The golden ratio in human anatomy. Fibonacci Golden Ratio

30.04.2019

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The proportionality of the earth is closely related to the proportionality of man.

As in nature there are five-petalled flowers, plants, the bodies of sea stars, since the human figure also has a five-ray structure: a head, two arms, two legs, a human figure can be inscribed in PENTA WURF, with the following parameters: the posture of a sprawled person with arms spread 180 degrees and legs turned 90 degrees. And although in pentagrams the figures do not fit into ideally equilateral circles a little, nevertheless there is a ratio everywhere - 1.618 - i.e. "Golden ratio"

The golden ratio and its secrets

(a + b) / b = b / a or x 2 -x-1 = 0;

And what happens if you mirror:

1 + x + x2 = x2-x-1;

1+0+0 2 =1;

1+1+1 2 =3

1+2+2 2 =7;

1+3+3 2 =12;

As you can see, with a mirror image, magic numbers 1,3,7,12, creating a series of "lucky numbers".

The golden ratio. Lecture. Part 1.

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How not to admire the device of the eyeball, where the distance to honey is projected by two points at the entrance and exit according to the law of proportional dependence of refraction in the lens and turning it up (real image) down (reflected image) onto the retina, where light-color perceptions of rods and cones encode spatial images outside world and with the help of bioelectrical impulses transmitted to the cerebral cortex. So in reality everything comes to our consciousness in an inverted form and then subconsciously turns over again, creating an imaginary reality.

In the eye there is, as it were, a pyramidal crystal created by rays of light, joined by vertices (time clock).

Proportionality, proportionality, similarity, commensurability.

It turns out that through visual perception the world reflected in our consciousness and this connects the thinking of our "I" by analogy with the real world to those around us. That is, the harmony of nature creates the harmony of consciousness. Disharmony gives rise to anxiety, worries, nervous tension, stress. This proves once again that man is a part of nature. Many scientists, philosophers, assume the awareness of the world as a consequence of the transition of the perception of images through abstraction and schematization, which leads to the logic of comparison, the development of accumulative memory and the emergence of reason, as it is, with the ensuing consequences.

The formation of one person keeps the memory of the formation of all mankind. With countdown meditation, you can achieve a memory of the past, when information is absorbed from each cell of each atom, by analogy with reflecting the essence of being. It was the brain-nature commensuration that gave rise to the concept of microcosm and macrocosm. The simplest comparison of a person's orientation in space is vertical and horizontal, which ideally creates the ubiquitous cross when crossing. But any consciousness cannot think in infinite categories. Reality requires completeness. The most complete relationship between vertical and horizontal is a kind of oblique line joining two lines, this will be the third diagonal, which gives rise to the relation:

a: b = c.

But why not a: a = c? Yes, because the eyeball itself precisely in the ratio of 3: 4: 5 gives rise to this ratio. This relationship is ideal for the physiological perception of the human eye.

The next step in reflecting similarity and commensurability is the realization by humanity of dualism a (duality of similarity). As there are 2 eyes, 2 ears, 2 arms, 2 legs and so on, that is, the cognition of symmetry is mirrored about the axis. This is clearly reflected in these drawings:

As you can see, 2 interconnected squares with a circle inscribed inside give rise to the proportionality of the golden ratio.

And now you will learn a schematic understanding of the birth of a pyramidal crystal from 4 elements:

The adherents of antiquity considered this combination of a square, a circle, and a pyramidal crystal to be a general cosmic phenomenon of harmony, a symbol of the macrocosm, and a person inscribed in these proportions was a symbol of the microcosm.

Subsequently, this attitude was improved by initiations in secret knowledge Agripp Nepeysheimer in the “pentasymmertric law”, where there were 4 elements, 4 squares, combined with a star by 5 primary elements. To some extent, this is a combination of Western and Eastern philosophy, running along the Western and Eastern axes of the world.

Also, the search for a reflection of the microworld was the laws derived by Vitruvius, Leonardo da Vinci, Durer.

If you look closely, the starting points for the new circle and square where the human figure is inscribed are taken from the original 4 squares, divided into 16 parts. This system splitting into equal parts (doubling), corresponds to doubling into fractions,

½, ¼, 1/8, 1/16, 1/32 (setata)

So the secret construction of the connection of 4 elements and 5 primary elements eloquently confirms the direct relationship of the formula of the universe into one whole through the pyramidal crystal.

Example: Ancient adepts in the form of rectangles built along the golden ratio gave their names.

Conclusion: the harmony of man and nature was the main super task, the solution of which was an end in itself for all civilizations. The tools for this were the foundations of global thinking through the interconnection of the macrocosm and the microcosm through the construction of a pyramidal crystal where the elements and primary elements are at the heart.

Great pyramids are in the form of a semi-octahedron, where the lower semi-octahedron is implied. The shape of the upper semi-actahedron of the pyramid is not opposed to the surrounding space, but connects the zenith and the horizontal, unshakably and majestically rising above the earthly space, being an eternal symbol of harmony.

The secret knowledge, uniting the macro and microcosm due to the change of generations of formations, religions, was gradually forgotten, separated and modified, which led to different interpretations the same basics. For example: antique greeks as the Egyptians used decimal (as well as modern society) the number system (10, 100, 1000, and so on), which led to the gradation.

Man - 1, God-Man - 10, Accordingly, the construction of the temple suggested a tenfold increase in the columns, cornices (Parthenon temple) If you look closely, the proportionality of the Greek buildings was a reflection of the harmony of the human body. In Russia, Byzantine masters who came with baptism also introduced ancient traditions and proportions, which was reflected in the architecture and interior.

The Middle Ages introduced a new understanding of the divine, where the root cause of everything (GOD) was presented as an ethereal spirit and invisibly present everywhere. Therefore, changes in the spiritual principle were made in the temples of the Byzantine school. Narrow windows, loopholes, where the rays of the sun fell, under a certain angle, and at a certain time.

The temple of antiquity, filled with sculptures, was replaced by an interior interior of murals that expand the space, that is, creating the illusion of a supernatural space - a part of the divine space - a temple of some unearthly capsule of a spaceship that descended into the world of everyday life.

Indeed, even during big wars, the one who managed to get inside the monastery church was, as it were, guarded by God himself and to punish the church hiding in the church was considered an insult to GOD, sacrilege. And even in modern times many, having committed many sins, try to hide in the church from punishment and atone for their sins there.

The architecture of Christianity, Islam, resembles the images of heroes-warriors defending their faith, that is, the same associative images of God-man, but on a different level than in ancient Greece or Rome. If in ancient Greece temples with a number of columns are a whole galaxy of gods, then Byzantine Christianity landed, grew into the earth, temples became closed in themselves, many loopholes appeared, which speaks of civil strife and long-term wars between different denominations, and although the proportionality classic relationship survived, but the wisdom of the past was dismembered and gradually disunited, fading away in the contradictions of future generations.

The philosophy of the globality of the pyramidal crystal, left over from the supercivilization of the Atlanteans and at first supported by the initiated priests, was gradually forgotten due to lack of education ordinary people and the secrecy of the adepts. The barbarians did not understand the initiates, which led to natural destruction, the inability to understand the meaning of everything and everyone. Humanity is still gradually returning to the concepts of globality, scientific and technical progress modifies concepts, abstracts and accelerates the comprehensiveness of the understanding of the world for one generation of accumulated knowledge of the past. If earlier the movement went from the general to the particular by the method of trial and error, now there is a process from the particular to the general, and the more and deeper knowledge about nature and man, the closer is the approximation to the formula of the universe, knowledge from ancient times encoded in various forms transmitted to us from the past.

The new millennium, where the 21st century is its beginning, and our generation - a monad - a unit (1), the beginning, presents a chance to define a certain global plan - the basis of future generations. Indeed, by and large, the process of global understanding of the understanding of the past by the present for the future has begun.

Even what is still fantastic in practice is already there in theory. As in antiquity, mankind realized, but rather rediscovered by trial and error the concept of universal human values, ecological balance, sustainable development, dead-end and progressive methods of the state, methods of state development, the creation of a global information network philosophy, which appears and comes to the surface from the depths of nothingness, when civilization begins to mature and provide a global nature of changes, harm or benefit brought by the economy, politics, sociology. Only by knowing your mistakes can you become wiser and the 21st century is the era of wise men, especially among the leaders of civilized countries, where a high standard of living contributes to a thrifty attitude towards environment, preservation natural resources, a work of art, ancient monuments, the development of savings systems, the development of computer networks.

The modern world is not an endless space, but a closed organism, where a part can harm the interests of the whole. For example: 1 terrorist attack at a nuclear power plant can destroy hundreds of thousands of people and contaminate a number of states with radionuclides. It was the ancient adepts who supported the theory of globalization and were waiting in the wings.

Each of you can become initiated and enlightened. And what difference does it make in which way you reach the top of the pyramid of true knowledge and skills. You are part of the whole, and you are responsible for it.

When a person is part of a whole, he is not afraid for his fate, because fate leads it by itself, the main thing is not to violate the eco-equilibrium everywhere and everywhere.

The culture of the ancient world is a multi-level system created by the genius of man, who combined various sciences and arts, where the master was at the same time a thinker and nature was the subject for a logical analysis of the habitat, where the categories of space, philosophy of personality, forms and content of matter, spiritual entities based on combinations musical sounds, mathematical numbers, picturesque flowers, sculptural forms, architectural structures and all kinds of miracles created by a creative genius in a creative person.

Until recently, many attempts were made to comprehend the laws of nature differentially, in parts, but the separate perception of physicists, chemists, biologists, mathematicians, natural phenomena led to the fact that the whole was divided into multiple, one-sided professionalism, due to its unnecessary, destroyed the integral perception of the world, and so that to introduce the general law of harmony, it is necessary to find a certain discrete symbol, which is the basis of everything. The particle that everything else is made of. The most suitable for this math modeling any shape through geometric abstraction. Where there is a combination of systematic time and space. Form and content. Statics and dynamics. Clarity and schematization.

But nature itself tells us that the most ideal method that gives life is a dichotomy - a system of splitting the whole into + and -, as well as the reverse phenomenon of merging + and - (male and female cells into one whole).

Examples:

Stage 1:

1. - motherhood

Stage 2 +1/2 and -1/2 - division into 2 parts.

Stage 3: +1/2: 2 = 1/4 and -1/2: 2 = -1 / 4. Dividing 12 again into 2 parts.

Stage 4: + 1/2 + 1/4 = 3/4 A new quality is created that is not similar to stages 1, 2 or 3.

Conclusion: this example shows how from 1 by dividing in half and then joining in different variants gives rise to an infinite difference in combinations.

Let's introduce a mathematical concept: additivity - where in the number series, each previous term of the series is equal to the sum of two consecutive

Drawing

2. Multiplicativity - where, in a number series, all the members of the series are connected in geometric progression.

Drawing

But in nature there is only 1 combination of additivity and multiplicativity of a global nature - this is the ratio: a + b = c (additivity) and a / b = b / c (multiplicativity), in which the whole c is represented by two parts a + b.

Conclusion: But a sign of any self-organized system is:

1. Regularity of growth.

2. Change of parameters.

That is, a single whole consists of two parts in a single organism, all parts grow according to the same law: the law of geometric progression. The idea of similarity appears, which forms the basis for the development of living nature on the basis of which the laws of genetics are based. Mathematically, this equation leads to the "golden ratio", the golden numbers of the descending series. This is 1; 1.618; 2.618; and to the numbers of the ascending series: 1; 0.618, 0.382;

Indeed, taking the integer c as the unit c = 1 from the equation a / b = b / c, we find that a = b 2 from the equation a + b = c

That b2 + b-1 = 0

Drawing

But these relationships are static, and all life is dynamic. Dynamism can be reflected by vector equations, where a vector is the movement of a point in space and time.

In stage 3, the diagonal dichotomy introduced linear incompatibility and inequality of angles.

Drawing

Let's create a "golden chain", model a shape in multidimensional space - time.

When introducing right angle the linear series of the golden section is transformed into a space of similarity symmetry.

Proof:

With the help of the obtained drawings of the asymmetric a-rhombus, the ascending branch of the golden ratio was obtained. A-rhombus is a form that evokes deep associations and interconnections of any life forms, as it has aesthetic and mathematical properties.

After the previous constructions of vector geometry, when proto-eggs, proto-apples, and proto-shells appear from the drawing, I would like to consider the secret of birth from the primary element of a living object. Why does an apple tree grow from an apple seed, and a man grows from a man's seed? Until then, it remains a mystery which bio-fields are embedded in the DNA gene helix, form a given formula, because there is some kind of system, framework, axis, which, as it develops in time, purposefully forms the necessary cells and tissues of living organisms. Since the form is a category of space, then the area of vector geometry can be applied to it, according to the principles of the pyramidal crystal of life, everything develops by analogy (what is above, then below).

Since a person is inherently egocentric, then from antiquity 2 fundamental terms came to us - macrocosm and microcosm (universe and man).

Modern cosmology assumes that by the time the universe was born, its mass, energy and pressure had already been concentrated in vacuum in advance. Something similar exists in the human cell, where all the data of the human genome is concentrated. Now nanotechnology is developing genetic engineering according to cosmological principles "shocks" from "nothing", when everything appeared everywhere and at the same time. Spectral analysis of stars confirms this. Galaxies that are close and distant by billions of light years are homogeneous in composition. According to vector geometry, where the mother's cell is complemented by the father's, a certain form formation occurs, inherent in the formation of these chromosomes, rigidly forming the frame of human likeness.

Samir forms of living nature indicate 2 tendencies of shaping in nature, which is confirmed by the previous geometric constructions:

1. There is a pronounced direction of growth along the up and down axis - the biological vertical (stems of many plants, cereals, various phallic forms

2. There is an expression "the direction of growth to spherical and rounded shapes"(Human skull, apple, egg, orange).

3. There is an interpretation of the flat shape (sunflower disc, Pecten shell leaves, that is, the pyramidal crystal of life)

This harmony is striking in its scale ...

Hello, friends!

Have you heard anything about Divine Harmony or the Golden Ratio? Have you ever thought about why something seems ideal and beautiful to us, but something repels?

If not, then you have successfully ended up on this article, because in it we will discuss the golden ratio, find out what it is, how it looks in nature and in humans. Let's talk about its principles, find out what the Fibonacci series is and much more, including the concept of a golden rectangle and a golden spiral.

Yes, the article contains a lot of images, formulas, after all, the golden ratio is also mathematics. But everything is described enough simple language, clearly. And also, at the end of the article, you will find out why everyone loves cats so much =)

What is the Golden Ratio?

In simple terms, the golden ratio is certain rule proportion that creates harmony? That is, if we do not violate the rules of these proportions, then we get a very harmonious composition.

The most capacious definition of the golden ratio says that the smaller part refers to the larger, as large to the whole whole.

But besides this, the golden ratio is mathematics: it has a specific formula and a specific number. Many mathematicians, in general, consider it a formula of divine harmony, and call it "asymmetric symmetry".

The golden ratio has come down to our contemporaries since the times Ancient Greece however, it is believed that the Greeks themselves had already spied on the golden ratio of the Egyptians. Because many works of art Ancient egypt clearly built according to the canons of this proportion.

It is believed that Pythagoras was the first to introduce the concept of the golden section. The works of Euclid have survived to this day (he built regular pentagons using the golden section, which is why such a pentagon is called "golden"), and the number of the golden section is named after the ancient Greek architect Phidias. That is, this is our number "phi" (denoted by the Greek letter φ), and it is equal to 1.6180339887498948482 ... Naturally, this value is rounded: φ = 1.618 or φ = 1.62, and in percentage terms the golden ratio looks like 62% and 38%.

What is the uniqueness of this proportion (and, believe me, it is)? Let's first try to figure it out on the example of a segment. So, we take a segment and divide it into unequal parts in such a way that its smaller part belongs to the larger one, as large to the whole whole. I understand that it is not very clear yet what is what, I will try to illustrate more clearly using the example of segments:

So, we take a segment and divide it into two others, so that the smaller segment a refers to the larger segment b, in the same way as the segment b refers to the whole, that is, to the entire line (a + b). Mathematically, it looks like this:

This rule works indefinitely, you can divide segments as long as you like. And you see how simple it is. The main thing is to understand once and that's it.

But now consider more complex example, which comes across very often, since the golden ratio is still represented in the form of a golden rectangle (the aspect ratio of which is φ = 1.62). This is a very interesting rectangle: if we "cut off" a square from it, we will again get a golden rectangle. And so many times. See:

But mathematics would not be mathematics if there were no formulas in it. So, friends, now it will be a little "painful". I hid the solution of the golden ratio under the spoiler, there are a lot of formulas, but I don’t want to leave the article without them.

Fibonacci series and the golden ratio

We continue to create and observe the magic of mathematics and the golden ratio. In the Middle Ages, there was such a friend - Fibonacci (or Fibonacci, they write differently everywhere). He loved mathematics and problems, and he had interesting problem with the breeding of rabbits =) But that's not the point. He discovered a numerical sequence, the numbers in it are called "Fibonacci numbers".

The sequence itself looks like this:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 ... and on to infinity.

In words, the Fibonacci sequence is a sequence of numbers, where each subsequent number is equal to the sum of the two previous ones.

What does the golden ratio have to do with it? You will see now.

Fibonacci spiral

To see and feel the full connection between the Fibonacci number series and the golden ratio, you need to look at the formulas again.

In other words, from the 9th term of the Fibonacci sequence, we begin to get the values of the golden ratio. And if we visualize this whole picture, we will see how the Fibonacci sequence creates rectangles closer and closer to the golden rectangle. Here is such a connection.

Now let's talk about the Fibonacci spiral, it is also called the "golden spiral".

The golden spiral is a logarithmic spiral whose growth rate is equal to φ4, where φ is the golden ratio.

All in all, mathematically speaking, the golden ratio is the perfect proportion. But this is where her miracles only begin. Almost the whole world is subordinated to the principles of the golden section, this proportion was created by nature itself. Even esotericists, and those, see numerical power in it. But we will definitely not talk about this in this article, so in order not to miss anything, you can subscribe to site updates.

The golden ratio in nature, man, art

Before we start, I would like to clarify a number of inaccuracies. First, the very definition of the golden ratio in this context is not entirely correct. The fact is that the very concept of "section" is a geometric term, always denoting a plane, but not a sequence of Fibonacci numbers.

And, secondly, the number series and the ratio of one to the other, of course, were turned into a kind of stencil that can be applied to anything that seems suspicious, and you can be very happy when there are coincidences, but nevertheless, you should not lose common sense.

However, "everything is mixed in our kingdom" and one has become synonymous with the other. So in general, the meaning of this is not lost. And now to the point.

You will be surprised, but the golden ratio, or rather the proportions as close as possible to it, can be seen almost everywhere, even in the mirror. Don't believe me? Let's start with this.

You know, when I was learning to draw, they explained to us how easier it is to build a person's face, his body, and so on. Everything must be calculated in relation to something else.

Everything, absolutely everything is proportional: bones, our fingers, palms, distances on the face, distance outstretched arms in relation to the body and so on. But even that's not all internal structure our organism, even it, is equated or almost equated with the golden ratio. Here are the distances and proportions:

from shoulders to crown to head size = 1: 1.618

from the navel to the crown to the segment from the shoulders to the crown = 1: 1.618

from the navel to the knees and from the knees to the feet = 1: 1.618

from the chin to the extreme point of the upper lip and from it to the nose = 1: 1.618

Isn't that amazing !? Harmony in pure form, both inside and outside. And that is precisely why, on some subconscious level, some people do not seem beautiful to us, even if they have a strong toned body, velvet skin, beautiful hair, eyes, and so on and everything else. But, all the same, the slightest violation of the proportions of the body, and the appearance is already slightly "hurts the eyes."

In short, the more beautiful a person seems to us, the closer his proportions are to ideal. And this, by the way, can be attributed not only to the human body.

The golden ratio in nature and its phenomena

The classic example of the golden ratio in nature is the shell of the mollusk Nautilus pompilius and ammonite. But that's not all, there are many more examples:

in the curls of the human ear, we can see a golden spiral;

its own (or close to it) in the spirals along which the galaxies twist;

and in the DNA molecule;

the center of the sunflower is arranged along the Fibonacci row, cones, the middle of flowers, pineapple and many other fruits grow.

Friends, there are so many examples that I will just leave a video here (it is just below) so as not to overload the article with text. Because if you dig this topic, you can delve into such a jungle: the ancient Greeks argued that the Universe and, in general, all space, was planned according to the principle of the golden ratio.

You will be surprised, but these rules can be found even in sound. See:

The highest point of a sound painful and the discomfort in our ears is 130 decibels.

We divide by the proportion 130 by the number of the golden ratio φ = 1.62 and we get 80 decibels - the sound of a human scream.

We continue to divide proportionally and we get, let's say, the normal loudness of human speech: 80 / φ = 50 decibels.

Well, and the last sound that we get thanks to the formula is a pleasant whispering sound = 2.618.

According to this principle, it is possible to determine the optimal-comfortable, minimum and maximum number of temperature, pressure, humidity. I have not tested it, and I do not know how correct this theory is, but, you see, it sounds impressive.

Absolutely in everything alive and not alive, one can read the highest beauty and harmony.

The main thing is just not to get carried away with it, because if we want to see something in something, we will see it, even if it is not there. For example, I drew attention to the design of the PS4 and saw the golden ratio there =) However, this console is so cool that I would not be surprised if the designer was really tricky about it.

The golden ratio in art

This is also a very large and extensive topic that should be considered separately. Here are just a few basic points. The most remarkable thing is that many works of art and architectural masterpieces of antiquity (and not only) are made according to the principles of the golden section.

Egyptian and Mayan pyramids, Notre Dame de Paris, Greek Parthenon and so on.

In the musical works of Mozart, Chopin, Schubert, Bach and others.

In painting (it can be clearly seen there): all the most famous paintings famous artists made taking into account the rules of the golden ratio.

These principles can be found both in Pushkin's poems and in the bust of the beautiful Nefertiti.

Even now, the rules of the golden ratio are used, for example, in photography. And, of course, in all other art, including cinematography and design.

Fibonacci golden cats

And finally, about cats! Have you ever wondered why everyone loves kitties so much? They have flooded the Internet! Seals are everywhere and it's wonderful =)

And the thing is, cats are perfect! Don't believe me? Now I will prove it to you mathematically!

See? The secret is revealed! Cats are ideal from the point of view of mathematics, nature and the Universe =)

* I'm kidding, of course. No, cats are really perfect) But no one measured them mathematically, probably.

On this, in general, everything, friends! We will see you in the next articles. Good luck to you!

P. S. Images taken from medium.com.

The human body and the golden ratio.

Artists, scientists, fashion designers, designers make their calculations, drawings or sketches based on the ratio of the golden ratio. They use measurements from the human body, also created according to the principle of the golden ratio. Leonardo Da Vinci and Le Corbusier, before creating their masterpieces, took the parameters of the human body, created according to the law of the Golden Ratio.

The most main book of all modern architects, E. Neufert's reference book "Building Design" contains the basic calculations of the parameters of the human body, containing the golden proportion.

Proportions different parts our body is a number that is very close to the golden ratio. If these proportions coincide with the formula of the golden ratio, then the appearance or body of a person is considered perfectly folded. The principle of calculating the golden measure on the human body can be depicted as a diagram:

M / m = 1.618

The first example of the golden ratio in the structure of the human body:
If we take the navel point as the center of the human body, and the distance between a person's feet and the navel point as a unit of measurement, then a person's height is equivalent to 1.618.

In addition, there are several more basic golden proportions of our body:

* the distance from the fingertips to the wrist to the elbow is 1: 1.618;

* distance from shoulder level to crown of head and head size is 1: 1.618;

* the distance from the navel point to the crown of the head and from the shoulder level to the crown of the head is 1: 1.618;

* the distance of the navel point to the knees and from the knees to the feet is 1: 1.618;

* the distance from the tip of the chin to the tip of the upper lip and from the tip of the upper lip to the nostrils is 1: 1.618;

* the distance from the tip of the chin to the upper line of the eyebrows and from the upper line of the eyebrows to the crown is 1: 1.618;

* the distance from the tip of the chin to the upper line of the eyebrows and from the upper line of the eyebrows to the crown is 1: 1.618:

The golden ratio in human facial features as a criterion for perfect beauty.

In the structure of human facial features, there are also many examples that approach the value of the golden ratio formula. However, do not rush immediately after the ruler to measure the faces of all people. Because exact correspondences to the golden ratio, according to scientists and people of art, artists and sculptors, exist only in people with perfect beauty. Actually, the exact presence of the golden ratio in a person's face is the ideal of beauty for the human eye.

For example, if we add up the width of the two front upper teeth and divide this amount by the height of the teeth, then, having received the Golden Ratio number, it can be argued that the structure of these teeth is ideal.

On the human face, there are other incarnations of the rule of the golden ratio. Here are some of these relationships:

* Face height / face width;

* The center point of the junction of the lips to the base of the nose / length of the nose;

* Face height / distance from the tip of the chin to the center point of the junction of the lips;

* Width of the mouth / width of the nose;

* Width of nose / distance between nostrils;

* Distance between pupils / distance between eyebrows.

Human hand.

It is enough just to bring your palm closer to you now and carefully look at forefinger, and you will immediately find the Golden Ratio formula in it. Each finger of our hand consists of three phalanges.

* The sum of the first two phalanges of the finger in relation to the entire length of the finger and gives the number of the golden ratio (except thumb);

* In addition, the ratio between middle finger and little finger is also equal to the golden ratio;

* A person has 2 hands, the fingers on each hand consist of 3 phalanges (excluding the thumb). Each hand has 5 fingers, that is, a total of 10, but with the exception of two biphalangeal thumbs, only 8 fingers are created according to the principle of the golden ratio. Whereas all these numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence:

Golden proportion in the structure of the human lungs.

American physicist B.D. West and Dr. A.L. Goldberger, during physical and anatomical studies, found that the golden ratio also exists in the structure of the human lungs.

The peculiarity of the bronchi that make up the human lungs lies in their asymmetry. The bronchi are made up of two main airways, one of which (left) is longer and the other (right) is shorter.

* It was found that this asymmetry continues in the branches of the bronchi, in all the smaller airways. Moreover, the ratio of the length of the short and long bronchi also makes up the golden ratio and is equal to 1: 1.618.

The structure of the golden orthogonal quadrilateral and spiral.

The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment refers to the larger part as much as the larger part itself refers to the smaller one; or in other words, a smaller segment relates to a larger one as a larger one to everything.

In geometry, a rectangle with this aspect ratio has come to be called the golden rectangle. Its long sides compare to short sides in a 1.168: 1 ratio.

The golden rectangle also has many amazing properties. The golden rectangle has many unusual properties. Cutting off a square from the gold rectangle, the side of which is equal to the smaller side of the rectangle, we again get a smaller gold rectangle. This process can be continued indefinitely. As we continue to cut out the squares, we will get smaller and smaller golden rectangles. Moreover, they will be located along a logarithmic spiral, which is important in mathematical models natural objects (for example, snail shells).

The spiral pole lies at the intersection of the diagonals of the initial rectangle and the first vertical cut to be cut. Moreover, the diagonals of all subsequent decreasing golden rectangles lie on these diagonals. Of course, there is also a golden triangle.

The English designer and esthetician William Charlton stated that people find spiral shapes pleasing to the eye and have been using them for millennia, explaining it this way:

"We like the look of the spiral, because visually we can see it with ease."

Let's find out what is common between the ancient Egyptian pyramids, the painting by Leonardo da Vinci "Mona Lisa", a sunflower, a snail, a pine cone and human fingers?

The answer to this question is hidden in amazing numbers that have been discovered the Italian mathematician of the Middle Ages Leonardo of Pisa, better known by the name of Fibonacci (born c. 1170 - died after 1228), italian mathematician ... Traveling in the East, I got acquainted with the achievements of Arab mathematics; contributed to their transfer to the West.

After his discovery, these numbers began to be called by the name of the famous mathematician. The amazing essence of the Fibonacci sequence is that each number in this sequence is obtained from the sum of the two previous numbers.

So, the numbers forming the sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, …

are called "Fibonacci numbers", and the sequence itself is called the Fibonacci sequence.

In Fibonacci numbers, there is one very interesting feature... When dividing any number from the sequence by the number in front of it in the row, the result will always be a value that fluctuates around the irrational value 1.61803398875 ... and through times it either rises or does not reach it. (Note: an irrational number, i.e. a number whose decimal representation is infinite and not periodic)

Moreover, after the 13th in the sequence, this division result becomes constant indefinitely ... It is this constant number of divisions in the Middle Ages that was called the Divine proportion, and nowadays it is called the golden ratio, golden mean or golden proportion ... In algebra, this number is denoted by the Greek letter phi (Ф)

So, the Golden Ratio = 1: 1.618

233 / 144 = 1,618

377 / 233 = 1,618

610 / 377 = 1,618

987 / 610 = 1,618

1597 / 987 = 1,618

2584 / 1597 = 1,618

The human body and the golden ratio

The most important book of all modern architects, E. Neufert's reference book "Building Design" contains the basic calculations of the parameters of the human body, containing the golden proportion.

The proportions of the various parts of our body make up a number very close to the golden ratio. If these proportions coincide with the formula of the golden ratio, then the appearance or body of a person is considered perfectly folded. The principle of calculating the golden measure on the human body can be depicted as a diagram:

M / m = 1.618

The first example of the golden ratio in the structure of the human body:
If we take the navel point as the center of the human body, and the distance between a person's feet and the navel point as a unit of measurement, then a person's height is equivalent to 1.618.

In addition, there are several more basic golden proportions of our body:

* the distance from the fingertips to the wrist to the elbow is 1: 1.618;

* distance from shoulder level to crown of head and head size is 1: 1.618;

* the distance from the navel point to the crown of the head and from the shoulder level to the crown of the head is 1: 1.618;

* the distance of the navel point to the knees and from the knees to the feet is 1: 1.618;

* the distance from the tip of the chin to the tip of the upper lip and from the tip of the upper lip to the nostrils is 1: 1.618;

* the distance from the tip of the chin to the upper line of the eyebrows and from the upper line of the eyebrows to the crown is 1: 1.618;

* the distance from the tip of the chin to the upper line of the eyebrows and from the upper line of the eyebrows to the crown is 1: 1.618:

The golden ratio in human facial features as a criterion for perfect beauty.

On the human face, there are other incarnations of the rule of the golden ratio. Here are some of these relationships:

* Face height / face width;

* The center point of the junction of the lips to the base of the nose / length of the nose;

* Face height / distance from the tip of the chin to the center point of the junction of the lips;

* Width of the mouth / width of the nose;

* Width of nose / distance between nostrils;

* Distance between pupils / distance between eyebrows.

Human hand

It is enough just to bring your palm closer to you now and carefully look at the index finger, and you will immediately find the formula of the golden ratio in it. Each finger of our hand consists of three phalanges.

* The sum of the first two phalanges of the finger in relation to the entire length of the finger and gives the number of the golden ratio (excluding the thumb);

* In addition, the ratio between middle finger and little finger is also equal to the golden ratio;

The golden proportion in the structure of the human lungs

American physicist B.D. West and Dr. A.L. Goldberger, during physical and anatomical studies, found that the golden ratio also exists in the structure of the human lungs.

The peculiarity of the bronchi that make up the human lungs lies in their asymmetry. The bronchi are made up of two main airways, one of which (left) is longer and the other (right) is shorter.

The structure of the golden orthogonal quadrilateral and spiral

In geometry, a rectangle with this aspect ratio has come to be called the golden rectangle. Its long sides compare to short sides in a 1.168: 1 ratio.

The English designer and esthetician William Charlton stated that people find spiral shapes pleasing to the eye and have been using them for millennia, explaining it this way:

"We like the look of the spiral, because visually we can easily see it."

In nature

* The rule of the golden ratio underlying the structure of the spiral is found in nature very often in creations that are incomparable in beauty. Most illustrative examples- the spiral shape can be seen in the arrangement of sunflower seeds, and in pine cones, in pineapples, cacti, the structure of rose petals, etc.;

* Botanists have established that in the arrangement of leaves on a branch, sunflower seeds or pine cones, the Fibonacci series is clearly manifested, and therefore, the law of the golden section is manifested;

The Supreme Lord established a special measure and proportionality for each of His creation, which is confirmed by examples found in nature. A great many examples can be cited when the growth process of living organisms occurs in strict accordance with the shape of a logarithmic spiral.

All the springs in the coil have the same shape. Mathematicians have found that even with an increase in the size of the springs, the shape of the spiral remains unchanged. There is no other form in mathematics that has the same unique properties as a spiral.

The structure of sea shells

Scientists who have studied the internal and external structure of the shells of soft-bodied mollusks living at the bottom of the seas stated:

“The inner surface of the shells is impeccably smooth, while the outer surface is covered with roughness and irregularities. The mollusk was in the shell, and for this the inner surface of the shell had to be perfectly smooth. External corners-bends of the shell increase its strength, hardness and thus increase its strength. The perfection and amazing intelligence of the structure of the shell (snail) is amazing. The spiral idea of shells is a perfect geometric shape and is astonishing in its polished beauty. "

In most snails that have shells, the shell grows in a logarithmic spiral. However, there is no doubt that these unreasonable creatures do not have an idea not only of a logarithmic spiral, but do not even have the simplest mathematical knowledge to create a spiral shell for themselves.

But then how could these unreasonable beings determine and choose for themselves the ideal form of growth and existence in the form of a spiral shell? Could these living creatures, which the scientists of the world call primitive forms of life, calculate that the logarithmic form of a shell would be ideal for their existence?

Of course not, because such a plan cannot be realized without the presence of reason and knowledge. But neither primitive mollusks, nor unconscious nature, which, however, some scientists call the creator of life on earth (?!)

Trying to explain the origin of such even the most primitive form of life by a random coincidence of certain natural circumstances is at least absurd. It is clear that this project is a conscious creation.

Biologist Sir D'arkey Thompson calls this type of growth of sea shells "The growth form of the gnomes."

Sir Thompson makes the following comment:

“There is no simpler system than the growth of seashells, which grow and expand proportionately, keeping the same shape. The shell, most surprisingly, grows, but never changes shape. "

The nautilus, a few centimeters in diameter, is the most dramatic example of the gnome type of growth. S. Morrison describes this process of the growth of the nautilus in the following way, which is rather difficult to plan even by the human mind:

“Inside the nautilus shell there are many compartments-rooms with mother-of-pearl partitions, and the shell itself inside is a spiral expanding from the center. As the nautilus grows, another room grows in front of the shell, but already large sizes than the previous one, and the partitions of the room left behind are covered with a layer of mother-of-pearl. Thus, the spiral is expanding proportionally all the time. "

Here are just some types of spiral shells with a logarithmic growth in accordance with their scientific names:
Haliotis Parvus, Dolium Perdix, Murex, Fusus Antiquus, Scalari Pretiosa, Solarium Trochleare.

All discovered fossil remains of shells also had a developed spiral shape.

However, the logarithmic form of growth is found in the animal kingdom not only in molluscs. The horns of antelopes, wild goats, rams and other similar animals also develop in the form of a spiral according to the laws of the golden ratio.

The golden ratio in the human ear

In the inner ear of a person there is an organ called Cochlea ("Snail"), which performs the function of transmitting sound vibration. This bony structure is filled with fluid and is also created in the form of a snail, containing a stable logarithmic spiral shape = 73º 43 ’.

Horns and tusks of animals that develop in a spiral shape

The tusks of elephants and extinct mammoths, the claws of lions and the beaks of parrots are logarithmic shapes and resemble the shape of an axis that tends to turn into a spiral. Spiders always spin their webs in a logarithmic spiral. The structure of microorganisms such as plankton (species globigerinae, planorbis, vortex, terebra, turitellae and trochida) are also spiral-shaped.

The golden ratio in the structure of microworlds

Geometric shapes are not limited to just triangles, squares, pentagons, or hexagons. If we connect these figures in various ways with each other, then we get new three-dimensional geometric figures... Examples of this are shapes such as a cube or a pyramid. However, besides them, there are also other three-dimensional figures that we did not have to meet in Everyday life, and whose names we hear, perhaps for the first time. These three-dimensional figures include a tetrahedron (a regular four-sided figure), an octahedron, a dodecahedron, an icosahedron, etc. The dodecahedron consists of 13 pentagons, the icosahedron of 20 triangles. Mathematicians note that these figures are mathematically very easily transformed, and their transformation occurs in accordance with the formula for the logarithmic spiral of the golden ratio.

In the microcosm, three-dimensional logarithmic forms built according to golden proportions are widespread everywhere. ... For example, many viruses have three-dimensional geometric shape icosahedron. Perhaps the most famous of these viruses is the Adeno virus. The protein coat of the adeno virus is formed from 252 units of protein cells arranged in a specific sequence. In each corner of the icosahedron there are 12 units of protein cells in the form of a pentagonal prism, and spike-like structures extend from these corners.

For the first time, the golden ratio in the structure of viruses was discovered in the 1950s. scientists from London Birkbeck College A. Klug and D. Kaspar. 13 Polyo virus was the first to appear in the logarithmic form. The form of this virus was found to be similar to that of the Rhino 14 virus.

The question arises, how do viruses form such complex three-dimensional forms, the structure of which contains the golden ratio, which even our human mind is quite difficult to construct? The discoverer of these forms of viruses, virologist A. Klug, gives the following comment:

“Dr. Kaspar and I have shown that for the spherical envelope of the virus, the most optimal shape is symmetry, such as the shape of the icosahedron. This order minimizes the number of connecting elements ... Most of Buckminster Fuller's geodesic hemispherical cubes are constructed according to a similar geometric principle. 14 The installation of such cubes requires an extremely accurate and detailed explanatory diagram. Whereas unconscious viruses themselves construct such a complex shell of elastic, flexible protein cell units. "

The most capacious definition of the golden ratio says that the smaller part refers to the larger, as large to the whole whole. Its approximate value is 1.6180339887. In a rounded percentage, the proportions of parts of a whole will relate as 62% to 38%. This relationship operates in the forms of space and time. The ancients saw in the golden ratio a reflection of the cosmic order, and Johannes Kepler called it one of the treasures of geometry. Modern science considers the golden ratio as "asymmetric symmetry", calling it in broad sense a universal rule reflecting the structure and order of our world order.

History

The ancient Egyptians had an idea of the golden proportions, they knew about them in Russia, but for the first time the golden ratio was explained by the monk Luca Pacioli in the book "Divine Proportion" (1509), the illustrations of which were supposedly made by Leonardo da Vinci. Pacioli saw the divine trinity in the golden section: the small segment personified the Son, the large one - the Father, and the whole - the Holy Spirit. The name of the Italian mathematician Leonardo Fibonacci is directly related to the rule of the golden section. As a result of solving one of the problems, the scientist came up with a sequence of numbers, now known as Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 etc. Kepler drew attention to the relation of this sequence to the golden ratio: “It is arranged in such a way that the two lowest terms of this endless proportion add up to the third term, and any two last terms, if added, give the next term, and the same proportion remains indefinitely ". Now the Fibonacci series is an arithmetic basis for calculating the proportions of the golden ratio in all its manifestations. Leonardo da Vinci also devoted a lot of time to studying the features of the golden ratio, most likely it was he who owns the term itself. His drawings of a stereometric solid formed by regular pentagons prove that each of the rectangles obtained by cutting gives aspect ratios in gold division. Over time, the rule of the golden ratio turned into an academic routine, and only the philosopher Adolph Zeising gave him a second life in 1855. He brought the proportions of the golden section to the absolute, making them universal for all phenomena of the surrounding world. However, his "mathematical aesthetics" drew a lot of criticism.

Nature

Without even going into the calculations, the golden ratio can be easily found in nature. So, the ratio of the tail and body of the lizard, the distance between the leaves on the branch, there is a golden ratio and in the form of an egg, if a conditional line is drawn through its widest part, fall under it. The Belarusian scientist Eduard Soroko, who studied the forms of gold divisions in nature, noted that everything growing and striving to take its place in space is endowed with the proportions of the golden section. In his opinion, one of the most interesting forms is spiral twisting. Even Archimedes, paying attention to the spiral, derived an equation based on its shape, which is still used in technology. Later, Goethe noted the gravitation of nature to spiral forms, calling the spiral "the curve of life." Modern scientists have found that such manifestations of spiral forms in nature such as the snail shell, the arrangement of sunflower seeds, the patterns of the cobweb, the movement of the hurricane, the structure of DNA and even the structure of galaxies contain the Fibonacci series.

Human

Fashion designers and clothing designers make all calculations based on the proportions of the golden ratio. Man is a universal form for testing the laws of the golden ratio. Of course, by nature, not all people have ideal proportions, which creates certain difficulties with the selection of clothes. In Leonardo da Vinci's diary, there is a drawing of a naked man inscribed in a circle, in two superimposed positions. Building on research by the Roman architect Vitruvius, Leonardo In a similar way tried to establish the proportions of the human body. Later, the French architect Le Corbusier, using Leonardo's Vitruvian Man, created his own scale “ harmonic proportions”, Which influenced the aesthetics of architecture of the XX century. Adolf Zeising, investigating the proportionality of man, has done a tremendous job. He measured about two thousand human bodies, as well as many antique statues and deduced that the golden ratio expresses the average law. In a person, almost all parts of the body are subordinate to him, but the main indicator of the golden ratio is the division of the body by the navel point. As a result of measurements, the researcher found that the proportions male body 13: 8 is closer to the golden ratio than the proportions of the female body - 8: 5.

The art of spatial forms

The artist Vasily Surikov used to say that "there is an immutable law in the composition, when nothing can be removed or added in a picture, even an extra point cannot be put, this is real mathematics." For a long time, artists have been following this law intuitively, but after Leonardo da Vinci, the process of creating picturesque canvas can no longer do without solving geometric problems. For example, Albrecht Durer used a proportional compass invented by him to determine the points of the golden section. The art critic FV Kovalev, having examined in detail the painting by Nikolai Ge "Alexander Sergeevich Pushkin in the village of Mikhailovskoye," notes that every detail of the canvas, be it a fireplace, a bookcase, an armchair or the poet himself, is strictly inscribed in golden proportions. Researchers of the Golden Ratio tirelessly study and measure the masterpieces of architecture, claiming that they became such because they were created according to the golden canons: in their list are the Great Pyramids of Giza, the Cathedral Notre dame de paris, St. Basil's Cathedral, Parthenon. And today, in any art of spatial forms, they try to follow the proportions of the golden section, since, according to art critics, they facilitate the perception of the work and form an aesthetic feeling for the viewer.

Word, sound and filmstrip

Temporary art forms in their own way demonstrate to us the principle of the golden division. Literary scholars, for example, have noticed that the most popular number of lines in poems late period Pushkin's creativity corresponds to the Fibonacci series - 5, 8, 13, 21, 34. The rule of the golden section also applies in individual works of the Russian classic. So climax « The Queen of Spades”Is the dramatic scene of Hermann and the Countess, ending with the death of the latter. There are 853 lines in the story, and the culmination is on line 535 (853: 535 = 1.6) - this is the point of the golden section. The Soviet musicologist E.K. Rosenov notes the amazing accuracy of the golden ratio in the strict and free forms of the works of Johann Sebastian Bach, which corresponds to the thoughtful, concentrated, technically verified style of the master. This is also true of the outstanding works of other composers, where the most striking or unexpected musical decision usually falls on the golden section. Film director Sergei Eisenstein deliberately coordinated the script of his film "Battleship Potemkin" with the rule of the golden section, dividing the tape into five parts. In the first three sections, the action takes place on the ship, and in the last two - in Odessa. Going to scenes in the city is golden mean film.