In a person's life, whether he realizes it or not, incredible things happen. The signs of the universe, like a unique fingerprint, surround every living creature on earth. The PHI number is the Fibonacci numerical code that personifies what any soul is drawn to. Harmony, balance and tranquility.
Fibonacci numbers can be seen in the most unexpected things. The natural codes displayed on the planet and hidden in the universe are a manifestation of forces that are not so easy to comprehend. How can the value of FI for self-knowledge help a person?
The meaning of numbers in people's lives
Harmony is the true goal of any philosopher and thinker. On an intuitive level, every living being seeks that unshakable invisible balance that brings a feeling of serene happiness. PI numbers are used in science, art, philosophy. A unique set of numbers allows you to create masterpieces, unexplained even by skeptics and atheists. What magical powers are associated with the golden number?
The meaning of FI has a complicated history. Like a person, a unique code went a long way before the curious mind of a person did not find a worthy use for it. The mysterious number has its own specific numerical framework, a conventional meaning that is taught in school. Numbers are the pillar of the economy, and indeed the world as a whole. Numerology, as the most exact science of all esoteric ones, has been helping humanity for hundreds of years.
The role of the Golden Number in our lives
How did it all start? Each person is born under a certain position of the planets. Lunar, standard and magic calendars allow you to create a unique map for any soul. No one is born twice, and no one repeats the path of another. It is on this code, received at birth and unchanged throughout life, that numerologists determine fate. No special attributes are required for magical action. Only accurate, dry calculations. Even those who do not believe in the existence of otherworldly forces listen to the predictions of numerologists. The numbers are unbiased, they are indifferent to the motives of people and their spiritual qualities.
The FI code works on the same principle, looking for a pattern in the universal chaos. By nature, man is incapable of comprehending the whole meaning. PI numbers are a small part of what an ordinary mortal does not even suspect. The formula of the golden ratio helps not only to look into the human soul, but also to understand at least a little bit of the universe. The PI numbers are called the numbers of God, that invisible, but incredible, unconscious force by the human brain. What are the properties of the golden ratio?
Numerology and accurate predictions by numbers
Can you trust numerologists? The numbers from the formula, which is named after the inimitable Fibonacci, are only a small part of what a person can get by examining the series of consecutive numbers. Numbers are everywhere. The lottery, bank account, phone number and sequence number in the queue are all manifestations of order, assembled from hundreds of threads of endless chaos. Before the derivation of the golden ratio formula, similar codes were created by scientists around the world. Year after year, people tried to understand their own soul through the discoveries of the external world, through the universe - so distant and incomprehensible. A drunk FI is a pattern without a hint of chance. In numerology, there is such a concept of "natural chance" that speaks of how little a person understands the nature of the events taking place.
Fear arises from anxiety and uncertainty, and it pushes a person to terrible, sometimes irreversible actions. The harmony that the golden ratio describes is between the knowledge of one's own nature and the realization that is not subject to human will.
The FI code digits derived by Fibonacci are not a simple set of numbers.
This is harmony that a person can see with his own eyes, to which he is able to touch and believe, even if limited consciousness does not allow the exercise of faith. The properties of the PI number are studied to this day, like a riddle that escapes right from under the nose. How to find your own destiny by the Fibonacci code? The meaning of the derived formulas is quite simple. All a person needs is to show a little patience and be open to everything new and exciting.
What is the FI number?
What are PI numbers? The numerical value of the sequence is 1.61803398. It means proportion, the same harmony, only in two abstract concepts - more and less. FI means that the smaller part always refers to the larger, and the larger to the whole. Calculation means more than any other coefficient derived by a mathematician or physicist. As a percentage, the proportion is displayed as 62% to 38%. Why is this proportion needed? The golden ratio has spatial and temporal characteristics. The FI code is a cosmic order, it is a product of chaos that has become consistent. In mathematics and geometry, digital representation of harmony is viewed as "asymmetric symmetry." The world order and the meaning of FI are closely intertwined. Regularity, open to the human eye, is a reflection of everything that can happen in the fate of an individual.
Where is the FI number traced? The Golden Ratio consists of a definite, unchanging sequence. The proportion that can be traced in everything that exists on Earth:
- in the human body;
- in nature;
- in the Universe;
- in natural phenomena.
It is not so important to discern a pattern in your own body or in the world around you. The FI sequence exists regardless of whether a person believes in it or not. Why does a person need to know the meaning of proportion? FI is the key to how every living cell in the universe arises. This is knowledge that the highest minds of humanity could not even dream of. The divine principle and possibly the answer to the question that worries every believing soul and atheist: where did man come from, and what is the purpose of his stay on earth?
History of the FI number
Words, like numerical values, surround an adult and a child. The history and meaning of FI will make it possible to understand what it is for, and how the proportion will help change fate? Divine measure, as adepts of different cultures call the golden ratio, associate such earthly miracles: "Mona Lisa", the Egyptian pyramids and an ordinary pine cone. FI unites seemingly incoherent things, phenomena, manifestations of those events that a person does not think about at all.
Leonard of Pisa discovered the first proportions that have remained unchanged to this day. In mathematics, a set of numbers is honored, because they describe the fundamental function for various calculations. A sequence consists of numbers, the sum of which determines the previous value. The first values of the sequence: 0, 1, 1, 2, 3, 5, 8, 13. 4 drops out, because the first four digits of the sequence complete the chain of any other possible pattern. It is impossible to confuse the golden ratio in the code with other functions or proportions.
Such a concept as the golden ratio appeared in ancient Rome. The Egyptians, the Greek Empire, even on the territory of ancient Russia, thinkers tried to understand the essence of the world around them. The meaning of the amazing and interesting proportion was first explained by the monk Luca Pacioli. The divine trinity was considered as the basis of all that exists. Interesting facts about the numerical code amaze people to this day. In mathematics, sequence is used to calculate the most complex formulas. An amazing function describes many changes taking place in the body and soul. It challenges long-studied postulates and opens up a new view of the universe.
FIBONACCI NUMBERS AMAZING REGULARITY [PHI Number & Golden Ratio]
The mystery of the number 1.618034 - the most IMPORTANT number in the world
GOLDEN SECTION
The magic of the FI number
There are many uses for the golden ratio in the modern world. In the human body, in his spirit world and environment. Symmetry is used in literally every art form. Calculations and proportions are the main thing for the future masterpiece. Before the world recognized the power of abstraction, asymmetry was considered a manifestation of ignorance and a lack of talent for the artist. Body measurements for future sculpture, facial features for a painting - the initial sketches are drawn up exclusively thanks to the golden ratio. The golden proportion, as Leonardo da Vinci himself called it, made it possible to recreate the smallest details of the human body. To date, da Vinci's creations are the most detailed, believable and skillfully executed. The methods used by the artist are studied at any university of art.
The human body is the most reliable proof of the existence of the golden ratio. Not only in mathematics, using formulas, you can see the sequence. It can be seen with the naked eye in facial features, in the structure of the body of an adult or a child:
- equivalent height and center point of the navel;
- the distance between the hands (fingertips, wrists, hands to the elbows);
- head size and distance from neck to crown;
- the distance from the center (navel) of the body to the knees, and from the knee caps to the feet;
- symmetry of parts of a human face.
The proportions of various parts in the human body are the conditional golden ratio. Physiognomy is a new science based on old methods of studying the structure of the human body, helping to determine fate by only one facial features or curves of body parts. To have such knowledge means to have incredible insight. To understand the people around you means to love them. Even Aristotle remembered the golden ratio in his works. A code equal to a sequence of numbers, which is named after the brilliant Fibonacci, is placed in the diagram. Such a table allows you to make the most complex calculations in mathematics or design complex buildings.
Beauty formula
They say about the golden ratio as a formula of ideal beauty. From childhood, the child is taught that beauty and harmony comes from within. As an invisible and unshakable force. The symmetry of the body has long been considered an indicator of the divine culture of an adult or a child. The most beautiful people were said to be sent by heaven, deities. It was believed that such people are not so much born as they are born. The modern table, taking into account all the proportions of the human body in accordance with the number of PIs, indicates the likely traits of an ideal woman or man.
Symmetrical and proportional, such people cannot but evoke love and admiration. Beauty and perfection are just words. There are no perfect people. This is the conclusion reached by scientists, thinkers and creators of masterpieces. Only a few reach the standard of the golden ratio, but even they - the creations of God, there are deviations from the section formula. The ideal of beauty occurs in nature in the most unpretentious forms. Snail, auricle, sunflower seeds. Echoes of the Fibonacci proportion surround people day in and day out. This is where true beauty lies.
What features of the human body are considered to be maximally proportional? The standard of beauty includes people with a size ratio:
- height and width of the face;
- the length of the nose and the point of the junction of the lips;
- distance from the point of the chin to the lips and the length of the entire face;
- the width of the mouth and the width of the lips;
- the distance from one nostril to the other and the length of the entire nose;
- distance between eyebrows and pupils.
Striving to comply with the golden ratio is unwise and dangerous. Each person is individual, unique, created not according to a model, but as a result of a million years of evolution, development and mutations. Man is the crown of God's creation, who seeks answers to questions about his own origin. The proportions discovered by scientists or mathematicians are just codes with which a person cognizes not only the body, but also the soul.
Echoes of the PI number in nature
The universe affects every living thing on the planet. Even the oceans change their own motion from the heavenly bodies, and the cosmic energy is so boundless that it sometimes frightens people. The patterns and proportions that can be traced in nature often go unnoticed by the little man.
The pinecone and the arrangement of the seeds in the sunflower is a prime example of the golden ratio. It will be possible to find similar shapes in the peel of a pineapple, in the arrangement of rose petals and in the shape of seashells. Not every person, not every creator of incredible paintings or sculptures can see the wonders of the universe in a simple snail.
Conclusion
Knowledge of the nature of things leads a person to self-knowledge, without discoveries the growth of the human soul and consciousness will not happen. Anyone who knows how to see the special in the usual things, he knows the value of life and the world around him.
There are still many unsolved mysteries in the universe, some of which scientists have already been able to identify and describe. Fibonacci numbers and the golden ratio form the basis for solving the world around, constructing its shape and optimal visual perception by a person, with the help of which he can feel beauty and harmony.
Golden ratio
The principle of determining the size of the golden section lies at the basis of the perfection of the whole world and its parts in its structure and functions, its manifestation can be seen in nature, art and technology. The doctrine of the golden ratio was founded as a result of studies by ancient scientists of the nature of numbers.
It is based on the theory of the proportions and ratios of divisions of segments, which was made by the ancient philosopher and mathematician Pythagoras. He proved that when dividing a segment into two parts: X (smaller) and Y (larger), the ratio of the larger to the smaller will be equal to the ratio of their sum (the entire segment):
The result is the equation: x 2 - x - 1 = 0, which is solved as x = (1 ± √5) / 2.
If we consider the ratio 1 / x, then it is equal to 1,618…
Evidence of the use of the golden ratio by ancient thinkers is given in Euclid's book "Beginnings", written back in the 3rd century. BC, who applied this rule to construct regular 5-gons. Among the Pythagoreans, this figure is considered sacred, since it is both symmetrical and asymmetrical. The pentagram symbolized life and health.
Fibonacci numbers
The famous book Liber abaci by a mathematician from Italy, Leonardo of Pisa, who later became known as Fibonacci, was published in 1202. In it, the scientist for the first time cites the regularity of numbers, in a row of which each number is the sum of 2 previous digits. The sequence of Fibonacci numbers is as follows:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc.
The scientist also cited a number of patterns:
- Any number from the series, divided by the next, will be equal to a value that tends to 0.618. Moreover, the first Fibonacci numbers do not give such a number, but as we move from the beginning of the sequence, this ratio will become more and more accurate.
- If we divide the number from the series by the previous one, then the result will rush to 1.618.
- One number divided by the next one after one will show a value tending to 0.382.
The application of the connection and the laws of the golden ratio, the Fibonacci number (0.618) can be found not only in mathematics, but also in nature, in history, in architecture and construction, and in many other sciences.
Archimedes spiral and golden rectangle
Spirals, which are very common in nature, were investigated by Archimedes, who even derived its equation. The spiral shape is based on the laws of the golden ratio. When it is untwisted, the length is obtained, to which the proportions and Fibonacci numbers can be applied, the step increases evenly.
The parallel between the Fibonacci numbers and the Golden Ratio can be seen by constructing a “golden rectangle” with sides proportional to 1.618: 1. It is constructed, passing from a large rectangle to small ones so that the lengths of the sides will be equal to the numbers from the row. Its construction can be done in reverse order, starting with the box "1". When the corners of this rectangle are connected by lines in the center of their intersection, a Fibonacci spiral or logarithmic spiral is obtained.
The history of the use of golden proportions
Many ancient architectural monuments of Egypt were erected using golden proportions: the famous pyramids of Cheops and others. Architects of Ancient Greece widely used them in the construction of architectural objects such as temples, amphitheaters, stadiums. For example, such proportions were used in the construction of the ancient temple of the Parthenon, (Athens) and other objects that have become masterpieces of ancient architecture, demonstrating harmony based on mathematical laws.
In later centuries, interest in the Golden Ratio subsided, and the patterns were forgotten, but again resumed in the Renaissance, together with the book of the Franciscan monk L. Pacioli di Borgo "Divine Proportion" (1509). It contained illustrations by Leonardo da Vinci, who consolidated the new name "golden ratio". Also, 12 properties of the golden ratio were scientifically proven, and the author talked about how it manifests itself in nature, in art and called it "the principle of building the world and nature."
Vitruvian Man Leonardo
The drawing, which Leonardo da Vinci used to illustrate the book of Vitruvius in 1492, depicts a human figure in 2 positions with arms spread apart. The figure is inscribed in a circle and a square. This drawing is considered to be the canonical proportions of the human body (male), described by Leonardo based on his study in the treatises of the Roman architect Vitruvius.
The navel is considered the center of the body as an equidistant point from the end of the arms and legs, the length of the arms is equal to the height of a person, the maximum shoulder width = 1/8 of the height, the distance from the top of the chest to the hair = 1/7, from the top of the chest to the top of the head = 1/6 etc.
Since then, the drawing has been used as a symbol to show the internal symmetry of the human body.
Leonardo used the term "Golden Ratio" to refer to proportional relationships in the figure of a person. For example, the distance from the waist to the feet is related to the same distance from the navel to the crown as well as the height to the first length (from the waist down). This calculation is done similarly to the ratio of the segments when calculating the golden ratio and tends to 1.618.
All of these harmonious proportions are often used by artists to create beautiful and impressive pieces.
Studies of the Golden Ratio in the 16th-19th Centuries
Using the golden ratio and Fibonacci numbers, research on proportions has been going on for centuries. In parallel with Leonardo da Vinci, the German artist Albrecht Durer was also developing the theory of the correct proportions of the human body. For this, he even created a special compass.
In the 16th century. the question of the connection between the Fibonacci number and the golden ratio was the subject of the works of the astronomer I. Kepler, who was the first to apply these rules to botany.
A new "discovery" awaited the golden ratio in the 19th century. with the publication of "Aesthetic Research" by the German scientist Professor Zeisig. He raised these proportions to absolute and announced that they are universal for all natural phenomena. He conducted studies of a huge number of people, or rather their bodily proportions (about 2 thousand), based on the results of which conclusions were drawn about statistically confirmed patterns in the ratios of various parts of the body: the length of the shoulders, forearms, hands, fingers, etc.
Objects of art (vases, architectural structures), musical tones, dimensions when writing poems were also studied - Zeisig reflected all this through the lengths of segments and numbers, he also introduced the term "mathematical aesthetics". After receiving the results, it turned out that a Fibonacci series is obtained.
Fibonacci number and the golden ratio in nature
In the plant and animal world, there is a tendency to form formation in the form of symmetry, which is observed in the direction of growth and movement. Division into symmetrical parts, in which the golden proportions are observed, is a pattern inherent in many plants and animals.
The nature around us can be described using Fibonacci numbers, for example:
- the location of the leaves or branches of any plants, as well as the distances, are related to the number of given numbers 1, 1, 2, 3, 5, 8, 13 and further;
- sunflower seeds (scales on cones, pineapple cells), arranged in two rows along twisted spirals in different directions;
- the ratio of the length of the tail and the whole body of the lizard;
- the shape of the egg, if you draw a line conditionally through its wide part;
- the ratio of the size of the fingers on a person's hand.
And, of course, the most interesting shapes are the spiraling snail shells, the patterns on the cobwebs, the movement of the wind inside the hurricane, the double helix in DNA and the structure of galaxies - all of which include a sequence of Fibonacci numbers.
The use of the golden ratio in art
Researchers looking for examples of the use of the golden ratio in art are examining various architectural objects and paintings in detail. Famous sculptural works, the creators of which adhered to the golden proportions, are known - statues of Olympian Zeus, Apollo Belvedere and
One of the creations of Leonardo da Vinci - "Portrait of Mona Lisa" - has been the subject of research by scientists for many years. They found that the composition of the work entirely consists of "golden triangles" combined together to form a regular pentagon-star. All da Vinci's works are evidence of how deep his knowledge was in the structure and proportions of the human body, thanks to which he was able to catch the incredibly mysterious smile of La Gioconda.
Golden ratio in architecture
As an example, scientists have studied architectural masterpieces created according to the rules of the "golden section": the Egyptian pyramids, the Pantheon, the Parthenon, Notre Dame de Paris Cathedral, St. Basil's Cathedral, etc.
The Parthenon, one of the most beautiful buildings in Ancient Greece (5th century BC), has 8 columns and 17 on different sides, the ratio of its height to the length of the sides is 0.618. The protrusions on its facades are made according to the "golden ratio" (photo below).
One of the scientists who invented and successfully applied the improvement of the modular system of proportions for architectural objects (the so-called "modulator") was the French architect Le Corbusier. The modulator is based on a measuring system associated with the conditional division into parts of the human body.
The Russian architect M. Kazakov, who built several residential buildings in Moscow, as well as the buildings of the Senate in the Kremlin and the Golitsyn Hospital (now the 1st Clinical named after N.I. Pirogov), was one of the architects who used laws in the design and construction about the golden ratio.
Applying proportions in designs
In clothing design, all fashion designers make new images and models taking into account the proportions of the human body and the rules of the golden ratio, although by nature not all people have ideal proportions.
When planning landscape design and creating volumetric park compositions using plants (trees and shrubs), fountains and small architectural objects, the laws of "divine proportions" can also be applied. After all, the composition of the park should be focused on creating an impression on the visitor, who can freely navigate in it and find a compositional center.
All the elements of the park are in such proportions that with the help of the geometric structure, mutual arrangement, illumination and light, to give a person the impression of harmony and perfection.
Application of the Golden Ratio in Cybernetics and Engineering
The laws of the golden ratio and Fibonacci numbers are also manifested in energy transitions, in processes occurring with elementary particles that make up chemical compounds, in space systems, in the genetic structure of DNA.
Similar processes occur in the human body, manifesting themselves in the biorhythms of his life, in the action of organs, for example, the brain or vision.
Algorithms and patterns of golden proportions are widely used in modern cybernetics and computer science. One of the simple tasks that beginner programmers are given to solve is to write a formula and determine the sum of Fibonacci numbers up to a certain number using programming languages.
Modern research on the theory of the golden ratio
Since the middle of the 20th century, interest in the problems and the influence of the patterns of golden proportions on human life has been growing sharply, and on the part of many scientists of various professions: mathematicians, ethnos researchers, biologists, philosophers, medical workers, economists, musicians, etc.
Since the 1970s, The Fibonacci Quarterly magazine has been published in the United States, where works on this topic are published. In the press there are works in which the generalized rules of the golden ratio and the Fibonacci series are used in various branches of knowledge. For example, for coding information, chemical research, biological, etc.
All this confirms the conclusions of ancient and modern scientists that the golden ratio is multilaterally related to the fundamental issues of science and manifests itself in the symmetry of many creations and phenomena of the world around us.
The Phi number is recognized as the most beautiful in the universe ... Despite the mystical origin, the Phi number played a unique role - the role of the basic block in the construction of all living things. All plants, animals, and human beings correspond to physical proportions approximately equal to the root of the ratio of Phi to 1 ... Phi is 1.618. The Phi number is obtained from the Fibonacci sequence, a mathematical progression known not only because the sum of two adjacent numbers in it is equal to the next number, but also because the quotient of two adjacent numbers has a unique property - closeness to the number 1.618, that is, to the Phi number! This omnipresence of Phi in nature indicates the connection of all living things. Sunflower seeds are arranged in spirals, counterclockwise and the ratio of the diameter of each of the spirals to the diameter of the next one is Phi. Spiral twisted corn cob leaves, arrangement of leaves on plant stems, segmental parts of insect bodies. And all of them in their structure obediently follow the law of "divine proportion". Drawing by Leonardo da Vinci depicting a naked man in a circle. No one better than da Vinci understood the divine structure of the human body, its structure. He was the first to show that the human body consists of "building blocks", the ratio of the proportions of which is always equal to our cherished number. If you measure the distance from the top of your head to the floor, then divide by your height, then we will see what the number will be. It is Fi - 1.618. The Fibonacci mathematician lived in the twelfth century (1175). He was one of the most famous scientists of his time. Among his greatest achievements is the introduction of Arabic numerals instead of Roman ones. He discovered the Fibonacci summation sequence. This mathematical sequence occurs when, starting from 1, 1, the next number is obtained by adding the previous two. This sequence tends asymptotically to some constant relation. However, this ratio is irrational, that is, it is a number with an infinite, unpredictable sequence of decimal digits in the fractional part. It is impossible to express it precisely. If any member of the Fibonacci sequence is divided by the one that precedes it (for example, 13: 8), the result will be a value that fluctuates around the irrational value of 1.61803398875 ... and once again surpasses or does not reach it. But, even having spent Eternity on this, it is impossible to know the ratio exactly, to the last decimal digit. When dividing any member of the Fibonacci sequence by the next one, you get just the inverse of 1.618 (1: 1.618). But this is also a very unusual, even remarkable phenomenon. Since the original ratio is an infinite fraction, this ratio should also have no end. Many have tried to unravel the secrets of the pyramid at Giza. Unlike other Egyptian pyramids, this is not a tomb, but rather an insoluble puzzle of number combinations. The remarkable ingenuity, skill, time and work of the architects of the pyramid, which they used in the construction of the eternal symbol, indicate the extreme importance of the message that they wanted to convey to future generations. Their era was pre-written, pre-hieroglyphic and symbols were the only means of recording discoveries. The key to the geometrical-mathematical secret of the pyramid in Giza, which had been a mystery to mankind for so long, was actually given to Herodotus by the temple priests, who informed him that the pyramid was built so that the area of each of its faces was equal to the square of its height. The area of the triangle is 356 * 440/2 = 78320. The area of the square is 280 * 280 = 78400. The facet length of the pyramid at Giza is 783.3 feet (238.7 m), the height of the pyramid is 484.4 feet (147.6 m). The length of the face divided by the height leads to the ratio Ф = 1.618. A height of 484.4 feet corresponds to 5813 inches (5-8-13) - these are numbers from the Fibonacci sequence. These interesting observations suggest that the design of the pyramid is based on the proportion Φ = 1.618. Modern scholars are inclined to interpret that the ancient Egyptians built it with the sole purpose of transmitting knowledge that they wanted to preserve for future generations. Intensive studies of the pyramid at Giza showed how extensive knowledge in mathematics and astrology was at that time. In all internal and external proportions of the pyramid, the number 1.618 plays a central role. Not only are the Egyptian pyramids built according to the perfect proportions of the golden ratio, the same phenomenon is found in the Mexican pyramids. It is thought that both the Egyptian and the Mexican pyramids were erected at about the same time by people of common descent.
Leonardo Fibonacci- one of the greatest mathematicians of the Middle Ages. In one of his works, "The Book of Calculations," Fibonacci described the Indo-Arabic system of calculus and the advantages of using it over the Roman one.Fibonacci numbers or Fibonacci sequence- a numerical sequence - with a number of properties. For example, the sum of two adjacent numbers of the sequence gives the value of the next one (for example, 1 + 1 = 2; 2 + 3 = 5, etc.), which confirms the existence of the so-called Fibonacci ratios, i.e. constant ratios.
The Fibonacci sequence starts like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 ...
Fibonacci sequence properties
1.
The ratio of each number to the next one more and more tends to 0.618 as the ordinal number increases. The ratio of each number to the previous one tends to 1.618 (inverse to 0.618). The number 0.618 is called (PI).
2.
When dividing each number by the next one, after one, the number 0.382 is obtained; on the contrary - respectively 2.618.
3.
Choosing the ratios in this way, we get the main set of Fibonacci coefficients:… 4.235, 2.618, 1.618, 0.618, 0.382, 0.236.
The connection between the Fibonacci sequence and the "golden ratio"
The Fibonacci sequence asymptotically (approaching more and more slowly) tends to some constant ratio. However, this ratio is rational, that is, it is a number with an infinite, unpredictable sequence of decimal digits in the fractional part. It is impossible to express it precisely.
If any member of the Fibonacci sequence is divided by the one that precedes it (for example, 13: 8), the result will be a value fluctuating around the irrational value of 1.61803398875 ... and once again rising, then not reaching it. But even having touched Eternity on it, it is impossible to know the ratio exactly, up to the last decimal digit. For the sake of hardness, we will translate it in the form of 1.618. Specific names for this ratio began to be given even before Luca Pacioli (a mid-century mathematician) called it the Divine Proportion. Among its modern names there are such as Golden Ratio, Golden Mean, and the ratio of rotating squares. Keplep called this relationship one of the "treasures of geometry". In algebra its designation by the Greek letter phi is generally accepted.
Ф = 1.618
Golden ratio- this is such a proportional division of a segment into unequal parts, in which the entire segment refers to the larger part in the same way as the larger part itself refers to the smaller one; or in other words, the smaller segment relates to the larger one as the larger one to everything.
The segments of the golden ratio are expressed by the infinite irrational fraction 0.618 ..., if AB is taken as one, AC = 0.382 .. As we already know the numbers 0.618 and 0.382 are the coefficients of the Fibonacci sequence.
Fibonacci and Golden Ratios in nature and history
It is important to note that Fibonacci, as it were, reminded his sequence to humanity. She was known even to the ancient Greeks and Egyptians. Indeed, since then in nature, architecture, fine arts, mathematics, physics, astronomy, biology and many other fields, patterns have been found described by the Fibonacci coefficients. It's amazing how many constants can be calculated using the Fibonacci sequence, and how its members appear in a huge number of combinations. However, it would not be an exaggeration to say that this is not just a game with numbers, but the most important mathematical expression of natural phenomena ever discovered.
The examples below show some interesting applications of this mathematical sequence.
1. Spiral shell. If you unfold it, you get a length slightly inferior to the length of the snake. A small ten-centimeter shell has a spiral 35 cm long. The shape of the spirally curled shell attracted the attention of Archimedes. The point is that the ratio of the measurements of the shell curls is constant and equal to 1.618. Archimedes studied the spiral of shells and derived the equation for the spiral. The spiral drawn from this equation is named after him. The increase in her step is always uniform. Currently, the Archimedes spiral is widely used in technology.
2. Plants and animals. Even Goethe emphasized the tendency of nature to spiral.
The helical and spiral arrangement of leaves on tree branches was noticed long ago. The spiral was seen in the arrangement of sunflower seeds, in pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch of sunflower seeds, pine cones, the Fibonacci series manifests itself, and therefore the law of the golden ratio manifests itself. The spider weaves the web in a spiral manner. A hurricane is spinning in a spiral. A frightened herd of reindeer scatters in a spiral. The DNA molecule is twisted in a double helix. Goethe called the spiral "the curve of life".
In both the plant and the animal world, the formative tendency of nature is persistently breaking through - symmetry with respect to the direction of growth and movement. Here, the golden ratio appears in the proportions of parts perpendicular to the direction of growth. Nature has carried out the division into symmetrical parts and golden proportions. In the parts, the repetition of the structure of the whole is manifested.
Pierre Curie at the beginning of this century formulated a number of profound ideas of symmetry. He argued that one cannot consider the symmetry of any body without considering the symmetry of the environment. The patterns of golden symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and space systems, in the genetic structures of living organisms. These patterns, as indicated above, are in the structure of individual organs of a person and the body as a whole, and are also manifested in biorhythms and the functioning of the brain and visual perception.
3. Space. It is known from the history of astronomy that I. Titius, a German astronomer of the 18th century, with the help of this series (Fibonacci) found the regularity and order in the distances between the planets of the solar system
However, one case that seemingly contradicted the law: there was no planet between Mars and Jupiter. Concentrated observation of this region of the sky led to the discovery of the asteroid belt. This happened after the death of Titius at the beginning of the 19th century.
The Fibonacci series is widely used: it is used to represent the architectonics of living beings, and man-made structures, and the structure of the Galaxies. These facts are evidence of the independence of the number series from the conditions of its manifestation, which is one of the signs of its universality.
4. Pyramids. Many have tried to unravel the secrets of the pyramid at Giza. Unlike other Egyptian pyramids, this is not a tomb, but rather an insoluble puzzle of number combinations. The remarkable ingenuity, skill, time and labor of the architects of the pyramid, which they used in the construction of the eternal symbol, indicate the extreme importance of the message that they wanted to convey to future generations. Their era was preliterate, pre-hieroglyphic and symbols were the only means of recording discoveries. The key to the geometrical-mathematical secret of the pyramid in Giza, which had been a mystery to mankind for so long, was actually given to Herodotus by the temple priests, who informed him that the pyramid was built so that the area of each of its faces was equal to the square of its height.
Some modern scholars are inclined to interpret that the ancient Egyptians built it with the sole purpose of transmitting knowledge that they wanted to preserve for future generations. Intensive studies of the pyramid at Giza showed how extensive knowledge in mathematics and astrology was at that time. In all internal and external proportions of the pyramid, the number 1.618 plays a central role.
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The second article from the planned series for the media.
So, if the Phi number - the number of the Golden Section is fixed in the structure of the entire multitude of Terrestrial forms - from the simplest to mammals - it means that a certain fundamental law of Terrestrial form-creation is expressed through this number.
It takes energy to create any shape.
To create a living form, you need the energy of Life.
Our wise ancestors believed that both people and everything on Earth came from Light, the Primordial Light, to which they gave the name RA. This is not the visible light available to our eye, but Light as the highest, ultrafine form of energy, when it is compacted, that is, the vibrational-frequency characteristic is lowered, turning into visible light, color, sound and, finally, dense matter, and thus creating our entire visible and invisible world in its endless variety of forms and mutual transition of energies.
We are the children of Light and we carry a particle of it through our lives - “with the Light around the world” - this is how we greeted each other when they met buffoons, who were not “pea jesters”, as it is customary to depict them in modern “art”, but messengers, magi - wanderers, magicians in a simple and clear sense - "who can, he is a magician."
The Phi number determines the Law of refraction of Life energy - the energy of the Primordial Light - into less energy-saturated types of energy, including dense matter.
When it comes to the Golden Ratio, it is customary to illustrate the story roughly as follows:
(a + c) / c = c / a = Phi
What do we see? ONE segment, divided into TWO in a certain ratio.
There is nothing more deceiving than the so-called self-evident things.
Without going into mathematical details (you can see them in LJ ), let's say that solving the proportion of the Golden Section in order to determine the value of the Phi number leads us to the conclusion that the Golden Ratio is not ONE or TWO segments, but a TRIUNITY of segments combined into a very remarkable right-angled triangle, which we will call the Golden Triangle.
We will talk about the Sacred Egyptian Triangle later, but let's pay attention - the figure clearly shows that these two triangles are LIKE.
If b = Phi is the vertical energy of Life, the energy of the Primordial Light, then a =√ Phi is the maximum possible refraction of the energy of Life in the process of shaping, that is, the refraction of subtle energy into a denser one, in other words, the creation of a form derived from the Primordial Light.
At present, the fact that the Egyptian pyramids are not the tombs of the pharaohs, but powerful energy converters, one of the functions of which is to fix the Earth in its orbit, which has been constant for a significant time interval, is fairly well known. The pyramids, among their other amazing qualities, have one particularly remarkable - they were practically not subject to destruction from the moment of their creation until very recently, this property is recorded in the proverb "Everything in the world is afraid of Time, but even Time is afraid of the Pyramids."
So, all three pyramids of Giza in their section have (and I want to say - they rely on) Golden triangles, and the accuracy of the correspondence of the angle of inclination of the giant faces to the angle of the Golden Triangle, the value of which51 ° 50" , boggles the imagination.
Interestingly enough, for the pyramids of Cheops and Mykerinus, the base-to-height ratio corresponds to Phi with great accuracy.
Separately, I would like to note that it is always said about CONFORMITY, not EQUALITY. The equality of quantities does not exist, since in this case there would be no need for the manifestation of one of the quantities.
There is proportionality and consonance that arise when each quantity is tuned to the energy of Life within oneself, and then through tuning to the Unified stream of Life the quantities become consonant and proportional to each other.
Consonance is resonance, sounding in unison, it does not imply a complete coincidence of vibrations - the string begins to sound in response to the other string, although their characteristics are different. This "difference" can be within plus or minus 3%. The music of the Universe is generated by the sound of different strings, there are many of them, but they are united into groups by the principle of consonance.
Consonance and proportionality are the conditions for the conduction and transformation of energies.
Let us consider in more detail the Sacred Egyptian triangle is a right-angled triangle, the larger leg of which is 4, the smaller leg - 3, the hypotenuse - 5. The ratios of the corresponding sides of the Sacred Egyptian and Golden triangles a "/ a, c" / c, b "/ b differ no more than by 3%, that is, the Sacred Egyptian is a triangle similar to the Golden one and having integer values of the sides, that is, it is an easy-to-manufacture model of the Golden Triangle that allows you to create volumes with maximum energy filling through the use of the Law of Refraction of Energy.
Let's take a look at what a number is from this point of view. It would seem that what is common between any number, for example, 5, and the Golden Ratio?
Indeed, modern education has taught us to look at any number or as a meaningless abstraction - well, who thinks about the essence of the number 3 at the time of its use? - or as an expression of some amount - 2 apples, 100 rubles ...
What does the huge Universe and the world of any person consist of? From an infinite set of commensurate quantities that interact with each other.
Any number is a reflection of one of the many interactions, this is a ratio or ratio of two quantities, all these ratios are derivatives of the energy of Life, they carry the energy of Life, that is, they are Gold-plated, are some degree or part of the Phi number, which means that irrational, like the Phi number. Whole numbers do not really exist, any integer is an irrational number, rounded to the nearest whole, just like any fractional number is an irrational number, rounded to a certain degree of precision. As with the Phi number - you can ACCEPT its values equal to 1.618, but from this it WILL NOT BECOME equal to 1.618.
In the light of the above, let us consider the essence and origin of the concepts of NUMBER and NUMBER.
Our ancestors used the concept of DIGITS, the concept of "number" somehow imperceptibly appeared later and also imperceptibly became commonly used, and "digital" began to be considered obsolete ... equivalent?
NUMBER - the energy of the Primordial Light, refracted through Phi.
The essence of the concept of DIGITS is irrational and endless, it lies beyond the boundaries of Earth time, in a different space-time determination.
The concept of NUMBER is the named (pronounced) energy.
Number is a rational concept, driven into the framework of a word, truncated, truncated in its endless striving for Perfection.
The architects of Ancient Russia worked not with numbers, but with numbers, and, even rounding off values in accordance with the possibilities of their practical reproduction, they thought not in limited segments, but in processes of endless change, eternally striving for Perfection.
