Interesting facts about the “golden ratio” - a little bit of good stuff. Golden ratio and symmetry

It is often said that mathematics has its own beauty, but by the middle of the 5th century BC. e. or even much earlier it became known that in beauty a large number of mathematics.

Phi number

Calculating the Golden Ratio

There are a large number of ways to express the golden ratio mathematically, and all of these methods have their own certain simplicity, precision and charm. Euclid described it as “a section in extreme and mean ratio.” A more “mathematical” expression looks like this: if the golden ratio equals x, then . Or like this: x/1 = 1/x -1. In words, the golden ratio is defined as the proportion in which “the length of the entire line relates to its larger part in the same way as that to the smaller.”

Interesting fact about the golden ratio No. 3. Golden rectangles can be divided into an infinite number of golden rectangles decreasing in size, “cutting off” parts of them along the shortest line. In the terminology of the Greek school of mathematicians, this property makes the golden rectangle a gnomon - an object capable of maintaining its shape as it grows (or shrinks).

A good example of the golden ratio is a credit card that has uniform standard sizes worldwide. According to the rules of the golden ratio, the ratio of its short side to its long side is the same as the ratio of its long side to the sum of the lengths of the short and long sides. This makes the credit card a golden rectangle. This shape was chosen for its balanced appearance - it does not appear too long or too wide. One way to check whether a rectangle is golden is to place two rectangles side by side, one “standing” vertically on a small edge, the other “standing” touching the first on a long edge. If the diagonal passing through the corners of a horizontal rectangle, continuing, reaches top corner rectangle arranged vertically, the rectangles are golden. Much more often this principle is seen in architecture. Thus, the golden rectangle is the facade of the UN building in New York.

Mathematics in Art and Nature

There is something prosaic about the golden ratio - at least for those who are not mathematically inclined. We are talking about its numerical expression. The value of x in the algebraic expression x 2 – x – 1 = 0 is 1.6180339887... and so on endlessly. However, the golden ratio has the most direct relation to Western art. To a large extent, this connection appeared thanks to the works of Luca Pacioli at the turn of the 16th century. Pacioli was a contemporary, and some of the maestro's drawings - including the most famous image of the Vitruvian Man - appear in Pacioli's book De Divina Proportione (The Divine Proportione), published in 1509. This book lays down the basic geometric rules of beauty, and inspired the creator number phi. Thus, in the perfect proportions of the human body, the ratio of height to the navel and full growth there is gold. Unfortunately, actual measurements indicate that in reality there are actually no “perfect” bodies. In the 20th century the golden ratio was looked for in natural forms. Those who did this persistently enough found it in the proportions of leaves, the distribution of buds on the stem (natural patterns rather roughly obey the principle of the Fibonacci sequence), and also in the dive trajectory of a hunting hawk. For some, this was evidence in favor of the existence of a certain plan, in accordance with which nature itself is organized. For others, it meant that our perception of beauty (or at least pleasing proportionality to the eye) was dictated by the mathematics of growth, which represents structures increasing in size without losing their overall shape.

Interesting fact #5. Actual measurements indicate that in reality there are actually no “perfect” bodies that satisfy the golden section rule.

Golden spiral

A spiral unfolding in accordance with the principle of the golden ratio can be built using a series of golden rectangles. This is a special case of a logarithmic spiral diverging from an axis point at a constant angle (Mathematically, it is more correct to formulate this way: a curve whose tangent forms the same angle with the radius vector at each point). This spiral is associated with the name of Jacob Bernoulli (despite the fact that he was the first to outline it), the main researcher of its properties. Bernoulli also wanted such a spiral engraved on his tombstone, but the mason, poorly versed in geometry, reproduced the Archimedean spiral there with a flatter divergence trajectory.

Golden ratio is a universal manifestation of structural harmony. It is found in nature, science, art - in everything that a person can come into contact with. Once having become acquainted with the golden rule, humanity no longer betrayed it.

Definition.

The most comprehensive definition of the golden ratio states that the smaller part relates to the larger, as the larger part relates to the whole. Its approximate value is 1.6180339887. In a rounded percentage value, the proportions of the parts of the whole will correspond as 62% to 38%. This relationship in the forms of space and time operates.

The ancients saw the golden ratio as a reflection of cosmic order, and Johannes Kepler called it one of the treasures of geometry. Modern science considers the golden ratio as “Asymmetrical Symmetry”, calling it in a broad sense a universal rule that reflects the structure and order of our world order.

Story.
The ancient Egyptians had an idea about the golden proportions, they knew about them in Rus', but for the first time the golden ratio was scientifically explained by the monk Luca Pacioli in the book “Divine Proportion” (1509), illustrations for 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 segment the father, and the whole the holy spirit.

The name of the Italian mathematician Leonardo Fibonacci is directly associated with the golden ratio rule. As a result of solving one of the problems, the scientist came up with a sequence of numbers now known as the Fibonacci series: 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. Kepler drew attention to the relationship of this sequence to the golden ratio : “It is arranged in such a way that the two Younger Members of This Infinite Proportion in the Sum give the Third Member, and any two Last Members, If Added, Give the Next Member, Moreover, the same Proportion is Preserved to Infinity.” Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden ratio in all its manifestations

Fibonacci numbers are a harmonic division, a measure of beauty. The golden ratio in nature, man, art, architecture, sculpture, design, mathematics, music https://psihologiyaotnoshenij.com/stati/zolotoe-sechenie-kak-eto-rabotaet

Leonardo da Vinci also devoted a lot of time to studying the features of the golden ratio; most likely, the term itself belongs to him. His drawings of a stereometric body formed by regular pentagons prove that each of the rectangles obtained by section gives the aspect ratio in the golden division.

Over time, the golden ratio rule became an academic routine, and only the philosopher Adolf Zeising gave it 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” caused a lot of criticism.

Nature.
Even without going into calculations, the golden ratio can be easily found in nature. So, the ratio of the tail and body of a lizard, the distances between the leaves on a branch fall under it, there is a golden ratio in the shape of an egg, if a conditional line is drawn through its widest part.

The Belarusian scientist Eduard Soroko, who studied the forms of golden 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.
Archimedes, paying attention to the spiral, derived an equation based on its shape, which is still used in technology. Later, Goethe noted the attraction of nature to spiral forms, calling the spiral the “Curve of Life.” Modern scientists have found that such manifestations of spiral forms in nature as a snail shell, the arrangement of sunflower seeds, spider web patterns, the movement of a 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.

Adolf Zeising, exploring the proportionality of a person, did a colossal job. He measured about two thousand human bodies, as well as many ancient statues, and concluded that the golden ratio expresses the average statistical law. In a person, almost all parts of the body are subordinate to it, but main indicator The golden ratio is the division of the body by the navel point.
As a result of measurements, the researcher found that the proportions of the male body 13:8 are closer to the golden ratio than the proportions of the female body - 8:5.

The art of spatial forms.
The artist Vasily Surikov said, “that in a Composition there is an Immutable Law, When in a Picture You Can’t Remove or Add Anything, You Can’t Even Put an Extra Point, This is Real Mathematics.” For a long time artists followed this law intuitively, but after Leonardo da Vinci, the process of creating a painting can no longer be accomplished without solving geometric problems. For example, Albrecht Durer used the proportional compass he invented to determine the points of the golden section.

Art critic F. v. Kovalev, having examined in detail Nikolai Ge’s painting “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 architectural masterpieces, claiming that they became such because they were created according to the golden canons: their list includes the great pyramids of Giza, Notre Dame Cathedral, St. Basil's Cathedral, and the 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 in the viewer.

Word, sound and film.
Forms are temporary? The Go arts, 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 of the late period of Pushkin’s work 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. Thus, the climax of “The Queen of Spades” is the dramatic scene of Herman and the Countess, ending with the death of the latter. The story has 853 lines, and the climax occurs on line 535 (853: 535 = 1, 6) - this is the point of the golden ratio.

Soviet musicologist E. K. Rosenov notes the amazing accuracy of the relationships of the golden section 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 solution usually occurs at the golden ratio point.
Film director Sergei Eisenstein deliberately coordinated the script of his film “Battleship Potemkin” with the golden ratio rule, dividing the film into five parts. In the first three sections the action takes place on the ship, and in the last two - in Odessa. The transition to scenes in the city is the golden middle of the film.

The golden ratio is a universal manifestation of structural harmony. It is found in nature, science, art - in everything that a person can come into contact with. Once having become acquainted with the golden rule, humanity no longer betrayed it.

Definition

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

Story

The ancient Egyptians had an idea about the golden proportions, they knew about them in Rus', but for the first time the golden ratio was scientifically explained by the monk Luca Pacioli in the book “Divine Proportion” (1509), illustrations for which were supposedly made by Leonardo da Vinci. Pacioli saw in the golden section the divine trinity: the small segment personified the Son, the large segment the Father, and the whole the Holy Spirit.

The name of the Italian mathematician Leonardo Fibonacci is directly associated with the golden ratio rule. As a result of solving one of the problems, the scientist came up with a sequence of numbers now known as the Fibonacci series: 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. Kepler drew attention to the relationship of this sequence to the golden proportion: “It is arranged in such a way that the two lower terms of this never-ending proportion add up to the third term, and any two last terms, if added, give the next term, and the same proportion is maintained ad infinitum " Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden ratio in all its manifestations.

Over time, the golden ratio rule became an academic routine, and only the philosopher Adolf Zeising gave it 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” caused a lot of criticism.

Nature

Even without going into calculations, the golden ratio can be easily found in nature. So, the ratio of the tail and body of a lizard, the distances between the leaves on a branch fall under it, there is a golden ratio in the shape of an egg, if a conditional line is drawn through its widest part.

Archimedes, paying attention to the spiral, derived an equation based on its shape, which is still used in technology. Goethe later noted nature’s attraction to spiral forms, calling the spiral the “curve of life.” Modern scientists have found that such manifestations of spiral forms in nature as a snail shell, the arrangement of sunflower seeds, spider web patterns, the movement of a 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. Based on the research of the Roman architect Vitruvius, Leonardo similarly tried to establish the proportions of the human body. Later, the French architect Le Corbusier, using Leonardo’s “Vitruvian Man,” created his own scale of “harmonic proportions,” which influenced the aesthetics of 20th-century architecture.
Adolf Zeising, studying the proportionality of a person, did a colossal job. He measured about two thousand human bodies, as well as many ancient statues, and concluded that the golden ratio expresses the average statistical law. In a person, almost all parts of the body are subordinate to it, 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 of the male body 13:8 are closer to the golden ratio than the proportions of the female body - 8:5.

The art of spatial forms

The artist Vasily Surikov said “that in composition there is an immutable law, when in a picture you cannot remove or add anything, you cannot even add an extra point, this is real mathematics.” For a long time, artists have followed this law intuitively, but after Leonardo da Vinci, the process of creating a painting is no longer complete without solving geometric problems. For example, Albrecht Durer used the proportional compass he invented to determine the points of the golden section.

Art critic F.V. Kovalev, having examined in detail Nikolai Ge’s painting “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 architectural masterpieces, claiming that they became such because they were created according to the golden canons: on their list are the Great Pyramids of Giza, the Cathedral Notre Dame of 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 in the viewer.

Word, sound and film

The forms of temporary art 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 of the late period of Pushkin’s work 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. Thus, the climax of “The Queen of Spades” is the dramatic scene of Herman and the Countess, ending with the death of the latter. The story has 853 lines, and the climax occurs on line 535 (853:535 = 1.6) - this is the point of the golden ratio.

Film director Sergei Eisenstein deliberately coordinated the script of his film “Battleship Potemkin” with the rule of the golden ratio, dividing the film into five parts. In the first three sections the action takes place on the ship, and in the last two - in Odessa. The transition to scenes in the city is the golden middle of the film.

Taras Repin

Original post and comments at

Definition
The most comprehensive definition of the golden ratio states that the smaller part is related to the larger one, as the larger part is to the whole. Its approximate value is 1.6180339887. In a rounded percentage value, the proportions of the parts of the whole will correspond as 62% to 38%. This relationship operates in the forms of space and time.

The ancients saw the golden ratio as a reflection of cosmic order, and Johannes Kepler called it one of the treasures of geometry. Modern science considers the golden ratio as “asymmetrical symmetry”, calling it in a broad sense a universal rule reflecting the structure and order of our world order.

Story
The ancient Egyptians had an idea about the golden proportions, they knew about them in Rus', but for the first time the golden ratio was scientifically explained by the monk Luca Pacioli in the book “Divine Proportion” (1509), illustrations for which were supposedly made by Leonardo da Vinci. Pacioli saw in the golden section the divine trinity: the small segment personified the Son, the large segment the Father, and the whole the Holy Spirit.

The name of the Italian mathematician Leonardo Fibonacci is directly associated with the golden ratio rule. As a result of solving one of the problems, the scientist came up with a sequence of numbers now known as the Fibonacci series: 0, 1, 1, 2, 3... etc. Kepler drew attention to the relationship of this sequence to the golden proportion: “It is arranged in such a way that the two lower terms of this never-ending proportion add up to the third term, and any two last terms, if added, give the next term, and the same proportion is maintained ad infinitum " Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden ratio in all its manifestations.

Over time, the golden ratio rule became an academic routine, and only the philosopher Adolf Zeising gave it 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” caused a lot of criticism.

Nature
Even without going into calculations, the golden ratio can be easily found in nature. So, the ratio of the tail and body of a lizard, the distances between the leaves on a branch fall under it, there is a golden ratio in the shape of an egg, if a conditional line is drawn through its widest part.

The Belarusian scientist Eduard Soroko, who studied the forms of golden 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.
Archimedes, paying attention to the spiral, derived an equation based on its shape, which is still used in technology. Goethe later noted nature’s attraction to spiral forms, calling the spiral the “curve of life.” Modern scientists have found that such manifestations of spiral forms in nature as a snail shell, the arrangement of sunflower seeds, spider web patterns, the movement of a 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.

Adolf Zeising, studying the proportionality of a person, did a colossal job. He measured about two thousand human bodies, as well as many ancient statues, and concluded that the golden ratio expresses the average statistical law. In a person, almost all parts of the body are subordinate to it, 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 of the male body 13:8 are closer to the golden ratio than the proportions of the female body - 8:5.

The art of spatial forms
The artist Vasily Surikov said “that in composition there is an immutable law, when in a picture you cannot remove or add anything, you cannot even add an extra point, this is real mathematics.” For a long time, artists have followed this law intuitively, but after Leonardo da Vinci, the process of creating a painting is no longer complete without solving geometric problems. For example, Albrecht Durer used the proportional compass he invented to determine the points of the golden section.

Researchers of the golden ratio tirelessly study and measure architectural masterpieces, claiming that they became such because they were created according to the golden canons: their list includes the Great Pyramids of Giza, Notre Dame Cathedral, St. Basil's Cathedral, and the 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 in the viewer.

Word, sound and film
The forms of temporary art 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 of the late period of Pushkin’s work 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. Thus, the climax of “The Queen of Spades” is the dramatic scene of Herman and the Countess, ending with the death of the latter. The story has 853 lines, and the climax occurs on line 535 (853:535 = 1.6) - this is the point of the golden ratio.

Soviet musicologist E.K. Rosenov notes the amazing accuracy of the ratios of the golden section 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 for the outstanding works of other composers, where the most striking or unexpected musical solution usually occurs at the golden ratio point
Film director Sergei Eisenstein deliberately coordinated the script of his film “Battleship Potemkin” with the rule of the golden ratio, dividing the film into five parts. In the first three sections the action takes place on the ship, and in the last two - in Odessa. The transition to scenes in the city is the golden middle of the film.

Golden ratio- this is such a proportional division of a segment into unequal parts, in which the smaller segment is related to the larger one, as the larger one is to the whole.

a: b = b: c or c: b = b: a.

This proportion is:

For example, in the correct five-pointed star, each segment is divided by a segment intersecting it in the golden ratio (i.e., the ratio of the blue segment to the green, red to blue, green to violet is equal 1.618

It is generally accepted that the concept of the golden ratio was introduced into scientific use by Pythagoras. There is an assumption that Pythagoras borrowed his knowledge from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamun indicate that Egyptian craftsmen used the ratios of the golden division when creating them.

In 1855, the German researcher of the golden ratio, Professor Zeising, published his work "Aesthetic Research".
Zeising measured about two thousand human bodies and came to the conclusion that the golden ratio expresses the average statistical law.

Golden proportions in parts of the human body

Dividing the body by the navel point - the most important indicator golden ratio. The proportions of the male body fluctuate within the average ratio of 13: 8 = 1.625 and are somewhat closer to the golden ratio than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 = 1.6.

In a newborn the proportion is 1:1, by the age of 13 it is 1.6, and by the age of 21 it is equal to that of a man.
The proportions of the golden ratio also appear in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.
Zeising tested the validity of his theory on Greek statues. He developed the proportions of Apollo Belvedere in the most detail. Greek vases have been examined architectural structures different eras, plants, animals, bird eggs, musical tones, poetic meters.

Zeising gave a definition to the golden ratio and showed how it is expressed in straight line segments and in numbers. When the figures expressing the lengths of the segments were obtained, Zeising saw that they amounted to Fibonacci series.

A series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. known as the Fibonacci series. The peculiarity of the sequence of numbers is that each of its members, starting from the third, equal to the sum of the previous two 2 + 3 = 5; 3 + 5 = 8; 5 + 8 = 13, 8 + 13 = 21; 13 + 21 = 34, etc., and the ratio of adjacent numbers in the series approaches the ratio of the golden division.

So, 21: 34 = 0.617, and 34: 55 = 0,618. (or 1.618 , if divided larger number to less).

Fibonacci series could have remained only a mathematical incident, if not for the fact that all researchers of the golden division in the plant and animal world, not to mention art, invariably came to this series as an arithmetic expression of the law of the golden section.

Golden ratio in art

Back in 1925, art critic L.L. Sabaneev, having analyzed 1,770 musical works by 42 authors, showed that the vast majority of outstanding works can be easily divided into parts either by theme, or by intonation structure, or by modal structure, which are in relation to each other golden ratio.

Moreover, the more talented the composer, the more golden sections are found in his works. In Arensky, Beethoven, Borodin, Haydn, Mozart, Scriabin, Chopin and Schubert, golden sections were found in 90% of all works. According to Sabaneev, the golden ratio leads to the impression of a special harmony of a musical composition.

In cinema, S. Eisenstein artificially constructed the film Battleship Potemkin according to the rules of the “golden ratio”. He broke the tape into five parts. In the first three, the action takes place on a ship. In the last two - in Odessa, where the uprising is unfolding. This transition to the city occurs exactly at the golden ratio point. And each part has its own fracture, which occurs according to the law of the golden ratio.

Golden ratio in architecture, sculpture, painting

One of the most beautiful works ancient greek architecture is the Parthenon (5th century BC).

The figures show a number of patterns associated with the golden ratio. The proportions of the building can be expressed through various degrees numbers Ф=0.618...

On the floor plan of the Parthenon you can also see the “golden rectangles”:

We can see the golden ratio in the building of Notre Dame Cathedral (Notre Dame de Paris) and in the Pyramid of Cheops:

Not only the Egyptian pyramids were built in accordance with the perfect proportions of the golden ratio; the same phenomenon was found in the Mexican pyramids.

The golden proportion was used by many ancient sculptors. Known golden ratio statues of Apollo Belvedere: the height of the person depicted is divided by the umbilical line in the golden ratio.

Moving on to examples of the “golden ratio” in painting, one cannot help but focus on the work of Leonardo da Vinci. Let's look closely at the painting "La Gioconda". The composition of the portrait is based on “golden triangles”.

Golden ratio in fonts and household items

Golden ratio in nature

Biological studies have shown that, starting with viruses and plants and ending with the human body, the golden proportion is revealed everywhere, characterizing the proportionality and harmony of their structure. The golden ratio is recognized as a universal law of living systems.

It was found that the numerical series of Fibonacci numbers characterizes the structural organization of many living systems. For example, the helical leaf arrangement on a branch is a fraction (number of revolutions on the stem/number of leaves in a cycle, eg 2/5; 3/8; 5/13), corresponding to the Fibonacci series.

The “golden” proportion of five-petaled flowers of apple, pear and many other plants is well known. The carriers of the genetic code - DNA and RNA molecules - have a double helix structure; its dimensions almost completely correspond to the numbers of the Fibonacci series.

Goethe emphasized nature's tendency toward spirality.

The spider weaves its web in a spiral pattern. A hurricane is spinning like a spiral. A frightened herd of reindeer scatters in a spiral.

Goethe called the spiral the “curve of life.” The spiral was seen in the arrangement of sunflower seeds, pine cones, pineapples, cacti, etc.

Flowers and seeds of sunflowers, chamomiles, scales in pineapple fruits, conifer cones are “packed” in logarithmic (“golden”) spirals, curling towards each other, and the numbers of “right” and “left” spirals are always related to each other, like neighboring numbers Fibonacci.

Consider a chicory shoot. A shoot has formed from the main stem. The first leaf was located right there. The shoot makes a strong ejection into space, stops, releases a leaf, but this time it is shorter than the first one, again makes an ejection into space, but with less force, releases a leaf of an even smaller size and is ejected again.

If the first emission is taken as 100 units, then the second is equal to 62 units, the third – 38, the fourth – 24, etc. The length of the petals is also subject to the golden proportion. In growing and conquering space, the plant maintained certain proportions. The impulses of its growth gradually decreased in proportion to the golden ratio.

In many butterflies, the ratio of the sizes of the thoracic and abdominal parts of the body corresponds to the golden ratio. Folding its wings, the moth forms a regular equilateral triangle. But if you spread your wings, you will see the same principle of dividing the body into 2,3,5,8. The dragonfly is also created according to the laws of the golden proportion: the ratio of the lengths of the tail and body is equal to the ratio of the total length to the length of the tail.

In a lizard, the length of its tail is related to the length of the rest of the body as 62 to 38. You can notice the golden proportions if you look closely at a bird's egg.