbers
The famous book Liber abaci by the Italian mathematician Leonardo of Pisa, who later became known as Fibonacci, was published in 1202. In it, the scientist for the first time gives a pattern of numbers, in a series of which each number is the sum of the 2 previous digits. The sequence of Fibonacci numbers is as follows:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc.
The scientist also cited a number of patterns:
- Any number from the series, divided by the next, will be equal to a value that tends to 0.618. Moreover, the first Fibonacci numbers do not give such a number, but as you move from the beginning of the sequence, this ratio will be more and more accurate.
- If you divide the number from the series by the previous one, then the result will tend to 1.618.
- One number divided by the next one will show a value tending to 0.382.
The application of the connection and patterns of the golden section, the Fibonacci number (0.618) can be found not only in mathematics, but also in nature, in history, in architecture and construction, and in many other sciences.
Spiral of Archimedes and golden rectangle
Spirals, very common in nature, were explored by Archimedes, who even derived her equation. The shape of the spiral is based on the laws of the golden ratio. When it is untwisted, a length is obtained to which proportions and Fibonacci numbers can be applied, the step increase occurs evenly.
The parallel between the Fibonacci numbers and the golden ratio can also be seen by constructing a "golden rectangle" whose sides are proportional as 1.618:1. It is built by moving from a larger rectangle to smaller ones so that the lengths of the sides will be equal to the numbers from the row. Its construction can be done in the reverse order, starting with the square "1". When connecting the corners of this rectangle with lines in the center of their intersection, a Fibonacci or logarithmic spiral is obtained.
The history of the use of golden proportions
Many ancient architectural monuments of Egypt were erected using golden proportions: the famous pyramids of Cheops and others. The architects of Ancient Greece widely used them in the construction of architectural objects, such as temples, amphitheatres, stadiums. For example, such proportions were used in the construction of the ancient Parthenon temple (Athens) and other objects that became masterpieces of ancient architecture, demonstrating harmony based on mathematical regularity.
In later centuries, interest in the golden ratio subsided, and the patterns were forgotten, but again resumed in the Renaissance, along with the book of the Franciscan monk L. Pacioli di Borgo "Divine Proportion" (1509). It included illustrations by Leonardo da Vinci, who fixed the new name "golden section". Also, 12 properties of the golden ratio were scientifically proven, and the author talked about how it manifests itself in nature, in art and called it "the principle of building the world and nature."
Vitruvian Man Leonardo
The drawing by which Leonardo da Vinci illustrated the book of Vitruvius in 1492 depicts a figure of a man in 2 positions with arms extended to the sides. The figure is inscribed in a circle and a square. This drawing is considered to be the canonical proportions of the human body (male), described by Leonardo based on their study in the treatises of the Roman architect Vitruvius.
The center of the body as an equidistant point from the end of the arms and legs is the navel, the length of the arms is equal to the height of a person, the maximum width of the shoulders = 1/8 of the height, the distance from the top of the chest to the hair = 1/7, from the top of the chest to the top of the head = 1/6 etc.
Since then, the drawing has been used as a symbol showing the internal symmetry of the human body.
The term "Golden Ratio" was used by Leonardo to denote proportional relationships in the human figure. For example, the distance from the waist to the feet is related to the same distance from the navel to the top of the head in the same way as the height to the first length (from the waist down). This calculation is done similarly to the ratio of the segments when calculating the golden ratio and tends to 1.618.
All these harmonious proportions are often used by artists to create beautiful and impressive works.
Studies of the golden ratio in the 16th-19th centuries
Using the golden ratio and Fibonacci numbers, research work on the issue of proportions has been going on for more than one century. In parallel with Leonardo da Vinci, the German artist Albrecht Dürer was also developing the theory of the correct proportions of the human body. For this, he even created a special compass.
In the 16th century the question of the connection between the Fibonacci number and the golden section was devoted to the work of the astronomer I. Kepler, who first applied these rules to botany.
A new "discovery" awaited the golden ratio in the 19th century. with the publication of "Aesthetic Research" by the German scientist Professor Zeisig. He raised these proportions to the absolute and announced that they are universal for all natural phenomena. He conducted studies of a huge number of people, or rather their bodily proportions (about 2 thousand), as a result of which conclusions were drawn about statistically confirmed patterns in the ratios of various parts of the body: the length of the shoulders, forearms, hands, fingers, etc.
Art objects (vases, architectural structures), musical tones, sizes when writing poems were also studied - Zeisig displayed all this through the lengths of segments and numbers, he also introduced the term "mathematical aesthetics". After receiving the results, it turned out that the Fibonacci series is obtained.
Fibonacci number and golden ratio in nature
In the plant and animal world, there is a tendency to form in the form of symmetry, which is observed in the direction of growth and movement. The division into symmetrical parts in which golden proportions are observed is a pattern inherent in many plants and animals.
The nature around us can be described using Fibonacci numbers, for example:
- the arrangement of leaves or branches of any plants, as well as the distances, are related to the series of given numbers 1, 1, 2, 3, 5, 8, 13 and so on;
- sunflower seeds (scales on cones, pineapple cells), arranged in two rows in twisted spirals in different directions;
- the ratio of the length of the tail and the entire body of the lizard;
- the shape of the egg, if you draw a line conditionally through its wide part;
- the ratio of the size of the fingers on the human hand.
And, of course, the most interesting forms are the spiraling snail shells, the patterns on the web, the movement of the wind inside a hurricane, the double helix in DNA, and the structure of galaxies - all of which include a sequence of Fibonacci numbers.
The use of the golden ratio in art
Researchers looking for examples of the use of the golden section in art examine in detail various architectural objects and paintings. Famous sculptural works are known, the creators of which adhered to golden proportions - the statues of Olympian Zeus, Apollo Belvedere and
One of the creations of Leonardo da Vinci - "Portrait of Mona Lisa" - has been the subject of research by scientists for many years. They found that the composition of the work entirely consists of "golden triangles", united together into a regular pentagon-star. All the works of da Vinci are evidence of how deep his knowledge of the structure and proportions of the human body was, thanks to which he was able to catch the incredibly mysterious smile of the Mona Lisa.
The golden ratio in architecture
As an example, scientists studied architectural masterpieces created according to the rules of the "golden section": the Egyptian pyramids, the Pantheon, the Parthenon, Notre Dame de Paris Cathedral, St. Basil's Cathedral, etc.
The Parthenon, one of the most beautiful buildings in Ancient Greece (5th century BC), has 8 columns and 17 on different sides, the ratio of its height to the length of the sides is 0.618. The protrusions on its facades are made according to the "golden section" (photo below).
One of the scientists who invented and successfully applied the improvement of the modular system of proportions for architectural objects (the so-called "modulor") was the French architect Le Corbusier. The modulor is based on a measuring system associated with a conditional division into parts of the human body.
The Russian architect M. Kazakov, who built several residential buildings in Moscow, as well as the buildings of the Senate in the Kremlin and the Golitsyn Hospital (now the 1st Clinical named after N.I. Pirogov), was one of the architects who used laws in the design and construction about the golden ratio.
Applying proportions in design
In fashion design, all fashion designers make new images and models, taking into account the proportions of the human body and the rules of the golden ratio, although by nature not all people have ideal proportions.
When planning landscape design and creating voluminous park compositions with the help of plants (trees and shrubs), fountains and small architectural objects, the patterns of "divine proportions" can also be applied. After all, the composition of the park should be focused on creating an impression on the visitor, who will be able to freely navigate in it and find the compositional center.
All elements of the park are in such proportions that, with the help of geometric structure, mutual arrangement, lighting and light, they give an impression of harmony and perfection to a person.
Application of the golden section in cybernetics and technology
The laws of the golden section and Fibonacci numbers are also manifested in energy transitions, in processes occurring with elementary particles that make up chemical compounds, in space systems, in the DNA gene structure.
Similar processes occur in the human body, manifesting itself in the biorhythms of his life, in the action of organs, for example, the brain or vision.
Algorithms and patterns of golden proportions are widely used in modern cybernetics and informatics. One of the simple tasks that beginner programmers are given to solve is to write a formula and determine the sum of Fibonacci numbers up to a certain number using programming languages.
Modern research on the theory of the golden ratio
Since the middle of the 20th century, interest in the problems and influence of the laws of the golden proportions on human life has increased dramatically, and from many scientists of various professions: mathematicians, ethnos researchers, biologists, philosophers, medical workers, economists, musicians, etc.
Since the 1970s, The Fibonacci Quarterly has been published in the United States, where works on this topic are published. Works appear in the press in which the generalized rules of the golden section and the Fibonacci series are used in various branches of knowledge. For example, for coding information, chemical research, biological, etc.
All this confirms the conclusions of ancient and modern scientists that the golden ratio is multilaterally connected with the fundamental issues of science and is manifested in the symmetry of many creations and phenomena of the world around us.
