However, this is not all that can be done with the golden ratio. If we divide one by 0.618, we get 1.618; if we square it, we get 2.618; if we cube it, we get 4.236. These are the Fibonacci expansion ratios. The only thing missing here is the number 3,236, which was proposed by John Murphy.
What do experts think about consistency?
Some would say that these numbers are already familiar because they are used in technical analysis programs to determine the magnitude of corrections and extensions. In addition, these same series play an important role in Eliot's wave theory. They are its numerical basis.
Our expert Nikolay is a proven portfolio manager at the Vostok investment company.
- — Nikolay, do you think that the appearance of Fibonacci numbers and its derivatives on the charts of various instruments is accidental? And is it possible to say: “Fibonacci series practical application” takes place?
- — I have a bad attitude towards mysticism. And even more so on stock exchange charts. Everything has its reasons. in the book “Fibonacci Levels” he beautifully described where the golden ratio appears, that he was not surprised that it appeared on stock exchange quote charts. But in vain! In many of the examples he gave, Pi appears frequently. But for some reason it is not included in the price ratios.
- — So you don’t believe in the effectiveness of Eliot’s wave principle?
- - 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, in your opinion, are the reasons for the appearance of the golden ratio on stock charts?
- — The correct answer to this question may earn you the Nobel Prize in Economics. For now we can guess about the true reasons. They are clearly not in harmony with nature. There are many models of exchange pricing. They do not explain the designated phenomenon. But not understanding the nature of a phenomenon should not deny the phenomenon as such.
- — And if this law is ever opened, will it be able to destroy the exchange process?
- — As the same wave theory shows, the law of changes 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.
Material provided by webmaster Maxim's blog.
The coincidence of the fundamental principles of mathematics in a variety of theories seems incredible. Maybe it's fantasy or customized for the final result. Wait and see. Much of what was previously considered unusual or was not possible: 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 over time. What was previously unnecessary will, in the hands of an experienced analyst, become a powerful tool for predicting future behavior.
Fibonacci numbers in nature.
Look
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 previous two. 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 when dividing other pairs decreased from the first to the last, which allows us to say that if this series continues indefinitely, then we will get a number equal to the golden ratio.
Thus, Fibonacci numbers do not stand out in any way. There are other sequences of numbers, of which there are an infinite number, that as a result of the same operations give the golden number phi.
Fibonacci was not an esotericist. He didn't want to put any mysticism into the numbers, he was simply solving an ordinary problem about rabbits. And he wrote a sequence of numbers that followed from his problem, in the first, second and other months, how many rabbits there would be after breeding. Within a year, he received that same sequence. And I didn't do a relationship. There was no talk of any golden proportion or divine relation. All this was invented after him during the Renaissance.
Compared to mathematics, the advantages of Fibonacci 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. It disappeared after the French Revolution. Fibonacci has nothing to do with the golden ratio.
There are an infinite number of spirals, the most popular are: the natural logarithm spiral, the Archimedes spiral, and the hyperbolic spiral.
Now let's take a look at the Fibonacci spiral. This piecewise composite unit consists of several quarter 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, such practice exists.
Huge masses of people act according to the Fibonacci line, which is found in many user terminals. Therefore, whether we like it or not: Fibonacci numbers influence , and we can take advantage of this influence.
Be sure to read the article -.
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Fibonacci numbers and the golden ratio form the basis for understanding the surrounding world, constructing its form and optimal visual perception by a person, with the help of which he can feel beauty and harmony.
The principle of determining the dimensions of the golden ratio 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 proportion was founded as a result of research by ancient scientists into the nature of numbers.
Evidence of the use of the golden ratio by ancient thinkers is given in Euclid’s book “Elements,” written back in the 3rd century. BC, who applied this rule to construct regular pentagons. Among the Pythagoreans, this figure is considered sacred because it is both symmetrical and asymmetrical. The pentagram symbolized life and health.
Fibonacci numbers
The famous book Liber abaci by Italian mathematician Leonardo of Pisa, who later became known as Fibonacci, was published in 1202. In it, the scientist for the first time cites the pattern of numbers, in a series of which each number is the sum of 2 previous digits. The Fibonacci number sequence 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 one will be equal to a value that tends to 0.618. Moreover, the first Fibonacci numbers do not give such a number, but as we move from the beginning of the sequence, this ratio will become more and more accurate.
If you divide the number from the series by the previous one, the result will rush to 1.618.
One number divided by the next by one will show a value tending to 0.382.
The application of the connection and patterns of the golden ratio, the Fibonacci number (0.618) can be found not only in mathematics, but also in nature, history, architecture and construction, and in many other sciences.
For practical purposes, they are limited to the approximate value of Φ = 1.618 or Φ = 1.62. In a rounded percentage value, the golden ratio is the division of any value in the ratio of 62% and 38%.
Historically, the golden section was originally called the division of segment AB by point C into two parts (smaller segment AC and larger segment BC), so that for the lengths of the segments AC/BC = BC/AB was true. In simple words, the golden ratio divides a segment into two unequal parts so that the smaller part is related to the larger one, just as the larger part is related to the entire segment. Later this concept was extended to arbitrary quantities.
The number Φ is also called golden number.
The golden ratio has many wonderful properties, but in addition, many fictitious properties are attributed to it.
Now the details:
The definition of GS is the division of a segment into two parts in such a ratio in which the larger part is related to the smaller one, as their sum (the entire segment) is to the larger one.
That is, if we take the entire segment c as 1, then segment a will be equal to 0.618, segment b - 0.382. Thus, if we take a building, for example, a temple built according to the 3S principle, then with its height, say, 10 meters, the height of the drum with the dome will be equal to 3.82 cm, and the height of the base of the structure will be 6.18 cm (it is clear that the numbers taken flat for clarity)
What is the connection between ZS and Fibonacci numbers?
The Fibonacci sequence numbers are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597…
The pattern of numbers is that each subsequent number is equal to the sum of the two previous numbers.
0 + 1 = 1;
1 + 1 = 2;
2 + 3 = 5;
3 + 5 = 8;
5 + 8 = 13;
8 + 13 = 21, etc.,
and the ratio of adjacent numbers approaches the ratio of ZS.
So, 21: 34 = 0.617, and 34: 55 = 0.618.
That is, the GS is based on the numbers of the Fibonacci sequence.
It is believed that the term “Golden Ratio” was introduced by Leonardo Da Vinci, who said, “let no one who is not a mathematician dare to read my works” and showed the proportions of the human body in his famous drawing “Vitruvian Man”. “If we tie a human figure - the most perfect creation of the Universe - with a belt and then measure the distance from the belt to the feet, then this value will relate to the distance from the same belt to the top of the head, just as the entire height of a person relates to the length from the waist to the feet.”
The Fibonacci number series is visually modeled (materialized) in the form of a spiral.
And in nature, the GS spiral looks like this:
At the same time, the spiral is observed everywhere (in nature and not only):
The seeds in most plants are arranged in a spiral
- The spider weaves a web in a spiral
- A hurricane is spinning like a spiral
- A frightened herd of reindeer scatters in a spiral.
- The DNA molecule is twisted in a double helix. The DNA molecule is made up of two vertically intertwined helices, 34 angstroms long and 21 angstroms wide. The numbers 21 and 34 follow each other in the Fibonacci sequence.
- The embryo develops in a spiral shape
- Cochlear spiral in the inner ear
- The water goes down the drain in a spiral
- Spiral dynamics shows the development of a person’s personality and his values in a spiral.
- And of course, the Galaxy itself has the shape of a spiral
Thus, it can be argued that nature itself is built according to the principle of the Golden Section, which is why this proportion is more harmoniously perceived by the human eye. It does not require “correction” or addition to the resulting picture of the world.
Movie. God's number. Irrefutable proof of God; The number of God. The incontrovertible proof of God.
Golden proportions in the structure of the DNA molecule
All information about the physiological characteristics of living beings is stored in a microscopic DNA molecule, the structure of which also contains the law of the golden proportion. The DNA molecule consists of two vertically intertwined helices. The length of each of these spirals is 34 angstroms and the width is 21 angstroms. (1 angstrom is one hundred millionth of a centimeter).
21 and 34 are numbers following each other in the sequence of Fibonacci numbers, that is, the ratio of the length and width of the logarithmic spiral of the DNA molecule carries the formula of the golden ratio 1:1.618
Golden ratio in the structure of microcosms
Geometric shapes are not limited to just a triangle, square, pentagon or hexagon. If we connect these figures with each other in different ways, we get new three-dimensional geometric figures. Examples of this are figures such as a cube or a pyramid. However, besides them, there are also other three-dimensional figures that we have not encountered in everyday life, and whose names we hear, perhaps for the first time. Among such three-dimensional figures are the tetrahedron (regular four-sided figure), octahedron, dodecahedron, icosahedron, etc. The dodecahedron consists of 13 pentagons, the icosahedron consists of 20 triangles. Mathematicians note that these figures are mathematically very easily transformed, and their transformation occurs in accordance with the formula of the logarithmic spiral of the golden ratio.
In the microcosm, three-dimensional logarithmic forms built according to golden proportions are ubiquitous. For example, many viruses have the three-dimensional geometric shape of an icosahedron. Perhaps the most famous of these viruses is the Adeno virus. The protein shell of the Adeno virus is formed from 252 units of protein cells arranged in a specific sequence. At each corner of the icosahedron there are 12 units of protein cells in the shape of a pentagonal prism and spike-like structures extend from these corners.
The golden ratio in the structure of viruses was first discovered in the 1950s. scientists from Birkbeck College London A. Klug and D. Kaspar. 13 The Polyo virus was the first to display a logarithmic form. The form of this virus turned out to be similar to the form of the Rhino 14 virus.
The question arises, how do viruses form such complex three-dimensional shapes, the structure of which contains the golden ratio, which are quite difficult to construct even with our human mind? The discoverer of these forms of viruses, virologist A. Klug, gives the following comment:
“Dr. Kaspar and I showed that for the spherical shell of the virus, the most optimal shape is symmetry such as the icosahedron shape. This order minimizes the number of connecting elements... Most of Buckminster Fuller's geodesic hemispherical cubes are built on a similar geometric principle. 14 Installation of such cubes requires an extremely accurate and detailed explanatory diagram. Whereas unconscious viruses themselves construct such a complex shell from elastic, flexible protein cellular units.”
About numbers and formulas that occur in nature. Well, a few words about these same numbers and formulas.
Numbers and formulas in nature are a stumbling block between those who believe in the creation of the universe by someone and those who believe in the creation of the universe itself. Because the question is: “If the universe arose on its own, then wouldn’t almost all living and inanimate objects be built according to the same scheme, according to the same formulas?”
Well, we won’t answer this philosophical question here (the site’s format is not the same 🙂), but we’ll voice the formulas. And let's start with the Fibonacci and Golden Spiral numbers.
Thus, Fibonacci numbers are elements of a number sequence in which each subsequent number is equal to the sum of the two previous numbers. That is, 0 +1=1, 1+1=2, 2+1=3, 3+2=5 and so on.
Total, we get the series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946
Another example of the Fibonacci series: 0, 2, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178 and so on. You can experiment yourself :)
How do Fibonacci numbers appear in nature? Very simple:
- The leaf arrangement of plants is described by the Fibonacci sequence. Sunflower seeds, pine cones, flower petals, and pineapple cells are also arranged according to the Fibonacci sequence.
- The lengths of the phalanges of human fingers are approximately the same as the Fibonacci numbers.
- The DNA molecule is made up of two vertically intertwined helices, 34 angstroms long and 21 angstroms wide. The numbers 21 and 34 follow each other in the Fibonacci sequence.
Using Fibonacci numbers you can build a Golden Spiral. So, let's draw a small square with a side of, say, 1. Next, let's remember school. What is 1 2? This will be 1. So, let's draw another square next to the first one, right next to it. Next, the next Fibonacci number is 2 (1+1). What is 2 2? This will be 4. Let's draw another square close to the first two squares, but now with a side of 2 and an area of 4. The next number is the number 3 (1+2). The square of the number 3 is 9. Draw a square with side 3 and area 9 next to the ones already drawn. Next we have a square with side 5 and area 25, a square with side 8 and area 64 - and so on, ad infinitum.
It's time for the golden spiral. Let's connect the border points between the squares with a smooth curved line. And we will get that same golden spiral, on the basis of which many living and inanimate objects in nature are built.
And before moving on to the golden ratio, let’s think. Here we have built a spiral based on the squares of the Fibonacci sequence (sequence 1, 1, 2, 3, 5, 8 and squares 1, 1, 4, 9, 25, 64). But what happens if we use not the squares of numbers, but their cubes? The cubes will look like this from the center:
And on the side:
Well, when constructing a spiral, it will turn out volumetric golden spiral:
This is what this voluminous golden spiral looks like from the side:
But what if we don’t take cubes of Fibonacci numbers, but move to the fourth dimension?.. This is a puzzle, right?
However, I have no idea how the volumetric golden ratio manifests itself in nature based on the cubes of Fibonacci numbers, much less numbers to the fourth power. Therefore, we return to the golden ratio on the plane. So, let's look at our squares again. Mathematically speaking, this is the picture we get:
That is, we get the golden ratio - where one side is divided into two parts in such a ratio that the smaller part is related to the larger one as the larger one is to the entire value.
That is, a: b = b: c or c: b = b: a.
On the basis of this ratio of magnitudes, among other things, a regular pentagon and a pentagram are built:
For reference: to build a pentagram you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Durer (1471...1528). Let O be the center of the circle, A a point on the circle, and E the midpoint of segment OA. The perpendicular to the radius OA, restored at point O, intersects the circle at point D. Using a compass, plot the segment CE = ED on the diameter. The side length of a regular pentagon inscribed in a circle is equal to DC. We plot the segments DC on the circle and get five points to draw a regular pentagon. We connect the corners of the pentagon through one another with diagonals and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.
In general, these are the patterns. Moreover, there are many more diverse patterns than have been described. And now, after all these boring numbers - the promised video clip, where everything is simple and clear:
As you can see, mathematics is indeed present in nature. And not only in the objects listed in the video, but also in many other areas. For example, when a wave hits the shore and spins, it spins along the Golden Spiral. And so on :)
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 being raised on a farm, all of which are considered females and the males are ignored. Rabbits begin breeding after they are two months old and then give birth to a rabbit every month. Rabbits never die.
We need to determine how many rabbits will be on the farm in n months, if at the initial time there was only one newborn rabbit.
Obviously, the farmer has one rabbit in the first month and one rabbit in the second month. By the third month there will be two rabbits, by the fourth month there will be three, etc. Let us denote the number of rabbits in n month like . Thus, 
,
,
,
,
,
…
It is possible to construct an algorithm to find 
at any n.
According to the problem statement, the total number of rabbits 
V n+1 month is divided into three components:
one-month-old rabbits incapable of reproducing, in the amount of
;
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 we accept 
And 
, then using formula (8.1) you 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 in steps of one step, or in steps 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. For n= 3 there are 3 options: three single steps, or one single and one double, or one double and one single.
In the following 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 the question asked at random n, let's denote the number of options as 
, and let's try to determine 
according to known 
And 
. If we start with 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. Total number of options for n+1 steps equals
. (8.2)
The resulting formula resembles formula (8.1) as a twin. However, this does not allow us to identify the number of combinations 
with Fibonacci numbers 
. We see, for example, that 
, But 
. However, the following dependence takes place:
.
This is true for n= 1, 2, and also true for everyone 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 several zeros in a row. Let's denote this number by 
. Obviously, 
, and words of length 2 that satisfy our constraint are: 10, 01, 11, i.e. 
. Let 
- such a word from n characters. If the symbol 
, That 
can be arbitrary ( 
)-literal word that does not contain several zeros in a row. This means that the number of words ending in one is 
.
If the symbol 
, then definitely 
, and the first 
symbol 
may be arbitrary, subject to the constraints considered. Therefore, there is 
words length
n with a zero at the end. Thus, the total number of words we are interested in is equal to
.
Considering 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 several zeros in a row.
You can think like this. Let 
the number of zeros in the words in question. Any word has 
spaces between 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, we get a word of length 
containing 
zeros. Any such word can be obtained in the indicated way from some (and only one) k-literal word containing 
zeros, no two of which are adjacent. This means that 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 
equal to Fibonacci numbers for any integer 
. Symbol 
stands for smallest integer greater than or equal to
. For example, if 
, That 
; what if 
, That 
ceil("ceiling"). There is also a symbol 
, which denotes largest integer less than or equal to
. In English this operation is called floor
("floor").
If 
, That 
. If 
, That 
. If 
, That 
.
Thus, for the cases considered, the sum is indeed equal to the Fibonacci numbers. Now we present the proof for the general case. Since the Fibonacci numbers can be obtained using the recurrence equation (8.1), the equality must be satisfied:
.
And it actually works:
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. General formula for determining the sum of the first n Fibonacci numbers have the form:
Let's prove this using the method of mathematical induction. To do this, let's write:
This amount should be equal 
.
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 will 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 1.618 (if we ignore the first few values). This property of Fibonacci numbers resembles the terms of a geometric progression. Let's accept 
,
(
). Then the expression
converted to
which after simplifications looks like this
.
We have obtained a quadratic equation whose roots are equal:
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 recurrence equation:
For n=0 this difference gives 
, that is: 
. However, when n=1 we have 
. To get 
, you must accept: 
.
Now we have two sequences: 
And 
, which start with the same two numbers and satisfy the same recurrence formula. They must be equal: 
. The theorem has been proven.
When increasing n member 
becomes very large while 
, and the role of the member 
the difference is reduced. Therefore, at large n we can approximately write
.
We ignore 1/2 (since Fibonacci numbers increase to infinity as n ad infinitum).
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 ratio, 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. First class numbers are called simple, and the second – composite. Prime numbers within the first three tens: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
The properties of prime numbers and their relationship with all natural numbers were studied by Euclid (3rd century BC). If you write down prime numbers in a row, you will notice that their relative density decreases. For the first ten there are 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 numbers, the difference between which is equal to two (the so-called simple twins), however, 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 the factors). Thus, prime numbers form a multiplicative basis of the natural series.
The study of the distribution of prime numbers led to the creation of an algorithm that allows one to obtain tables of prime numbers. Such an algorithm is sieve of Eratosthenes(3rd century BC). This method consists of eliminating (for example, by striking out) those integers of a given sequence 
, which are divisible by at least one of the prime numbers smaller 
.
Theorem 8 . 2 . (Euclidean theorem). The number of prime numbers is infinite.
Proof. We will prove Euclid’s theorem on the infinity of the number of prime numbers using 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 
, from which Euler’s identity follows:
.
Since when 
the series on the right diverges (harmonic series), then Euclid’s theorem follows from Euler’s identity.
Russian mathematician P.L. Chebyshev (1821–1894) derived a formula that determines the limits within which the number of prime numbers lies 
, not exceeding X:
,
Where 
,
.
based on materials from the book by B. Biggs “A hedger emerged from the fog”
About Fibonacci numbers and trading
As an introduction to the topic, let's briefly turn to technical analysis. In short, technical analysis aims to predict the future price movement of an asset based on past historical data. The most famous formulation of its supporters is that the price already includes all the necessary information. The implementation of technical analysis began with the development of stock market speculation and is probably not completely finished yet, since it potentially promises unlimited earnings. The most well-known methods (terms) in technical analysis are support and resistance levels, Japanese candlesticks, figures foreshadowing a price reversal, etc.
The paradox of the situation, in my opinion, lies in the following - most of the described methods have become so widespread that, despite the lack of evidence base on their effectiveness, they actually have the opportunity to influence market behavior. Therefore, even skeptics who use fundamental data should take these concepts into account simply because so many other players (“techies”) take them into account. Technical analysis can work well on history, but in practice almost no one manages to make stable money with its help - it is much easier to get rich by publishing a book in large quantities on “how to become a millionaire using technical analysis”...
In this sense, the Fibonacci theory stands apart, which is also used to predict prices for different periods. Her followers are usually called "wavers." It stands apart because it did not appear simultaneously with the market, but much earlier - as much as 800 years. Another feature of it is that the theory is reflected almost as a world concept for describing everything and everyone, and the market is only a special case for its application. The effectiveness of the theory and the period of its existence provide it with both new supporters and new attempts to create the least controversial and generally accepted description of the behavior of markets on its basis. But alas, the theory has not advanced beyond individual successful market predictions, which can be equated to luck.
The essence of Fibonacci theory
Fibonacci lived a long life, especially for his time, which he devoted to solving a number of mathematical problems, formulating them in his voluminous work “The Book of Abacus” (early 13th century). He was always interested in the mysticism of numbers - he was probably no less brilliant than Archimedes or Euclid. Problems related to quadratic equations were posed and partially solved before Fibonacci, for example by the famous Omar Khayyam, a scientist and poet; however, Fibonacci formulated the problem of the reproduction of rabbits, the conclusions from which brought him something that allowed his name not to be lost in the centuries.
Briefly, the task is as follows. A pair of rabbits was placed in a place fenced on all sides by a wall, and any pair of rabbits gives birth to another pair every month, starting from the second month of their existence. The reproduction of rabbits over time will be described by the sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, etc. From a mathematical point of view, the sequence turned out to be simply unique, since it had a number of outstanding properties:
the sum of any two consecutive numbers is the next number in the sequence;
the ratio of each number in the sequence, starting from the fifth, to the previous one is 1.618;
the difference between the square of any number and the square of a number two positions to the left will be the Fibonacci number;
the sum of the squares of adjacent numbers will be the Fibonacci number, which is two positions after the largest of the squared numbers
Of these findings, the second is the most interesting because it uses the number 1.618, known as the “golden ratio.” This number was known to the ancient Greeks, who used it during the construction of the Parthenon (by the way, according to some sources, the Central Bank served the Greeks). No less interesting is that the number 1.618 can be found in nature on both micro and macro scales - from the spiral turns on a snail’s shell to the large spirals of cosmic galaxies. The pyramids at Giza, created by the ancient Egyptians, also contained several parameters of the Fibonacci series during construction. A rectangle, one side of which is 1.618 times larger than the other, looks most pleasing to the eye - this ratio was used by Leonardo da Vinci for his paintings, and in a more everyday sense it was sometimes used when creating windows or doorways. Even a wave, as in the figure at the beginning of the article, can be represented as a Fibonacci spiral.
In living nature, the Fibonacci sequence appears no less often - it can be found in claws, teeth, sunflowers, spider webs and even the growth of bacteria. If desired, consistency is found in almost everything, including the human face and body. And yet, it is believed that many of the claims that find Fibonacci numbers in natural and historical phenomena are incorrect - this is a common myth that often turns out to be an inaccurate fit to the desired result.
Fibonacci numbers in financial markets
One of the first who was most closely involved in the application of Fibonacci numbers to the financial market was R. Elliot. His work was not in vain in the sense that market descriptions using Fibonacci theory are often called “Elliott waves.” The development of markets here was based on the model of human development from supercycles with three steps forward and two steps back. The fact that humanity is developing nonlinearly is obvious to almost everyone - the knowledge of Ancient Egypt and the atomistic teaching of Democritus was completely lost in the Middle Ages, i.e. after about 2000 years; The 20th century gave rise to such horror and insignificance of human life that it was difficult to imagine even in the era of the Punic Wars of the Greeks. However, even if we accept the theory of steps and their number as truth, the size of each step remains unclear, which makes Elliott waves comparable to the predictive power of heads and tails. The starting point and the correct calculation of the number of waves were and apparently will be the main weakness of the theory.
Nevertheless, the theory had local successes. Bob Pretcher, who can be considered a student of Elliott, correctly predicted the bull market of the early 1980s and saw 1987 as the turning point. This actually happened, after which Bob obviously felt like a genius - at least in the eyes of others, he certainly became an investment guru. Prechter's Elliott Wave Theorist subscription grew to 20,000 that year.however, it decreased in the early 1990s, as the further predicted "doom and gloom" of the American market decided to hold off a little. However, it worked for the Japanese market, and a number of supporters of the theory, who were “late” there for one wave, lost either their capital or the capital of their companies’ clients. In the same way and with the same success, they often try to apply the theory to trading in the foreign exchange market.
The theory covers a variety of trading periods - from weekly, which makes it similar to standard technical analysis strategies, to calculations for decades, i.e. gets into the territory of fundamental predictions. This is possible by varying the number of waves. The weaknesses of the theory, which were mentioned above, allow its adherents to speak not about the inconsistency of the waves, but about their own miscalculations among them and an incorrect definition of the starting position. It's like a labyrinth - even if you have the right map, you can only follow it if you understand exactly where you are. Otherwise the card is of no use. In the case of Elliott waves, there is every sign of doubting not only the correctness of your location, but also the accuracy of the map as such.
Conclusions
The wave development of humanity has a real basis - in the Middle Ages, waves of inflation and deflation alternated with each other, when wars gave way to a relatively calm peaceful life. The observation of the Fibonacci sequence in nature, at least in some cases, also does not raise doubts. Therefore, everyone has the right to give their own answer to the question of who God is: a mathematician or a random number generator. My personal opinion is that although all of human history and markets can be represented in the wave concept, the height and duration of each wave cannot be predicted by anyone.
At the same time, 200 years of observing the American market and more than 100 years of other markets make it clear that the stock market is growing, going through various periods of growth and stagnation. This fact is quite enough for long-term earnings in the stock market, without resorting to controversial theories and trusting them with more capital than should be within reasonable risks.
