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 . In this way,
,
,
,
,
, …

We can construct an algorithm to find for any n.

According to the condition of the problem, the total number of rabbits
in 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 a 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
, then
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
, then
; what if
, then
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 a
, then
. If a
, then
. If a
, then
.

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 proven.

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
,
.

Ecology of life. Cognitively: Nature (including Man) develops according to the laws that are laid down in this numerical sequence...

Fibonacci numbers - a numerical sequence where each subsequent member of the series is equal to the sum of the two previous ones, that is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, .. 75025, .. 3478759200, 5628750625, .. 260993908980000, .. 422297015649625, .. 19581068021641812000, .. studies a variety of professional scientists and amateurs of mathematics.

In 1997, several strange features of the series were described by the researcher Vladimir Mikhailov, who was convinced that Nature (including Man) develops according to the laws that are laid down in this numerical sequence.

A remarkable property of the Fibonacci number series is that as the numbers of the series increase, the ratio of two neighboring members of this series asymptotically approaches the exact proportion of the Golden Section (1: 1.618) - the basis of beauty and harmony in the nature around us, including in human relations.

Note that Fibonacci himself discovered his famous series, reflecting on the problem of the number of rabbits that should be born from one pair within one year. It turned out that in each subsequent month after the second, the number of pairs of rabbits exactly follows the digital series, which now bears his name. Therefore, it is no coincidence that man himself is arranged according to the Fibonacci series. Each organ is arranged according to internal or external duality.

Fibonacci numbers have attracted mathematicians because of their ability to appear in the most unexpected places. It has been noticed, for example, that the ratios of Fibonacci numbers, taken through one, correspond to the angle between adjacent leaves on the stem of plants, more precisely, they say what proportion of the turn this angle is: 1/2 - for elm and linden, 1/3 - for beech, 2/5 - for oak and apple, 3/8 - for poplar and rose, 5/13 - for willow and almond, etc. You will find the same numbers when counting seeds in sunflower spirals, in the number of rays reflected from two mirrors, in the number of options for crawling bees from one cell to another, in many mathematical games and tricks.

What is the difference between the Golden Ratio Spirals and the Fibonacci Spiral? The golden ratio spiral is perfect. It corresponds to the Primary source of harmony. This spiral has neither beginning nor end. She is endless. The Fibonacci spiral has a beginning, from which it starts “unwinding”. This is a very important property. It allows Nature, after the next closed cycle, to carry out the construction of a new spiral from “zero”.

It should be said that the Fibonacci spiral can be double. There are numerous examples of these double helixes found all over the place. So, sunflower spirals always correlate with the Fibonacci series. Even in an ordinary pinecone, you can see this double Fibonacci spiral. The first spiral goes in one direction, the second - in the other. If we count the number of scales in a spiral rotating in one direction and the number of scales in the other spiral, we can see that these are always two consecutive numbers of the Fibonacci series. The number of these spirals is 8 and 13. There are pairs of spirals in sunflowers: 13 and 21, 21 and 34, 34 and 55, 55 and 89. And there are no deviations from these pairs!..

In Man, in the set of chromosomes of a somatic cell (there are 23 pairs of them), the source of hereditary diseases are 8, 13 and 21 pairs of chromosomes ...

But why does this series play a decisive role in Nature? The concept of triplicity, which determines the conditions for its self-preservation, can give an exhaustive answer to this question. If the "balance of interests" of the triad is violated by one of its "partners", the "opinions" of the other two "partners" must be corrected. The concept of triplicity manifests itself especially clearly in physics, where “almost” all elementary particles were built from quarks. If we recall that the ratios of fractional charges of quark particles make up a series, and these are the first members of the Fibonacci series, which are necessary for the formation of other elementary particles.

It is possible that the Fibonacci spiral can also play a decisive role in the formation of the pattern of limitedness and closedness of hierarchical spaces. Indeed, imagine that at some stage of evolution, the Fibonacci spiral has reached perfection (it has become indistinguishable from the golden section spiral) and for this reason the particle must be transformed into the next “category”.

These facts once again confirm that the law of duality gives not only qualitative but also quantitative results. They make us think that the Macrocosm and the Microcosm around us evolve according to the same laws - the laws of hierarchy, and that these laws are the same for living and inanimate matter.

All this indicates that a series of Fibonacci numbers is a kind of encrypted law of nature.

The digital code for the development of civilization can be determined using various methods in numerology. For example, by converting complex numbers to single digits (for example, 15 is 1+5=6, etc.). Carrying out a similar addition procedure with all the complex numbers of the Fibonacci series, Mikhailov received the following series of these numbers: 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 8, 1, 9, then everything repeats 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 4, 8, 8, .. and repeats again and again... This series also has the properties of the Fibonacci series, each infinitely subsequent term is equal to the sum of the previous ones. For example, the sum of the 13th and 14th terms is 15, i.e. 8 and 8=16, 16=1+6=7. It turns out that this series is periodic, with a period of 24 terms, after which the whole order of numbers is repeated. Having received this period, Mikhailov put forward an interesting assumption - Isn't a set of 24 digits a kind of digital code for the development of civilization? published

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Fibonacci sequence, known to everyone from the film "The Da Vinci Code" - a series of numbers described as a riddle by the Italian mathematician Leonardo of Pisa, better known by the nickname Fibonacci, in the 13th century. Briefly, the essence of the riddle:

Someone placed a pair of rabbits in a certain closed space to find out how many pairs of rabbits would be born during the year, if the nature of the rabbits is such that every month a pair of rabbits produces another pair, and the ability to produce offspring appears on reaching two months old.

The result is a series of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 , where the number of pairs of rabbits in each of the twelve months is shown, separated by commas. It can be continued indefinitely. Its essence is that each next number is the sum of the previous two.

This series has several mathematical features that must be touched upon. It 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.

So the ratio of any member of the series to the one preceding it fluctuates around the number 1,618 , sometimes surpassing it, sometimes not reaching it. The ratio to the following similarly approaches the number 0,618 , which is inversely proportional 1,618 . If we divide the elements through one, then we get the numbers 2,618 and 0,382 , which are also inversely proportional. These are the so-called Fibonacci ratios.

Why all this? So we are approaching one of the most mysterious phenomena of nature. The savvy Leonardo, in fact, did not discover anything new, he simply reminded the world of such a phenomenon as Golden Section, which is not inferior in importance to the Pythagorean theorem.

We distinguish all the objects around us, including in form. We like some more, some less, some completely repulse the eye. Sometimes interest can be dictated by a life situation, and sometimes by the beauty of the observed object. The symmetrical and proportional shape contributes to the best visual perception and evokes a sense of beauty and harmony. A holistic image always consists of parts of different sizes, which are in a certain relationship with each other and the whole. golden ratio- the highest manifestation of the perfection of the whole and its parts in science, art and nature.

If on a simple example, then the Golden Section is the division of a segment into two parts in such a ratio in which the larger part relates to the smaller one, as their sum (the entire segment) to the larger one.

If we take the entire segment c per 1 , then the segment a will be equal to 0,618 , line segment b - 0,382 , only in this way the condition of the Golden Section will be met (0,618/0,382=1,618 ; 1/0,618=1,618 ) . Attitude c to a equals 1,618 , a With to b 2,618 . These are all the same, already familiar to us, Fibonacci coefficients.

Of course, there is a golden rectangle, a golden triangle, and even a golden cuboid. The proportions of the human body in many respects are close to the Golden Section.

Image: marcus-frings.de

But the most interesting begins when we combine the knowledge gained. The figure clearly shows the relationship between the Fibonacci sequence and the Golden Ratio. We start with two squares of the first size. From above we add a square of the second size. We paint next to a square with a side equal to the sum of the sides of the previous two, the third size. By analogy, a square of the fifth size appears. And so on until you get bored, the main thing is that the length of the side of each next square is equal to the sum of the lengths of the sides of the two previous ones. We see a series of rectangles whose side lengths are Fibonacci numbers, and oddly enough they are called Fibonacci rectangles.

If we draw a smooth line through the corners of our squares, we get nothing more than an Archimedes spiral, the increase in the pitch of which is always uniform.

Doesn't it remind you of anything?

A photo: ethanhein on Flickr

And not only in the shell of a mollusk you can find the spirals of Archimedes, but in many flowers and plants, they are just not so obvious.

Aloe multileaf:

A photo: brewbooks on Flickr

And then it's time to remember the Golden Section! Are any of the most beautiful and harmonious creations of nature depicted in these photographs? And that's not all. Looking closely, you can find similar patterns in many forms.

Of course, the statement that all these phenomena are built on the Fibonacci sequence sounds too loud, but the trend is on the face. And besides, she herself is far from perfect, like everything else in this world.

There is speculation that the Fibonacci series is nature's attempt to adapt to a more fundamental and perfect golden section logarithmic sequence, which is practically the same, just starts from nowhere and goes nowhere. Nature, on the other hand, definitely needs some kind of whole beginning, from which you can push off, it cannot create something out of nothing. The ratios of the first members of the Fibonacci sequence are far from the Golden Section. But the further we move along it, the more these deviations are smoothed out. To determine any series, it is enough to know three of its members, going one after another. But not for the golden sequence, two are enough for it, it is a geometric and arithmetic progression at the same time. You might think that it is the basis for all other sequences.

Each member of the golden logarithmic sequence is a power of the Golden Ratio ( z). Part of the row looks something like this: ... z -5 ; z-4; z-3; z-2; z -1 ; z0; z1; z2; z3; z4; z 5 ... If we round the value of the Golden Ratio to three decimal places, we get z=1.618, then the row looks like this: ... 0,090 0,146; 0,236; 0,382; 0,618; 1; 1,618; 2,618; 4,236; 6,854; 11,090 ... Each next term can be obtained not only by multiplying the previous one by 1,618 , but also by adding the two previous ones. Thus, exponential growth is achieved by simply adding two neighboring elements. This is a series without beginning and end, and it is precisely this that the Fibonacci sequence tries to be like. Having a well-defined beginning, it strives for the ideal, never reaching it. That is life.

And yet, in connection with everything seen and read, quite natural questions arise:
Where did these numbers come from? Who is this architect of the universe who tried to make it perfect? Was it ever the way he wanted it to be? And if so, why did it fail? Mutations? Free choice? What will be next? Is the coil twisting or untwisting?

Finding the answer to one question, you get the next. If you solve it, you get two new ones. Deal with them, three more will appear. Having solved them, you will acquire five unresolved ones. Then eight, then thirteen, 21, 34, 55...

Sources: ; ; ;

STATE EDUCATIONAL INSTITUTION

"KRIVLYANSKAYA SECONDARY SCHOOL"

ZHABINKO DISTRICT

FIBONACCI NUMBERS AND THE GOLDEN RATIO

Research work

Work completed:

10th grade student

Gardener Valeria Alekseevna

Supervisor:

Lavrenyuk Larisa Nikolaevna,

computer science teacher and

mathematics 1 qualifying

Fibonacci numbers and nature

A characteristic feature of the structure of plants and their development is helicity. Even Goethe, who was not only a great poet, but also a naturalist, considered helicity to be one of the characteristic features of all organisms, a manifestation of the innermost essence of life. The tendrils of plants twist in a spiral, tissue grows in a spiral in tree trunks, seeds in a sunflower are arranged in a spiral, spiral movements (nutations) are observed during the growth of roots and shoots.

At first glance, it may seem that the number of leaves, flowers can vary over a very wide range and take on any values. But such a conclusion turns out to be untenable. Studies have shown that the number of organs of the same name in plants is not arbitrary, there are values ​​that are often found and values ​​that are very rare.

In wildlife, forms based on pentagonal symmetry are widespread - starfish, sea urchins, flowers.

Photo 13. Buttercup

A chamomile has 55 or 89 petals.

Photo 14. Chamomile

Feverfew has 34 petals.

Phot. fifteen. Pyrethrum

Let's look at a pine cone. The scales on its surface are arranged in a strictly regular manner - along two spirals that intersect approximately at a right angle. The number of such spirals in pine cones is 8 and 13 or 13 and 21.

Photo 16. Cone

In sunflower baskets, seeds are also arranged in two spirals, their number is usually 34/55, 55/89.

Photo 17. Sunflower

Let's take a look at shells. If we count the number of "stiffening ribs" for the first shell taken at random - it turned out to be 21. Let's take the second, third, fifth, tenth shell - all will have 21 ribs on the surface. It can be seen that the mollusks were not only good engineers, they "knew" the Fibonacci numbers.

Photo 18. Shell

Here again we see a regular combination of Fibonacci numbers located side by side: 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89. Their ratio in the limit tends to the golden ratio, expressed by the number 0.61803 ...

Fibonacci numbers and animals

The number of rays in starfish corresponds to a series of Fibonacci numbers or very close to them and is equal to 5.8, 13.21.34.55.

Photo 19. Starfish

Modern arthropods are very diverse. The spiny lobster also has five pairs of legs, five feathers on the tail, the abdomen is divided into five segments, and each leg consists of five parts.

Phot. twenty. spiny lobster

In some insects, the abdomen consists of eight segments, there are three pairs of limbs, consisting of eight parts, and eight different antennae-like organs emerge from the mouth opening. Our well-known mosquito has three pairs of legs, the abdomen is divided into eight segments, and there are five antennae on the head. The mosquito larva is divided into 12 segments.

Phot. 21. Mosquito

In a cabbage fly, the abdomen is divided into five parts, there are three pairs of legs, and the larva is divided into eight segments. Each of the two wings is divided into eight parts by thin veins.

The caterpillars of many insects are divided into 13 segments, for example, in the skin-eater, flour-eater, Mauritanian booger. In most pest beetles, the caterpillar is divided into 13 segments. The structure of the legs of beetles is very characteristic. Each leg consists of three parts, as in higher animals - from the shoulder, forearm and paw. Thin, openwork paws of beetles are divided into five parts.

Openwork, transparent, weightless dragonfly wings are a masterpiece of "engineering" skill of nature. What proportions underlie the design of this tiny flying muscle car? The ratio of wingspan to body length in many dragonflies is 4/3. The body of a dragonfly is divided into two main parts: a massive body and a long thin tail. The body is divided into three parts: head, thorax, abdomen. The abdomen is divided into five segments, and the tail consists of eight parts. Here it is still necessary to add three pairs of legs with their division into three parts.

Phot. 22. Dragonfly

It is easy to see in this sequence of dividing the whole into parts the expansion of a series of Fibonacci numbers. The length of the tail, body and the total length of the dragonfly are interconnected by the golden ratio: the ratio of the lengths of the tail and body is equal to the ratio of the total length to the length of the tail.

It is not surprising that the dragonfly looks so perfect, because it is created according to the laws of the golden ratio.

The sight of a turtle against the background of a cracked takyr is an amazing phenomenon. In the center of the carapace there is a large oval field with large fused horny plates, and along the edges there is a border of smaller plates.

Phot. 23. Turtle

Take any turtle - from the swamp turtle close to us to the giant sea, soup turtle - and you will see that the pattern on the shell is similar: on the oval field there are 13 fused horn plates - 5 plates in the center and 8 - along the edges, and on the peripheral border about 21 plates (the Chilean tortoise has exactly 21 plates along the periphery of the shell). Turtles have 5 fingers on their paws, and the vertebral column consists of 34 vertebrae. It is easy to see that all of these quantities correspond to the Fibonacci numbers. Consequently, the development of the turtle, the formation of its body, the division of the whole into parts was carried out according to the law of a series of Fibonacci numbers.

Mammals are the highest type of animals on the planet. The number of ribs in many animal species is equal to or close to thirteen. In completely different mammals - a whale, a camel, a deer, a tour - the number of ribs is 13 ± 1. The number of vertebrae varies greatly, especially due to the tails, which can be of different lengths even in the same animal species. But in many of them the number of vertebrae is equal to or close to 34 and 55. So, 34 vertebrae in a giant deer, 55 in a whale.

The skeleton of the limbs of domestic animals consists of three identical bone links: the humerus (pelvic) bone, the bone of the forearm (shin) and the bone of the paw (foot). The foot, in turn, consists of three bone links.

The number of teeth in many domestic animals tends to Fibonacci numbers: a rabbit has 14 pairs, a dog, a pig, a horse has 21 ± 1 pairs of teeth. In wild animals, the number of teeth varies more widely: in one marsupial predator it is 54, in a hyena - 34, in one of the species of dolphins it reaches 233. The total number of bones in the skeleton of domestic animals (including teeth) in one group is close to 230, and the other - to 300. It should be noted that the small auditory ossicles and non-permanent ossicles are not included in the number of bones of the skeleton. Taking them into account, the total number of skeletal bones in many animals will become close to 233, while in others it will exceed 300. As you can see, the division of the body, accompanied by the development of the skeleton, is characterized by a discrete change in the number of bones in various organs of animals, and these numbers correspond to Fibonacci numbers or very close to them, forming a series of 3, 5, 8, 13, 21, 34, 55, 89, 144, 233. The size ratio for most chicken eggs is 4:3 (for some 3/2), pumpkin seeds - 3:2 , watermelon seeds - 3/2. The ratio of the length of pine cones to their diameter was found to be 2:1. The size of birch leaves on average is very close to, and acorns - 5:2.

It is believed that if it is necessary to divide a flower lawn into two parts (grass and flowers), then these strips should not be made equal in width, it will be more beautiful if you take them in a ratio of 5: 8 or 8: 13, i.e. use a ratio called the golden ratio.

Fibonacci numbers and photography

As applied to photographic art, the golden section rule divides the frame with two horizontal and two vertical lines into 9 unequal rectangles. To make it easier for themselves to shoot balanced images, photographers have simplified the task a little and began to divide the frame into 9 equal rectangles according to the Fibonacci numbers. So the rule of the golden section was transformed into the rule of thirds, which refers to one of the principles of composition.

Phot. 24. Frame and golden ratio

In the viewfinders of modern digital cameras, focus points are located at positions 2/8 or on imaginary lines dividing the frame according to the golden section rule.

Photo 25. Digital camera and focus points

Photo 26.

Photo 27. Photography and focus points

The rule of thirds applies to all subject compositions: you are shooting a landscape or a portrait, a still life or a reportage. As long as your sense of harmony has not become acquired and unconscious, following the simple rule of thirds will allow you to take pictures that are expressive, harmonious, balanced.

Photo 28. Photography and the ratio of heaven and earth 1 to 2.

The most successful example for demonstration is the landscape. The principle of composition is that the sky and land (or water surface) should have a ratio of 1:2. One third of the frame should be taken under the sky, and two thirds under the land, or vice versa.

Photo 29. Photo of a flower spiraling

Fibonacci and space

The ratio of water and land on planet Earth is 62% and 38%.

The dimensions of the Earth and the Moon are in the golden ratio.

Photo 30. Dimensions of the Earth and Moon

The figure shows the relative sizes of the Earth and the Moon to scale.

Let's draw the radius of the Earth. Let's draw a segment from the central point of the Earth to the central point of the Moon, the length of which will be equal). Let's draw a line to connect these two lines to form a triangle. We get the golden triangle.

Saturn shows the golden ratio in several of its dimensions

Photo 31. Saturn and its rings

The diameter of Saturn is very close in relation to the golden ratio with the diameter of the rings, as shown by the green lines.Radius inthe inside of the rings is in a ratio very close to the outer diameter of the rings, as shown by the blue line.

The distance of the planets from the Sun also obeys the golden ratio.

Photo 32. Distance of the planets from the Sun

The golden ratio in everyday life

The golden ratio is also used to add style and appeal to the marketing and design of everyday consumer products. There are many examples, but we will illustrate only a few.

Photo 33. EmblemToyota

Photo 34. The golden ratio and clothes

Photo 34. The Golden Ratio and Automotive Design

Photo 35. EmblemApple

Photo 36. EmblemGoogle

Practical research

Now we will apply the knowledge gained in practice. Let's first take measurements among students in grade 8.

If you look closely at the television picture, then its dimensions will be 16 to 9 or 16 to 10, which is also close to the golden ratio.

Carrying out measurements and constructions in CorelDRAW X4 and using a frame from the news channel Russia 24, you can find the following:

a) the ratio of the length to the width of the frame is 1.7.

b) the person in the frame is located exactly at the focus points located at a distance of 3/8.

Next, let's turn to the official microblog of the Izvestia newspaper, in other words, to the Twitter page. For a monitor screen with 4:3 sides, we see that the “header” of the page is 3/8 of the entire height of the page.

Looking closely at the caps of the military, you can find the following:

a) the cap of the Minister of Defense of the Russian Federation has the ratio of the indicated parts 21.73 to 15.52, equal to 1.4.

b) the cap of the border guard of the Republic of Belarus has the dimensions of the indicated parts 44.42 to 21.33, which is equal to 2.1.

c) the cap of the times of the USSR has the dimensions of the indicated parts 49.67 to 31.04, which is equal to 1.6.

For this model, the length of the dress is 113.13 mm.

If you “finish” the dress to the “ideal” length, we get this picture.

All measurements have some error, since they were taken from a photograph, which does not prevent us from seeing a trend - everything that is ideal contains the golden ratio to one degree or another.

Conclusion

The world of wildlife appears to us in a completely different way - mobile, changeable and surprisingly diverse. Life shows us a fantastic carnival of diversity and originality of creative combinations! The world of inanimate nature is, first of all, a world of symmetry, which gives stability and beauty to his creations. The world of nature is, first of all, a world of harmony, in which the "law of the golden section" operates.

“The Golden Ratio” appears to be that moment of truth, without which, in general, anything that exists is not possible. Whatever we take as an element of research, the "golden section" will be everywhere; even if there is no visible observance of it, then it necessarily takes place at the energy, molecular or cellular levels.

Indeed, nature turns out to be monotonous (and therefore uniform!) in the manifestation of its fundamental laws. The "most successful" solutions she found apply to the most diverse objects, to the most diverse forms of organization. The continuity and discreteness of the organization comes from the dual unity of matter - its corpuscular and wave nature, penetrates into chemistry, where it gives the laws of integer stoichiometry, chemical compounds of constant and variable composition. In botany, continuity and discreteness find their specific expression in phyllotaxis, discreteness quanta, growth quanta, unity of discreteness and continuity of space-time organization. And now, in the numerical ratios of plant organs, the “principle of multiple ratios” introduced by A. Gursky appears - a complete repetition of the basic law of chemistry.

Of course, the statement that all these phenomena are built on the Fibonacci sequence sounds too loud, but the trend is clear. And besides, she herself is far from perfect, like everything else in this world.

There is speculation that the Fibonacci series is nature's attempt to adapt to a more fundamental and perfect golden section logarithmic sequence, which is practically the same, just starts from nowhere and goes nowhere. Nature, on the other hand, definitely needs some kind of whole beginning, from which you can push off, it cannot create something out of nothing. The ratios of the first members of the Fibonacci sequence are far from the Golden Section. But the further we move along it, the more these deviations are smoothed out. To determine any series, it is enough to know three of its members, going one after another. But not for the golden sequence, two are enough for it, it is a geometric and arithmetic progression at the same time. You might think that it is the basis for all other sequences.

Each member of the golden logarithmic sequence is a degree of the Golden Ratio (). Part of the row looks something like this:... ; ; ; ; ; ; ; ; ; ; ... If we round the value of the Golden Ratio to three decimal places, we get=1,618 , then the row looks like this:... 0,090 0,146; 0,236; 0,382; 0,618; 1; 1,618; 2,618; 4,236; 6,854; 11,090 ... Each next term can be obtained not only by multiplying the previous one by1,618 , but also by adding the two previous ones. Thus, exponential growth is achieved by simply adding two neighboring elements. This is a series without beginning and end, and it is precisely this that the Fibonacci sequence tries to be like. Having a well-defined beginning, it strives for the ideal, never reaching it. That is life.

And yet, in connection with everything seen and read, quite natural questions arise:
Where did these numbers come from? Who is this architect of the universe who tried to make it perfect? Was it ever the way he wanted it to be? And if so, why did it fail? Mutations? Free choice? What will be next? Is the coil twisting or untwisting?

Finding the answer to one question, you get the next. If you solve it, you get two new ones. Deal with them, three more will appear. Having solved them, you will acquire five unresolved ones. Then eight, then thirteen, 21, 34, 55...

List of sources used

Vasyutinskiy, N. Golden proportion / Vasyutinskiy N, Moscow, Young Guard, 1990, - 238 p. - (Eureka).

Vorobyov, N.N. Fibonacci numbers,

Access mode: . Access date: 17.11.2015.

Access mode: . Access date: 16/11/2015.

Access mode: . Access date: 13. 11. 2015.

(Fibonacci numbers, English Fibonacci sequence, Fibonacci numbers) - a series of numbers derived by the famous mathematician Fibonacci. It has the following form: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, etc.

History of the Fibonacci Series

Leonardo from Pisa (Fibonacci) came to mathematics because of the practical need to establish business contacts. In his youth, Fibonacci traveled a lot, accompanied his father on various business trips, which allowed him to communicate with local scientists.

The series of numbers that today bears his name was derived from a problem with rabbits, which the author described in a book called "Liber abacci" (1202): a man put a pair of rabbits in a pen surrounded on all sides by a wall. Question: how many pairs of rabbits can this pair produce in a year, if it is known that every month, starting from the second month, each pair produces another pair of rabbits.

As a result, Fibonacci determined that the number of pairs of rabbits in each of the next twelve months would be, respectively:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

Where each subsequent number is the sum of the previous two. This is a series (numbers) of Fibonacci. This sequence has many properties that are interesting from a mathematical point of view. For example, if you divide a line into 2 segments so that the ratio between the smaller and larger segment is proportional to the ratio between the larger segment and the entire line, you get a proportionality factor known as the golden ratio. It is approximately equal to 0.618. Renaissance scientists believed that it was this proportion, if observed in architectural structures, that was most pleasing to the eye.

Application of the Fibonacci series

The Fibonacci series has found wide application in various fields of science and life. For example, in nature: in the structure of hurricanes, shells and even galaxies. The Forex currency market was no exception, where a sequential series of numbers began to be used to predict trends. It should be noted that there is an invariable relationship between these numbers. For example, as mentioned above, the ratio of the previous number to the next asymptotically approaches 0.618 (the golden ratio). The ratio of a certain number to the previous one also tends to the value of 0.618.

In addition to predicting trends, Fibonacci numbers in Forex are used to predict the direction of price movement. For example, a trend reversal along the golden ratio occurs at a level of about 61.8% of the previous price change (see Fig. 1). Accordingly, the most profitable option in this case would be to close the position just below this level. Based on the Fibonacci series, you can calculate the most profitable moments for closing and opening deals.

Also, one of the ways to use consecutive numbers of the Fibonacci series in the Forex market is to build arcs. The choice of the center for such an arc occurs at the point of an important bottom or ceiling. The radius of the arcs is calculated by multiplying the Fibonacci ratios by the value of the previous significant rise or fall in prices.

Selectable coefficients are 0.333, 0.382, 0.4, 0.5, 0.6, 0.618, 0.666. The location of the arcs determines their role: support or resistance. To get an idea also about the time of occurrence of price movements, arcs are usually used in conjunction with speed or fan lines.

The principle of their construction is similar: you need to select the points of past extremums and draw a horizontal line from the top of the first of them and a vertical line from the top of the second. Then you should divide the resulting vertical segment into parts corresponding to the coefficients, draw rays coming from the first point through the newly selected ones. When using ratios of 2/3 and 1/3, high-speed lines are obtained, with more stringent 0.618, 0.5 and 0.382 - fan lines. They all serve as support or resistance lines for the price trend (see Figure 2).

Crossings of fan arcs and lines serve as signals for determining the turning points of the trend - both in time and in price.

(Fig. 2 - Fibonacci series, drawing arcs)

More volatile currency pairs are characterized by reaching higher Fibonacci levels compared to less volatile ones. The maximum movements are recorded for the pairs Dollar/Frank and Pound/Dollar, followed by the Dollar/Yen and Euro/Dollar.

The use of the Fibonacci series in the Forex currency market has one feature - they can only be used for good impulse movements.