Finding Fibonacci numbers. Fibonacci numbers and the golden ratio: relationship

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 missing number here is 3,236, which was proposed by John Murphy.

What do experts think about consistency?

Some might 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, the number 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.

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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.

Algorithms,

Mathematics

• Translation

Introduction

Programmers should be fed up with Fibonacci numbers by now. Examples of their calculation are used throughout. This is because these numbers provide the simplest example of recursion. They are also a good example of dynamic programming. But is it necessary to calculate them like this in a real project? No need. Neither recursion nor dynamic programming are ideal options. And not a closed formula using floating point numbers. Now I will tell you how to do it correctly. But first, let's go through all the known solution options.

The code is intended for Python 3, although it should also work with Python 2.

To begin with, let me remind you of the definition:

F n = F n-1 + F n-2

And F 1 = F 2 =1.

Closed formula

We will skip the details, but those interested can familiarize themselves with the derivation of the formula. The idea is to assume that there is some x for which F n = x n and then find x.

What does it mean

Reduce x n-2

Solving the quadratic equation:

This is where the “golden ratio” ϕ=(1+√5)/2 grows. Substituting the original values ​​and doing some more calculations, we get:

Which is what we use to calculate Fn.

From __future__ import division import math def fib(n): SQRT5 = math.sqrt(5) PHI = (SQRT5 + 1) / 2 return int(PHI ** n / SQRT5 + 0.5)

Good:
Fast and easy for small n
The bad:
Floating point operations are required. Large n will require greater precision.
Evil:
Using complex numbers to calculate F n is beautiful from a mathematical point of view, but ugly from a computer point of view.

Recursion

The most obvious solution is one that you've seen many times before, most likely as an example of what recursion is. I will repeat it again for completeness. In Python it can be written in one line:

Fib = lambda n: fib(n - 1) + fib(n - 2) if n > 2 else 1

Good:
A very simple implementation that follows the mathematical definition
The bad:
Exponential execution time. For large n it is very slow
Evil:
Stack Overflow

Memorization

The recursion solution has a big problem: overlapping calculations. When fib(n) is called, fib(n-1) and fib(n-2) are counted. But when fib(n-1) is counted, it will count fib(n-2) again independently - that is, fib(n-2) is counted twice. If we continue the argument, we will see that fib(n-3) will be counted three times, etc. Too many intersections.

Therefore, you just need to remember the results so as not to count them again. This solution consumes time and memory in a linear manner. I'm using a dictionary in my solution, but a simple array could also be used.

M = (0: 0, 1: 1) def fib(n): if n in M: return M[n] M[n] = fib(n - 1) + fib(n - 2) return M[n]

(In Python, this can also be done using the decorator, functools.lru_cache.)

Good:
Just turn recursion into a memory solution. Converts exponential execution time into linear execution, which consumes more memory.
The bad:
Wastes a lot of memory
Evil:
Possible stack overflow, just like recursion

Dynamic programming

After solving with memorization, it becomes clear that we do not need all the previous results, but only the last two. Also, instead of starting from fib(n) and going backwards, you can start from fib(0) and going forward. The following code has linear execution time and fixed memory usage. In practice, the solution speed will be even higher, since there are no recursive function calls and associated work. And the code looks simpler.

This solution is often cited as an example of dynamic programming.

Def fib(n): a = 0 b = 1 for __ in range(n): a, b = b, a + b return a

Good:
Works fast for small n, simple code
The bad:
Still linear execution time
Evil:
Nothing special.

Matrix algebra

And finally, the least illuminated, but the most correct solution, wisely using both time and memory. It can also be extended to any homogeneous linear sequence. The idea is to use matrices. It is enough just to see that

And a generalization of this says that

The two values ​​for x that we obtained earlier, one of which was the golden ratio, are the eigenvalues ​​of the matrix. Therefore, another way to derive a closed formula is to use a matrix equation and linear algebra.

So why is this formulation useful? Because exponentiation can be done in logarithmic time. This is done through squaring. The point is that

Where the first expression is used for even A, the second for odd. All that remains is to organize the matrix multiplications, and everything is ready. This results in the following code. I created a recursive implementation of pow because it is easier to understand. See the iterative version here.

Def pow(x, n, I, mult): """ Returns x to the power of n. Assumes I is the identity matrix being multiplied with mult and n is a positive integer """ if n == 0: return I elif n == 1: return x else: y = pow(x, n // 2, I, mult) y = mult(y, y) if n % 2: y = mult(x, y) return y def identity_matrix(n): """Returns an n by n identity matrix""" r = list(range(n)) return [ for j in r] def matrix_multiply(A, B): BT = list(zip(*B) ) return [ for row_a in A] def fib(n): F = pow([, ], n, identity_matrix(2), matrix_multiply) return F

Good:
Fixed memory size, logarithmic time
The bad:
The code is more complicated
Evil:
You have to work with matrices, although they are not that bad

Performance comparison

It is worth comparing only the variant of dynamic programming and the matrix. If we compare them by the number of characters in the number n, it turns out that the matrix solution is linear, and the solution with dynamic programming is exponential. A practical example is calculating fib(10 ** 6), a number that will have more than two hundred thousand digits.

N=10**6
Calculating fib_matrix: fib(n) has only 208988 digits, the calculation took 0.24993 seconds.
Calculating fib_dynamic: fib(n) has only 208988 digits, the calculation took 11.83377 seconds.

Theoretical Notes

While not directly related to the code above, this remark still has some interest. Consider the following graph:

Let's count the number of paths of length n from A to B. For example, for n = 1 we have one path, 1. For n = 2 we again have one path, 01. For n = 3 we have two paths, 001 and 101 It can be shown quite simply that the number of paths of length n from A to B is exactly equal to Fn. Having written down the adjacency matrix for the graph, we get the same matrix that was described above. It is a well-known result from graph theory that, given an adjacency matrix A, occurrences in A n are the number of paths of length n in the graph (one of the problems mentioned in the movie Good Will Hunting).

Why are there such markings on the ribs? It turns out that when you consider an infinite sequence of symbols on a round-trip infinite sequence of paths on a graph, you get something called "finite type subshifts", which is a type of symbolic dynamics system. This particular subshift of a finite type is known as the “golden ratio shift,” and is specified by a set of “forbidden words” (11). In other words, we will get binary sequences that are infinite in both directions and no pairs of them will be adjacent. The topological entropy of this dynamical system is equal to the golden ratio ϕ. It's interesting how this number appears periodically in different areas of mathematics.

There are still many unsolved mysteries in the universe, some of which scientists have already been able to identify and describe. Fibonacci numbers and the golden ratio form the basis for unraveling the world around us, constructing its form 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 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.

It is based on the theory of proportions and ratios of divisions of segments, 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 (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 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 section, 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.

Archimedes spiral and golden rectangle

Spirals, very common in nature, were studied by Archimedes, who even derived its equation. The shape of the spiral is based on the laws of the golden ratio. When unwinding it, a length is obtained to which proportions and Fibonacci numbers can be applied; the step increases evenly.

The parallel between Fibonacci numbers and the golden ratio can 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 are equal to the numbers from the series. It can also be constructed in reverse order, starting with square “1”. When the corners of this rectangle are connected by lines at the center of their intersection, a Fibonacci or logarithmic spiral is obtained.

History of the use of golden proportions

Many ancient architectural monuments of Egypt were built using golden proportions: the famous pyramids of Cheops, etc. Architects of Ancient Greece widely used them in the construction of architectural objects such as temples, amphitheaters, and stadiums. For example, such proportions were used in the construction of the ancient temple of the Parthenon, (Athens) and other objects that became masterpieces of ancient architecture, demonstrating harmony based on mathematical patterns.

In later centuries, interest in the golden ratio subsided, and the patterns were forgotten, but it resumed again in the Renaissance with the book of the Franciscan monk L. Pacioli di Borgo “The Divine Proportion” (1509). It contained illustrations by Leonardo da Vinci, who established the new name “golden ratio”. 12 properties of the golden ratio were also 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, which Leonardo da Vinci used to illustrate the book of Vitruvius in 1492, depicts a human figure in 2 positions with arms spread 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 studying them 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 the 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.

Leonardo used the term “Golden Ratio” to designate 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 height is to the first length (from the waist down). This calculation is done similarly to the ratio of segments when calculating the golden proportion and tends to 1.618.

All these harmonious proportions are often used by artists to create beautiful and impressive works.

Research on the golden ratio in the 16th to 19th centuries

Using the golden ratio and Fibonacci numbers, research on the issue of proportions has been going on for centuries. In parallel with Leonardo da Vinci, the German artist Albrecht Durer also worked on developing the theory of correct proportions of the human body. For this purpose, he even created a special compass.

In the 16th century The question of the connection between the Fibonacci number and the golden ratio was devoted to the work of 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 the “Aesthetic Investigation” of the German scientist Professor Zeisig. He raised these proportions to absolutes and declared 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), based on the results 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.

Objects of art (vases, architectural structures), musical tones, and sizes when writing poems were also studied - Zeisig displayed all this through the lengths of segments and numbers, and he also introduced the term “mathematical aesthetics.” After receiving the results, it turned out that the Fibonacci series was obtained.

Fibonacci number and the golden ratio in nature

In the plant and animal world there is a tendency towards morphology in the form of symmetry, which is observed in the direction of growth and movement. Division into symmetrical parts in which golden proportions are observed - this pattern is 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 distances, correspond to a 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 along twisted spirals in different directions;
• the ratio of the length of the tail and the entire body of the lizard;
• the shape of an egg, if you draw a line through its wide part;
• ratio of finger sizes on a person's hand.

And, of course, the most interesting shapes include spiraling snail shells, patterns on spider webs, the movement of wind inside a hurricane, the double helix in DNA and the structure of galaxies - all of which involve the Fibonacci sequence.

Use of the golden ratio in art

Researchers searching for examples of the use of the golden ratio in art study in detail various architectural objects and works of art. There are famous sculptural works, the creators of which adhered to golden proportions - statues of Olympian Zeus, Apollo Belvedere and

One of Leonardo da Vinci’s creations, “Portrait of the Mona Lisa,” has been the subject of research by scientists for many years. They discovered that the composition of the work consists entirely of “golden triangles” united together into a regular pentagon-star. All of da Vinci’s works are evidence of how deep his knowledge was in the structure and proportions of the human body, thanks to which he was able to capture the incredibly mysterious smile of Mona Lisa.

Golden ratio in architecture

As an example, scientists examined architectural masterpieces created according to the rules of the “golden ratio”: Egyptian pyramids, Pantheon, 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 ratio” (photo below).

One of the scientists who came up with and successfully applied an improvement to the modular system of proportions for architectural objects (the so-called “modulor”) was the French architect Le Corbusier. The modulator is based on a measuring system associated with the conditional division into parts of the human body.

Russian architect M. Kazakov, who built several residential buildings in Moscow, as well as the Senate building in the Kremlin and the Golitsyn hospital (now the 1st Clinical named after N. I. Pirogov), was one of the architects who used the laws in design and construction about the golden ratio.

Applying proportions in design

In clothing design, all fashion designers create 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 three-dimensional park compositions with the help of plants (trees and shrubs), fountains and small architectural objects, the laws 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 it and find the compositional center.

All elements of the park are in such proportions as to create an impression of harmony and perfection with the help of geometric structure, relative position, illumination and light.

Application of the golden ratio in cybernetics and technology

The laws of the golden section and Fibonacci numbers also appear in energy transitions, in processes occurring with elementary particles that make up chemical compounds, in space systems, and in the genetic structure of DNA.

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 computer science. One of the simple tasks that novice 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 into the theory of the golden ratio

Since the mid-20th century, interest in the problems and influence of the laws of golden proportions on human life has increased sharply, and from many scientists of various professions: mathematicians, ethnic researchers, biologists, philosophers, medical workers, economists, musicians, etc.

In the United States, the magazine The Fibonacci Quarterly began publishing in the 1970s, where works on this topic were published. Works appear in the press in which the generalized rules of the golden ratio and the Fibonacci series are used in various fields of knowledge. For example, for information coding, chemical research, biological research, etc.

All this confirms the conclusions of ancient and modern scientists that the golden proportion is multilaterally related to fundamental issues of science and is manifested in the symmetry of many creations and phenomena of the world around us.

Hello, dear readers!

Golden ratio - what is it? Fibonacci numbers are? The article contains answers to these questions briefly and clearly, in simple words.

These questions have been exciting the minds of more and more generations for several millennia! It turns out that mathematics may not be boring, but exciting, interesting, and fascinating!

Other useful articles:

What are Fibonacci numbers?

The amazing fact is that when dividing each subsequent number in a numerical sequence by the previous one the result is a number tending to 1.618.

A lucky guy discovered this mysterious sequence medieval mathematician Leonardo of Pisa (better known as Fibonacci). Before him Leonardo da Vinci discovered a surprisingly repeating proportion in the structure of the human body, plants and animals Phi = 1.618. Scientists also call this number (1.61) the “Number of God.”

Before Leonardo da Vinci, this sequence of numbers was known in Ancient India and Ancient Egypt. Egyptian pyramids were built using proportions Phi = 1.618.

But that's not all, it turns out laws of nature of the Earth and Space in some inexplicable way they obey strict mathematical laws Fidonacci number sequences.

For example, both a shell on Earth and a galaxy in Space are built using Fibonacci numbers. The vast majority of flowers have 5, 8, 13 petals. In a sunflower, on plant stems, in swirling vortices of clouds, in whirlpools and even in Forex exchange rate charts, Fibonacci numbers work everywhere.

Watch a simple and entertaining explanation of the Fibonacci sequence and the Golden Ratio in this SHORT VIDEO (6 minutes):

What is the Golden Ratio or Divine Proportion?

So, what is the Golden Ratio or Golden or Divine Proportion? Fibonacci also discovered that the sequence that consists of the squares of Fibonacci numbers is an even bigger mystery. Let's try graphically represent the sequence in the form of an area:

1², 2², 3², 5², 8²…

If we inscribe a spiral into a graphical representation of the sequence of squares of Fibonacci numbers, we will get the Golden Ratio, according to the rules of which everything in the universe is built, including plants, animals, the DNA spiral, the human body, ... This list can be continued indefinitely.

Golden ratio and Fibonacci numbers in nature VIDEO

I suggest watching a short film (7 minutes) that reveals some of the mysteries of the Golden Ratio. When thinking about the law of Fibonacci numbers, as the primary law that governs living and inanimate nature, the question arises: Did this ideal formula for the macrocosm and microcosm arise on its own or did someone create it and successfully apply it?

What do you think about it? Let's think about this riddle together and maybe we will get closer to it.

I really hope that the article was useful to you and you learned what is the Golden Ratio * and Fibonacci Numbers? See you again on the blog pages, subscribe to the blog. The subscription form is below the article.

I wish everyone many new ideas and inspiration for their implementation!

Fibonacci sequence, known to everyone from the film "The Da Vinci Code" - a series of numbers described in the form of 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 enclosed space to find out how many pairs of rabbits would be born during the year, if the nature of rabbits is such that every month a pair of rabbits gives birth to another pair, and they become capable of producing offspring when they reach two months of age.

The result is a series of numbers like this: 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 two previous ones.

This series has several mathematical features that definitely need to 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 is impossible to express it precisely.

Thus, the ratio of any member of a series to the one preceding it fluctuates around the number 1,618 , sometimes exceeding it, sometimes not achieving it. The ratio to the following similarly approaches the number 0,618 , which is inversely proportional 1,618 . If we divide the elements by one, we get numbers 2,618 And 0,382 , which are also inversely proportional. These are the so-called Fibonacci ratios.

What is all this for? This is how we approach one of the most mysterious natural phenomena. The savvy Leonardo essentially did not discover anything new, he simply reminded the world of such a phenomenon as Golden Ratio, which is not inferior in importance to the Pythagorean theorem.

We distinguish all the objects around us by their shape. We like some more, some less, some are completely off-putting. Sometimes interest can be dictated by the life situation, and sometimes by the beauty of the observed object. The symmetrical and proportional shape promotes the best visual perception and evokes a feeling of beauty and harmony. A complete image always consists of parts of different sizes that are in a certain relationship with each other and the whole. Golden ratio- the highest manifestation of the perfection of the whole and its parts in science, art and nature.

To use a simple example, the Golden Ratio is the division of a segment into two parts in such a ratio that the larger part is related to the smaller one, as their sum (the entire segment) is to the larger one.

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

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

Image: marcus-frings.de

But the fun begins when we combine the knowledge we have gained. The figure clearly shows the relationship between the Fibonacci sequence and the Golden Ratio. We start with two squares of the first size. Add a square of the second size on top. Draw a square next to it with a side equal to the sum of the sides of the previous two, third size. By analogy, a square of size five appears. And so on until you get tired, the main thing is that the length of the side of each next square is equal to the sum of the lengths of the sides of the previous two. We see a series of rectangles whose side lengths are Fibonacci numbers, and, oddly enough, they are called Fibonacci rectangles.

If we draw smooth lines through the corners of our squares, we will get nothing more than an Archimedes spiral, the increment of which is always uniform.

Doesn't remind you of anything?

Photo: ethanhein on Flickr

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

Aloe multifolia:

Photo: brewbooks on Flickr

Photo: beart.org.uk
Photo: esdrascalderan on Flickr
Photo: manj98 on Flickr

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

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

There is an assumption that the Fibonacci series is an attempt by nature to adapt to a more fundamental and perfect golden ratio logarithmic sequence, which is almost the same, only it starts from nowhere and goes to nowhere. Nature definitely needs some kind of whole beginning from which it can start; it cannot create something out of nothing. The ratios of the first terms of the Fibonacci sequence are far from the Golden Ratio. But the further we move along it, the more these deviations are smoothed out. To define any series, it is enough to know its three terms, coming 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. One might think that it is the basis for all other sequences.

Each term of the golden logarithmic sequence is a power of the Golden Ratio ( z). Part of the series looks something like this: ... z -5 ; z -4 ; z -3 ; z -2 ; z -1 ; z 0 ; z 1 ; z 2 ; z 3 ; z 4 ; z 5... If we round the value of the Golden Ratio to three decimal places, we get z=1.618, then the series looks like this: ... 0,090 0,146; 0,236; 0,382; 0,618; 1; 1,618; 2,618; 4,236; 6,854; 11,090 ... Each next term can be obtained not only by multiplying the previous one by 1,618 , but also by adding the two previous ones. Thus, exponential growth is achieved by simply adding two adjacent elements. It's a series without beginning or end, and that's what the Fibonacci sequence tries to be like. Having a very definite beginning, she strives for the ideal, never achieving it. That is life.

And yet, in connection with everything we have seen and read, quite logical questions arise:
Where did these numbers come from? Who is this architect of the universe who tried to make it ideal? Was everything ever the way he wanted? And if so, why did it go wrong? Mutations? Free choice? What will be next? Is the spiral curling or unwinding?

Having found the answer to one question, you will get the next one. If you solve it, you'll get two new ones. Once you deal with them, three more will appear. Having solved them too, you will have five unsolved ones. Then eight, then thirteen, 21, 34, 55...

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