Golden ratio in the human body. Interesting facts about mathematics on the topic: The human body and the golden ratio
The human body and the golden ratio...
The golden ratio (golden ratio, division in extreme and average ratio) is the ratio of two quantities, equal to the ratio of their sum to the larger of the given quantities. The approximate value of the golden ratio is 1.6180339887.
All human bones are kept in proportion to the golden ratio.
The proportions of the various parts of our body are a number very close to the golden ratio. If these proportions coincide with the golden ratio formula, then the person’s appearance or body is considered ideally proportioned.
If we take the navel point as the center of the human body, and the distance between a person’s foot and the navel point as a unit of measurement, then a person’s height is equivalent to the number 1.618.
The distance from shoulder level to the top of the head and the size of the head is 1:1.618
The distance from the navel point to the top of the head and from shoulder level to the top 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
Actually, the exact presence of the golden proportion in a person’s face is the ideal of beauty for the human gaze.
The distance from the tip of the chin to the top line of the eyebrows and from the top line of the eyebrows to the crown is 1:1.618
Face height/face width
The central point where the lips connect 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 lips
Mouth width/nose width
Nose width / distance between nostrils
Interpupillary distance/eyebrow distance
It is enough just to bring your palm closer to you and look carefully at your 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 gives the number of the golden ratio (with the exception of the thumb).
In addition, the ratio between the 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 (except for the thumb). There are 5 fingers on each hand, that is, 10 in total, but with the exception of two two-phalanx 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.
Also worth noting is the fact that for most people, the distance between the ends of their outstretched arms is equal to their height.
The peculiarity of the bronchi that make up the human lungs lies in their asymmetry. The bronchi consist of two main airways, one of which (the left) is longer and the other (the right) is shorter.
It was found that this asymmetry continues in the branches of the bronchi, in all the smaller respiratory tracts.
Moreover, the ratio of the lengths of short and long bronchi is also the golden ratio and is equal to 1:1.618.
In the human inner ear 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 shaped like a snail, containing a stable logarithmic spiral shape = 73 43'.
Blood pressure changes as the heart works. It reaches its greatest value in the left ventricle of the heart at the moment of its compression (systole). In the arteries, during the systole of the ventricles of the heart, blood pressure reaches a maximum value equal to 115-125 mmHg in a young, healthy person. At the moment of relaxation of the heart muscle (diastole), the pressure decreases to 70-80 mm Hg. The ratio of maximum (systolic) to minimum (diastolic) pressure is on average 1.6, that is, close to the golden ratio.
If we take the average blood pressure in the aorta as a unit, then the systolic blood pressure in the aorta is 0.382, and the diastolic pressure is 0.618, that is, their ratio corresponds to the golden proportion. This means that the work of the heart in relation to time cycles and changes in blood pressure are optimized according to the same principle - the law of the golden proportion.
In the Universe, all galaxies known to mankind and all the bodies in them exist in the form of a spiral, corresponding to the formula of the golden ratio.
The proportionality of the earth is closely related to the proportionality of man.
Just as in nature there are five-petaled flowers, plants, the bodies of starfish, just as the human figure has a five-ray structure: a head, two arms, two legs, the human figure can be entered into PENTA WURF, with the parameters: the pose of a prone person with his arms spread 180 degrees and legs turned 90 degrees. And although the figures in pentagrams do not fit slightly into perfectly equilateral circles, the ratio is still present everywhere - 1.618 - i.e. "golden ratio"
Golden ratio and its secrets
(a+b)/b=b/a or x 2 -x-1=0;
What happens if you mirror it:
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 mirror reflection, magic numbers 1,3,7,12 are born, creating a number series of “lucky numbers”.
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Golden ratio
Golden ratio
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How not to admire the structure of the eyeball, where the distance between two points at the entrance and exit is projected according to the law of proportional dependence of refraction in the lens and turning up (real image) down (reflected image) onto the retina of the eye, where light-color perceptions of rods and cones encode spatial images of the external world and with the help of bioelectric impulses are transmitted to the cerebral cortex. So in reality, everything comes to our consciousness upside down and then turns upside down again subconsciously, creating an imaginary reality.
In the eye there is, as it were, a pyramidal crystal created by rays of light, joined by its vertices (a clock of time).
Proportionality, proportionality, analogy, commensurability.
It turns out that through visual perception the world around us is reflected in our consciousness and this connects the thinking of our “I” by analogy with the real world that surrounds us. That is, the harmony of nature creates the harmony of consciousness. Disharmony gives rise to anxiety, worry, nervous tension, and stress. This once again proves that man is part of nature. Many scientists and philosophers suggest 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 cumulative memory and the emergence of the mind as it is with the ensuing consequences.
The formation of one person preserves the memory of the formation of all humanity. With countdown meditation, one can achieve recollection of the past, when information is absorbed from every cell of every atom in the same way that reflects the essence of existence. It was the brain-nature comparison that gave birth to the concept of microcosm and macrocosm. The simplest measurement of a person’s orientation in space is the vertical and horizontal, which ideally creates an omnipresent cross at the intersection. But any consciousness cannot think in infinite categories. Reality requires completeness. The most complete relationship between the vertical and horizontal is a certain inclined line joining two lines, this will be the third diagonal, which gives rise to the relationship:
a: b = c.
But why not a:a=c? Yes, because the eyeball itself precisely in the relation 3:4:5 gives birth to this relation. This ratio is ideal for the perception of the human eye in physiological terms.
The next step in reflecting similarity and commensurability is humanity’s awareness of dualism (the binary nature of similarity). Just as there are 2 eyes, 2 ears, 2 arms, 2 legs and so on, that is, the knowledge of symmetry is a mirror image of 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:
Adepts 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 macroworld, and a person inscribed in these proportions was a symbol of the microworld. 
Subsequently, this relationship was improved by initiations into secret knowledge by Agrippa Nepaceheimer in the “pentasymmetric law”, where there were 4 elements, 4 squares, combined with a star by 5 primary elements. To some extent, this is a unification of Western and Eastern philosophy, running along the Western and Eastern axes of the world. 
Also, the search for a reflection of the microcosm was the laws derived by Vitruvius, Leonardo da Vinci, Dürer. 
If you look closely, the starting points for the new circle and the square where the human figure is inscribed are taken from the original 4 squares, divided into 16 parts. This system of crushing into equal parts (doubling) corresponds to doubling on a fraction,
½, ¼, 1/8, 1/16, 1/32 (setata)
So the secret construction of the connection between the 4 elements and 5 primary elements eloquently confirms the direct relationship of the formula of the universe into one whole through a pyramidal crystal. 
Example: Ancient adepts gave their names to the shape of rectangles built according to the golden ratio.
Conclusion: the harmony of man and nature was the main super task, the solution of which was the goal in itself of all civilizations. The tool for this was the fundamentals of global thinking through the relationship between the macrocosm and the microcosm through the construction of a pyramidal crystal based on the elements and primary elements.
The Great Pyramids are shaped like 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. 
Secret knowledge that unites the macro and microcosm due to changing generations of formations and religions was gradually forgotten, separated and modified, which led to different interpretations of the same fundamentals. For example: the ancient Greeks, like the Egyptians, used a decimal (like modern society) number system (10, 100, 1000, and so on), which led to gradation.
Man - 1, God-man - 10. Accordingly, the construction of the temple suggested a tenfold increase in columns, cornices (Parthenon Temple). If you look closely, the proportionality of Greek buildings was a reflection of the harmony of the human body. In Rus', 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 a disembodied spirit and invisibly present everywhere. Therefore, spiritual changes were made to the churches of the Byzantine school. Narrow windows, loopholes, where the rays of the sun fell, at a certain angle, and at a certain time.
The temple of antiquity filled with sculptures was replaced by an interior of paintings that expand the space, that is, creating the illusion of a supernatural space - part of the divine cosmos - the temple of some unearthly capsule of a spaceship descended into the world of everyday life. 
After all, even during major wars, anyone who managed to get inside the church of the monastery was, as it were, guarded by God himself, and to punish those who hid in the church temple 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 confess their sins there. 
The architecture of Christianity and Islam resembles the images of warrior warriors defending their faith, that is, the same associative images of the god-man, but on a different level than in ancient Greece or Rome. If in ancient Greece temples with a row of columns are a whole galaxy of gods, then Byzantine Christianity landed, grew into the ground, the temples became closed in on themselves, many loopholes-windows appeared, which speaks of civil strife and many years of wars between different faiths, and although the proportionality of the classical relations were preserved, but the wisdom of the past was dismembered and gradually became 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 dedicated priests, was gradually forgotten due to the lack of education of ordinary people and the secrecy of the adherents. 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 technological progress modifies concepts, abstracts and accelerates the comprehensive understanding of the world to one generation of accumulated knowledge of the past. If earlier the movement went from the general to the particular by trial and error, now there is a process from the particular to the general, and the greater and deeper the knowledge about nature and man, the closer the approach to the formula of the universe, knowledge encoded in various forms handed down to us from the past. 
The new millennium, where the 21st century is its beginning, and our generation is a monad - a unit (1), the beginning, represents a chance to determine a certain global plan - the basis of subsequent generations. After all, 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 currently fantastic in practice is already in theory. As in ancient times, humanity has realized, or 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 has been created, but the main thing is that there has been an awareness of recreating the global pyramidal philosophy, which appears and comes to the surface from the depths of non-existence, when civilization begins to mature and exert global changes, harm or benefit brought by economics, politics, sociology. Only by recognizing your mistakes can you become wiser and the 21st century is the era of sages, especially among the leaders of civilized countries, where a high standard of living contributes to careful management of the environment, conservation of natural resources, works of art, ancient monuments, development of savings systems, 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 adherents 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 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 the whole, he is not afraid for his fate, because... fate leads it itself, the main thing is not to disturb the eco-balance 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 a subject for logical analysis of the environment, where the categories of space, philosophy of personality, form and content of matter, spiritual essences, which were based on combinations of musical sounds, mathematical numbers, picturesque colors, sculptural forms, architectural structures and all kinds of miracles created by the creative genius of the creative person.
Until recently, many attempts were made to comprehend the laws of nature differentiatedly, in parts, but the separate perception of natural phenomena by physicists, chemists, biologists, mathematicians led to the fact that the whole was divided into multiple ones, one-sided professionalism, due to unnecessaryness, destroyed the holistic perception of the world, and so that To introduce the general law of harmony, you need to find a certain discrete symbol that is the basis of everything. The particle from which everything else is made. The most suitable for this is mathematical modeling of any shape through geometric abstraction. Where there is a combination of systematic time and space. Forms and contents. Statics and dynamics. Visualization and schematization. 
But nature itself tells us that the most ideal technique that gives life is dichotomy - a system of bifurcation of the whole into + and -, as well as the reverse phenomenon of merging + and - (male and female cells into one whole).
Examples:
Stage 1:
1. – maternal principle
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 into 2 parts again.
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 method of dividing in half and then combining in different options gives rise to an infinite variety of combinations. 
Let's introduce a mathematical concept: additivity - where in a number series each previous member of the series is equal to the sum of two consecutive
Drawing
2. Multiplicativity - where in a number series all members of the series are connected in a geometric progression.
Drawing 
But in nature there is only one combination of additivity and multiplicativity of a global nature - this is the relation: a+b=c (additivity) and a/b=b/c (multiplicativity), in which the whole c is represented as consisting of two parts a+b.
Conclusion: But a sign of any self-organized system is:
1. Growth pattern.
2. Changing 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 a 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 one 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, diagonal dichotomy introduced linear incompatibility and angular inequality.
Drawing
Let’s create a “golden chain” and model the shape in multidimensional space – time.
When a right angle is introduced, the linear series of the golden section is transformed into a space of symmetry of similarities.
Proof:
Using the resulting drawings of an asymmetrical a-rhombus, an ascending branch of the golden ratio was obtained. The A-rhombus is a shape that evokes deep associations and relationships of any form of life, 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 mystery of the birth of a living object from the primary element. Why does an apple tree grow from an apple seed, and a man from a human seed? Until then, it remains a mystery which bio-fields are embedded in the DNA gene spiral and form a given formula, because there is some kind of system, framework, axis that, as it develops over time, purposefully forms the necessary cells and tissues of living organisms. Since form is a category of space, areas 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 is also below).
Since man is inherently egocentric, two fundamental terms came to us from ancient times - macrocosm and microcosm (universe and man).
Modern cosmology assumes that by the time the universe was born, its mass, energy and pressure were already concentrated in the vacuum. Something similar exists in the human cell, where all the data of the human genome is concentrated. Now nanotechnologies are developing genetic engineering according to cosmological principles of “shocks” from “nothing”, when everything appeared everywhere and at the same time. Spectral analysis of stars confirms this. Nearby and billions of light-years distant galaxies are homogeneous in composition. According to vector geometry, where the mother’s cell is complemented by the father’s, a certain morphogenesis occurs inherent in the formation of these chromosomes, rigidly forming the framework of human-likeness.
Samir forms of living nature indicate 2 trends in shape formation in nature, which is confirmed by previous geometric constructions:
1. There is a pronounced direction of growth along the axis up and down - a biological vertical (stems of many plants, cereals, various phallic forms
2. There is an expression “the direction of growth towards spherical and rounded shapes” (human skull, apple, egg, orange).
3. There is an interpretation of flat shapes (sunflower disk, Pecten shell leaves, that is, the pyramidal crystal of life)
The golden ratio is the division of a segment into unequal parts, with the entire segment (A) related to the larger part (B), as this larger part (B) is related to the smaller part (C), or A: B = B: C, or C:B = B:A.
Segments golden ratio relate to each other through the infinite irrational number Ф = 0.618... If C taken as one, then A= 0.382. The numbers 0.618 and 0.382 are the coefficients of the Fibonacci sequence, on which the basic geometric figures are built.
Human bones are kept in proportion close to the golden ratio. And the closer the proportions are to the golden ratio formula, the more ideal a person’s appearance looks.
If the distance between a person's feet and the navel point = 1, then the person's height = 1.618.
The distance from shoulder level to the top of the head and the size of the head is 1:1.618.
The distance from the navel point to the top of the head and from shoulder level to the top of the head is 1:1.618.
The distance from 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 top line of the eyebrows and from the top line of the eyebrows to the crown is 1:1.618.
Other proportional relationships:
Face height/face width; center point of connection of lips to base of nose/nose length; face height / distance from the tip of the chin to the central point where the lips meet; mouth width/nose width; nose width/distance between nostrils; distance between pupils / distance between eyebrows.
The exact presence of the golden proportion in a person’s face is the ideal of beauty for the human gaze.
The formula of the golden ratio is visible when looking at the index finger. Each finger of the hand consists of three phalanges. The sum of the first two phalanges of the finger in relation to the entire length of the finger = the golden ratio (excluding the thumb). Middle finger/little finger ratio = golden ratio.
A person has 2 hands, the fingers on each hand consist of 3 phalanges (except for the thumb). There are 5 fingers on each hand, that is, 10 in total, but with the exception of two double-phalangeal thumbs, only 8 fingers are created according to the principle of the golden ratio (numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence).
Already in the Middle Ages, measures of parts of the human body were used as standards. When building cathedrals in France, a device was used consisting of 5 rods, which represented the lengths of a palm, a large and small span, a foot and an elbow. All these lengths were multiples of a smaller unit of length called line and was equal to 1/12 inch, i.e. about 2.5 mm. If we convert these numbers to the metric system, we can see that the quantities lines are numbers from the Fibonacci series. The ratio of each to the previous one is equal to F, which is even more surprising, because these units correspond to arbitrary parts of the human body.
From open spaces for educational purposes)
Let's find out what the ancient Egyptian pyramids, Leonardo da Vinci's painting "Mona Lisa", a sunflower, a snail, a pine cone and human fingers have in common?
The answer to this question is hidden in the amazing numbers that have been discovered Italian medieval mathematician Leonardo of Pisa, better known by the name Fibonacci (born about 1170 - died after 1228), Italian mathematician . Traveling around the East, he became acquainted with the achievements of Arab mathematics; contributed to their transfer to the West.
After his discovery, these numbers began to be called after the famous mathematician. The amazing essence of the Fibonacci number sequence is that 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. There is one very interesting feature about Fibonacci numbers. When dividing any number from the sequence by the number in front of it in the series, the result will always be a value that fluctuates around the irrational value 1.61803398875... and sometimes exceeds it, sometimes does not reach it. (Approx. irrational number, i.e. a number whose decimal representation is infinite and non-periodic)
Moreover, after the 13th number in the sequence, this division result becomes constant until the infinity of the series... It was this constant number of divisions that was called the Divine proportion in the Middle Ages, and is now called the golden ratio, the golden mean, or the golden proportion. . In algebra, this number is denoted by the Greek letter phi (Ф)
So, 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.
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, which was also created according to the principle of the golden ratio. Before creating their masterpieces, Leonardo Da Vinci and Le Corbusier took the parameters of the human body, created according to the law of the Golden Proportion.
The most important book of all modern architects, E. Neufert's reference book "Building Design", contains basic calculations of the parameters of the human torso, which contain the golden proportion.
The proportions of the various parts of our body are a number very close to the golden ratio. If these proportions coincide with the golden ratio formula, then the person’s appearance or body is considered ideally proportioned. The principle of calculating the gold measure on the human body can be depicted in the form of 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 foot and the navel point as a unit of measurement, then a person’s height is equivalent to the number 1.618.
In addition to this, 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;
* the distance from shoulder level to the top of the head and the size of the head is 1:1.618;
* the distance from the navel point to the crown of the head and from 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 top line of the eyebrows and from the top line of the eyebrows to the crown is 1:1.618:
The golden ratio in human facial features as a criterion of perfect beauty.
In the structure of human facial features there are also many examples that are close in value to the golden ratio formula. However, do not immediately rush for a ruler to measure the faces of all people. Because exact correspondences to the golden ratio, according to scientists and artists, artists and sculptors, exist only in people with perfect beauty. Actually, the exact presence of the golden proportion in a person’s face is the ideal of beauty for the human gaze.
For example, if we sum up the width of the two front upper teeth and divide this sum by the height of the teeth, then, having obtained the golden ratio number, we can say that the structure of these teeth is ideal.
There are other embodiments of the golden ratio rule on the human face. Here are a few of these relationships:
*Face height/face width;
* Central point of connection of the lips to the base of the nose / length of the nose;
* Face height / distance from the tip of the chin to the central point where the lips meet;
*Mouth width/nose width;
* Nose width / distance between nostrils;
* Distance between pupils / distance between eyebrows.
Human hand.
It is enough just to bring your palm closer to you and look carefully at your 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 gives the number of the golden ratio (with the exception of the thumb);
* In addition, the ratio between the 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 (except for the thumb). There are 5 fingers on each hand, that is, 10 in total, but with the exception of two two-phalanx thumbs, only 8 fingers are created according to the principle of the golden ratio. Whereas all these numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence:
The golden ratio in the structure of the human lungs.
American physicist B.D. West and Dr. A.L. Goldberger, during physical and anatomical studies, established 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 consist of two main airways, one of which (the left) is longer and the other (the 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 lengths of short and long bronchi is also the golden ratio and is equal to 1:1.618.
Structure of a golden orthogonal quadrangle and spiral.
The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part as the larger part itself is related to the smaller one; or in other words, the smaller segment is to the larger as the larger is to the whole.
In geometry, a rectangle with this aspect ratio came to be called the golden rectangle. Its long sides are in relation to its short sides in a ratio of 1.168:1.
The golden rectangle also has many amazing properties. The golden rectangle has many unusual properties. By cutting a square from the golden rectangle, the side of which is equal to the smaller side of the rectangle, we again obtain a golden rectangle of smaller dimensions. This process can be continued indefinitely. As we continue to cut off squares, we will end up with smaller and smaller golden rectangles. Moreover, they will be located in a logarithmic spiral, which is important in mathematical models of natural objects (for example, snail shells).
The pole of the spiral lies at the intersection of the diagonals of the initial rectangle and the first vertical one to be cut. Moreover, the diagonals of all subsequent decreasing golden rectangles lie on these diagonals. Of course, there is also the golden triangle.
English designer and esthetician William Charlton stated that people find spiral shapes pleasing to the eye and have been using them for thousands of years, explaining it this way:
“We like the look of the spiral because visually we can see it easily.”
In nature.
* The rule of the golden ratio, which underlies the structure of the spiral, is found in nature very often in creations of unparalleled beauty. The most obvious examples are that the spiral shape can be seen in the arrangement of sunflower seeds, pine cones, pineapples, cacti, the structure of rose petals, etc.;
* Botanists have found 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 ratio is manifested;
The Almighty Lord established a special measure for each of His creations and gave it proportionality, which is confirmed by examples found in nature. One can give a great many examples when the growth process of living organisms occurs in strict accordance with the shape of a logarithmic spiral.
All springs in the spiral 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 the spiral.
The structure of sea shells.
Scientists who 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, and the outer surface is completely covered with roughness and irregularities. The mollusk was in the shell and for this the inner surface of the shell had to be impeccably smooth. The outer corners-curves of the shell increase its strength, hardness and thus increase its strength. Perfection and the amazing intelligence of the structure of the shell (snail) delights. The spiral idea of shells is a perfect geometric shape and is amazing in its refined beauty."
In most snails that have shells, the shell grows in the shape of a logarithmic spiral. However, there is no doubt that these unreasonable creatures not only have no idea about the logarithmic spiral, but do not even have the simplest mathematical knowledge to create a spiral-shaped shell for themselves.
But then how were these unreasonable creatures able to determine and choose for themselves the ideal form of growth and existence in the form of a spiral shell? Could these living creatures, which the scientific world calls primitive life forms, calculate that the logarithmic shell shape would be ideal for their existence?
Of course not, because such a plan cannot be realized without intelligence and knowledge. But neither primitive mollusks nor unconscious nature possess such intelligence, 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 combination of certain natural circumstances is absurd, to say the least. It is clear that this project is a conscious creation.
Biologist Sir D'arky Thompson calls this type of growth of sea shells "growth form of dwarves."
Sir Thompson makes this comment:
"There is no simpler system than the growth of seashells, which grow and expand in proportion, maintaining the same shape. The shell, most amazingly, grows, but never changes shape."
The Nautilus, measuring several centimeters in diameter, is the most striking example of the gnome growth habit. S. Morrison describes this process of nautilus growth as follows, which seems quite difficult to plan even with the human mind:
"Inside the nautilus shell there are many compartments-rooms with partitions made of mother-of-pearl, and the shell itself inside is a spiral expanding from the center. As the nautilus grows, another room grows in the front part of the shell, but this time larger than the previous one, and the partitions remain behind the room are covered with a layer of mother-of-pearl. Thus, the spiral expands proportionally all the time."
Here are just some types of spiral shells with a logarithmic growth pattern 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 growth form is found in the animal world not only in mollusks. 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.
Golden ratio in the human ear.
In the human inner ear 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 shaped like a snail, containing a stable logarithmic spiral shape = 73º 43'.
Animal horns and tusks developing in a spiral shape.
The tusks of elephants and extinct mammoths, the claws of lions and the beaks of parrots are logarithmic in shape and resemble the shape of an axis that tends to turn into a spiral. Spiders always weave their webs in the form of a logarithmic spiral. The structure of microorganisms such as plankton (species globigerinae, planorbis, vortex, terebra, turitellae and trochida) also have a spiral shape.
The 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 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 certain 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 The installation of such cubes requires extremely precise and a detailed diagram-explanation. Whereas unconscious viruses themselves construct such a complex shell from elastic, flexible protein cellular units."
When we look at a beautiful landscape, we are embraced by everything around us. Then we pay attention to details. A murmuring river or a majestic tree. We see a green field. We notice how the wind gently hugs him and shakes the grass from side to side. We can feel the aroma of nature and hear the singing of birds... Everything is harmonious, everything is interconnected and gives a feeling of peace, a sense of beauty. Perception proceeds in stages in slightly smaller fractions. Where will you sit on the bench: on the edge, in the middle, or anywhere? Most will answer that it is a little further from the middle. The approximate number for the proportion of the bench from your body to the edge would be 1.62. It’s the same in the cinema, in the library, everywhere. We instinctively create harmony and beauty, which I call the “Golden Ratio” all over the world.
Golden ratio in mathematics
Have you ever wondered whether it is possible to determine the measure of beauty? It turns out that from a mathematical point of view it is possible. Simple arithmetic gives the concept of absolute harmony, which is reflected in impeccable beauty, thanks to the principle of the Golden Ratio. The architectural structures of other Egypt and Babylon were the first to begin to comply with this principle. But Pythagoras was the first to formulate the principle. In mathematics, this is a division of a segment slightly more than half, or more precisely 1.628. This ratio is presented as φ =0.618= 5/8. A small segment = 0.382 = 3/8, and the entire segment is taken as one.
A:B=B:C and C:B=B:A 
The principle of the golden ratio was used by great writers, architects, sculptors, musicians, people of art, and Christians who drew pictograms (five-pointed stars, etc.) with its elements in churches, fleeing from evil spirits, and people studying exact sciences, solving problems of cybernetics.
Golden ratio in nature and phenomena.
Everything on earth takes shape, grows upward, to the side or in a spiral. Archimedes paid close attention to the latter and composed an equation. According to the Fibonacci series, there is a cone, a shell, a pineapple, a sunflower, a hurricane, a spider’s web, a DNA molecule, an egg, a dragonfly, a lizard... 
Titirius proved that our entire Universe, space, galactic space - everything is planned based on the Golden Principle. One can read the highest beauty in absolutely everything living and non-living. 
Golden ratio in man.
The bones are also designed by nature according to the proportion 5/8. This eliminates people’s reservations about “wide bones.” Most body parts in ratios apply to the equation. If all parts of the body obey the Golden Formula, then the external data will be very attractive and ideally proportioned.
The segment from the shoulders to the top of the head and its size = 1:1 .618
The segment from the navel to the top of the head and from the shoulders to the top of the head = 1:1 .618
The segment from the navel to the knees and from them to the feet = 1:1 .618
The segment from the chin to the extreme point of the upper lip and from it to the nose = 1:1 .618 
All facial distances give a general idea of the ideal proportions that attract the eye.
Fingers, palm, also obey the law. It should also be noted that the length of the spread arms with the torso is equal to the height of a person. Why, all organs, blood, molecules correspond to the Golden Formula. True harmony inside and outside our space.
Parameters from the physical side of surrounding factors.
Sound volume. The highest point of sound, causing an uncomfortable feeling and pain in the auricle = 130 decibels. This number can be divided by the proportion 1.618, then it turns out that the sound of a human scream will be = 80 decibels.
Using the same method, moving further, we get 50 decibels, which is typical for the normal volume of human speech. And the last sound that we get thanks to the formula is a pleasant whisper sound = 2.618.
Using this principle, it is possible to determine the optimal-comfortable, minimum and maximum numbers of temperature, pressure, and humidity. The simple arithmetic of harmony is embedded in our entire environment.
Golden ratio in art.
In architecture, the most famous buildings and structures are: Egyptian pyramids, Mayan pyramids in Mexico, Notre Dame de Paris, Greek Parthenon, Peter's Palace, and others. 
In music: Arensky, Beethoven, Havan, Mozart, Chopin, Schubert, and others.
In painting: almost all the paintings of famous artists are painted according to the cross-section: the versatile Leonardo da Vinci and the inimitable Michelangelo, such relatives in writing as Shishkin and Surikov, the ideal of the purest art - the Spaniard Raphael, and the Italian Botticelli, who gave the ideal of female beauty, and many, many others.
In poetry: the ordered speech of Alexander Sergeevich Pushkin, especially “Eugene Onegin” and the poem “The Shoemaker”, the poetry of the wonderful Shota Rustaveli and Lermontov, and many other great masters of words.
In sculpture: a statue of Apollo Belvedere, Olympian Zeus, beautiful Athena and graceful Nefertiti, and other sculptures and statues.
Photography uses the “rule of thirds.” The principle is this: the composition is divided into 3 equal parts vertically and horizontally, key points are located either on the lines of intersection (horizon) or at the points of intersection (object). Thus the proportions are 3/8 and 5/8. 
According to the Golden Ratio, there are many tricks that are worth examining in detail. I will describe them in detail in the next one. 
