Quite often they say 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 mathematically express the golden ratio, and all of these methods have their own certain simplicity, accuracy and charm. Euclid described it as "a section in the extreme and average ratio." A more "mathematical" expression looks like this: if the golden ratio is x, then . Or like this: x / 1 \u003d 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 one."
Interesting fact about the golden ratio number 3. Golden rectangles can be divided into an endless number of golden rectangles decreasing in size, “cutting off” parts from them along the shortest line. In terminology Greek school mathematicians, this property makes the golden rectangle a gnomon - an object capable of maintaining its shape as it grows (or decreases).
A good example of the golden ratio is a credit card that has the same standard sizes worldwide. In accordance with the rules of the golden ratio, the ratio of its small side to the long one is the same as the ratio of the 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 because of its balanced appearance - it does not seem too long or too wide. One way to check if a rectangle is golden is to place two rectangles side by side, one "placed" vertically on a small edge, the other "placed" touching the first one for a long time. If the diagonal passing through the corners of the horizontal rectangle continues to reach the top corner of the vertical rectangle, the rectangles are golden. Much more often this principle is seen in architecture. So, the facade of the UN building in New York is a golden rectangle.
Mathematics in art and nature
There is something prosaic about the golden ratio - at least for those who do not own a mathematical mindset. We are talking about its numerical expression. The value of x in the algebraic expression x 2 - x - 1 \u003d 0 is 1.6180339887 ... and so on without end. However, the golden ratio is most directly related to Western art. To a large extent, this connection appeared due to the work 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 depiction of the Vitruvian Man - appear in Pacioli's De Divina Proportione ("Divine Proportion"), published in 1509. In this book, the basic geometric rules beauty, and the creator was inspired by the 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, the actual measurements show that in reality there are actually no “perfect” bodies. In the twentieth century the golden ratio was looked for in natural forms. Those who did it persistently enough found it in the proportions of the leaves, the distribution of buds on the stem (natural patterns rather approximately obey the Fibonacci sequence principle), and also in the diving 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 eye-pleasing proportion) is dictated by the mathematics of growth, which represents the increase in size of structures without losing their overall shape.
Interesting fact number 5. Actual measurements show that in reality there are no “perfect” bodies that satisfy the golden section rule.
golden spiral
A spiral that unfolds 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 the axial point at a constant angle (Mathematically, it is more correct to formulate as follows: a curve, the tangent to which 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 describe it), the main researcher of its properties. Bernoulli also wanted to have such a spiral engraved on his tombstone, but a mason poorly versed in geometry reproduced the Archimedean spiral there with a more gentle trajectory of divergence.
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 acquainted with the golden rule, humanity no longer cheated on it.
Definition.
The most capacious definition of the golden ratio says that the smaller part refers to the larger one, as the larger part refers to the whole. Its approximate value is 1.6180339887. Rounded percentage the proportions of the parts of the whole will correlate as 62% to 38%. This ratio in the forms of space and time operates.
The ancients saw the golden section as a reflection of the cosmic order, and Johannes Kepler called it one of the treasures of geometry. modern science considers the golden ratio as "Asymmetric Symmetry", calling it in a broad sense a universal rule that reflects the structure and order of our world order.
Story.
The ancient Egyptians had an idea of the golden proportions, they also knew about them in Russia, but for the first time the monk of the onion patcholi scientifically explained the golden ratio in the book "Divine Proportion" (1509), which was supposedly illustrated by Leonardo da Vinci. Pacioli saw the divine trinity in the golden section: the small segment personified the son, the large one the father, and the whole the holy spirit.
The name of the Italian mathematician Leonardo Fibonacci is directly connected with the golden section rule. As a result of solving one of the problems, the scientist came to 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 Junior Members of This Infinite Proportion in the Sum Give the Third Member, and Any Two Last Members, If Added, Give the Next Member, and the same Proportion is Preserved to Infinity." Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden section in all its manifestations
Fibonacci numbers - 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 golden division.
Over time, the rule of the golden ratio turned into an academic routine, and only the philosopher Adolf Zeising in 1855 brought it back to a second life. 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, it includes the ratio of the tail and body of the lizard, the distance between the leaves on the branch, there is a golden ratio and in the shape of an egg, if conditional line pass 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 proportions of the golden section. In his opinion, one of the most interesting forms is spiraling.
Even Archimedes, paying attention to the spiral, derived an equation based on its shape, which is still used in technology. Later, Goethe noted nature's attraction to spiral forms, calling the spiral "Crooked Life". Modern scientists have found that such manifestations of spiral forms in nature as a snail shell, the arrangement of sunflower seeds, 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 section. Man is a universal form for testing the laws of the golden section. Of course, by nature, not all people have ideal proportions, which creates certain difficulties with the selection of clothes.
In the diary of Leonardo da Vinci there is a drawing of a naked man inscribed in a circle, in two positions superimposed on each other. Based on the studies 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, exploring the proportionality of man, did a colossal job. He measured about two thousand human bodies, as well as many ancient statues and deduced that the golden ratio expresses the average law. In a person, almost all parts of the body are subordinate to him, but the main indicator of the golden section 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.
Art of spatial forms.
The artist Vasily Surikov said that "there is an Immutable Law in the Composition, when nothing can be removed or added to the picture, even an extra point cannot be put, this is Real Mathematics." For a long time artists 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 Dürer used the proportional compass invented by him to determine the points of the golden section.
Art critic F. v. Kovalev, having studied in detail the painting by Nikolai Ge "Alexander Sergeevich Pushkin in the Village of Mikhailovsky", notes that every detail of the canvas, whether it be a fireplace, a bookcase, an armchair or the poet himself, is strictly inscribed in golden proportions.
Researchers of the golden section tirelessly study and measure the masterpieces of architecture, claiming that they have become such because they were created according to the golden canons: they include the great pyramids of Giza, Notre Dame Cathedral, St. Basil's Cathedral, the Parthenon.
And today, in any art of spatial forms, they try to follow the proportions of the golden section, since, according to art historians, they facilitate the perception of the work and form an aesthetic sensation in the viewer.
Word, sound and film.
Forms temporarily? Go arts in their own way demonstrate to us the principle of golden division. Literary critics, for example, noticed that the most popular number of lines in the 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. So the climax of the "Queen of Spades" is the dramatic scene of Herman and the Countess, ending with the death of the latter. There are 853 lines in the story, and the culmination falls 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 golden section ratios in strict and free forms 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 golden ratio point usually accounts for the most striking or unexpected musical solution.
Film director Sergei Eisenstein deliberately coordinated the script for his film "Battleship Potemkin" with the rule of the golden section, dividing the tape into five parts. In the first three sections, the action takes place on a ship, and in the last two - in Odessa. The transition to the scenes in the city is the golden mean of the film.
"There is something more, made up of a combination and connection of three things (number, limitation and location), something that miraculously illuminates the whole face of beauty. We call this harmony, which, without a doubt, is the source of all charm and beauty. After all, the purpose and goal of harmony - to order the parts, generally speaking, different in nature, by some perfect ratio so that they correspond to each other, creating beauty ... It covers all human life, permeates the whole nature of things. For everything that nature produces, all this is measured by the law of harmony "And nature has no greater concern than that what it produces be perfect. This cannot be achieved without harmony, because without it the higher harmony of the parts breaks up."The Great Soviet Encyclopedia gives the following definition of the concept of "harmony":
"Harmony is the proportionality of parts and the whole, merging various components object into a single organic whole. In harmony, internal order and measure of being are externally revealed."Formulas of beauty" is already known a lot. For a long time in their creations, people prefer the right ones. geometric shapes- square, circle, isosceles triangle, pyramid, etc. In the proportions of structures, preference is given to integer ratios. Of the many proportions that people have long used when creating harmonic works, there is one, the only and inimitable, which has unique properties. This proportion was called differently - "golden", "divine", "golden section", "golden number", "golden mean". rice. one The "golden proportion" is a mathematical concept and its study is, first of all, the task of science. But it is also a criterion of harmony and beauty, and this is already a category of art and aesthetics. And our Museum, which is dedicated to the study of this unique phenomenon, is undoubtedly a scientific museum dedicated to the study of harmony and beauty from a mathematical point of view." On the website of A.P. Stakhov ( www. goldenmuseum . com) provides a lot of interesting and instructive information about the remarkable properties of the golden section. And this is not surprising. The concept of "golden section" is associated with the harmony of Nature. At the same time, as a rule, the principles of symmetry in animate and inanimate Nature are associated with harmony. Therefore, today you will not surprise anyone with the universality of the manifestation of the principle of the golden section. And each new discovery in the field of revealing another golden ratio no longer amazes anyone, except perhaps the very author of such a discovery. There is no doubt about the universality of this principle. Various reference books contain hundreds of formulas relating the Fibonacci series to the golden ratio, including a number of formulas that reflect interactions in the world of elementary particles. Among these formulas, I would like to note one - Newton's binomial for the golden ratio
where is the number of permutations.
And Newton's binomial, as is known, reflects the power function of the dual relation.
This formula binds the binomial of the golden ratio to the Unit.
Without this principle, in fact, it is impossible to consider a single fundamental problem. In milogia, this proportion is substantiated as the principle of self-sufficiency. And yet, despite the universality, the golden ratio is not always used in practice, and not everywhere.
2
. MONAD AND THE GOLDEN RATIO
The principles of symmetry underlie the theory of relativity, quantum mechanics, solid state physics, atomic and nuclear physics, elementary particle physics. It was shown above that symmetry is one of the manifestations of duality. Therefore, there is nothing surprising in the fact that these principles are most clearly expressed in the properties of the invariance of the laws of nature. It is shown that symmetry and asymmetry are not just interconnected with each other, but they are different forms of manifestation of the duality pattern.
The pattern of duality is one of the main mechanisms for the evolution of living and non-living matter. Indeed, the ability to reproduce in living organisms can be naturally explained only by the fact that in the process of its development the organism completely completes its shell and an attempt to further complicate the structure leads, due to the laws of limitedness and isolation, to transformation from an organism with internal duality into an organism with external duality, i.e., doubling, which is carried out by dividing the original. Then the process is repeated. The pattern of duality is responsible for the creation of duplicate organs in a living organism. This duplication is not a consequence of the evolution of living organisms.
The golden ratio is based on a simple proportion, which is clearly visible in the figure of the golden spiral:
The rules of the golden section were already known in Babylonia and ancient Egypt. The proportions of the pyramid of Cheops, objects from the tomb of Tutankhamun, other works ancient art eloquently testify to this, and the term “golden section” itself belongs to Leonardo da Vinci. Since then, many masterpieces of art, architecture and music have been made with strict observance of the golden ratio, which undoubtedly reflects the structure of our sensory membranes - eyes and ears, the brain - an analyzer of geometric, color, light, sound and other images.
The golden ratio has another secret. It hides the property self-rationing.
Academician Tolkachev V.K. In his book The Luxury of Systems Thinking, he writes about this important property of the golden ratio:
“Once upon a time, Claudius Ptolemy evenly divided a person’s height into 21 segments and singled out two main parts: a large one (major), consisting of 13 segments, and a smaller one (minor) - of 8. At the same time, it turned out that the ratio of the length of the entire human figure to the length of its larger part is equal to the ratio of the larger part to the smaller one....
The golden ratio can be illustrated as follows. If a unit segment is divided into two unequal parts (major and minor) so that the length of the entire segment (i.e. major + minor = 1) is related to major in the same way that major is related to minor:
(major + minor) / major = major / minor = F,
then such a problem has a solution in the form of the roots of the equation x 2 - x - 1 \u003d 0, the numerical value of which is:
X 1 \u003d - 0.618033989 ..., x 2 \u003d 1.618033989 ...,
The first root is denoted by the letter "
F", and second "- F
", but we will use other notation:
F
\u003d 1.618033989 ..., and F -1 \u003d 0.618033989 ...
It is the only number that has the property of being exactly one greater than its inverse ratio."
Note that another equation
X 2
-
y- 1 =
xy
turns into an identity for the following values
X 1
= + 0,618033989...,
y 1
=- 1,618033989...,
x 2
= -1,618033989...,
y 2
= 0,618033989...,
Maybe be in the aggregate these roots and give rise to the life-giving cross
- a cross of the golden section?
The golden section equation
Ф 2 -Ф \u003d 1
whereF 1 \u003d -F -1 \u003d - 0.618033989 ...,
andF 2 \u003d F 1 \u003d 1.618033989 ...,
satisfy the property self-rationing, allowing you to build more complex "constructions" according to "
image and likeness
".
Substituting the roots into the equation
X
(
x-1)=1,we'll get
F 1 (F 1 -1) = 1.618..*1.618..-1.618..=2.618..-1.618..=1
Ф -2 -(-Ф -1)=0.382...+0.6181=1.
Thus, this equation reflects not only the principle self-rationing, arising from the Unified law of evolution of the dual relation (monad), but also the connection of the golden section with Newton's binomial (with the monad).
It is easy to show that the following identities hold
Ф -2 = 0.382...;
Ф -1 =0.618...;
F 1
=1,618...;
F 2
=2,618...;
From where you can directly see that
equation rootsФ 2 -Ф \u003d 1They also have other wonderful properties.
F 1 F -1 \u003d F 0
=1
and
F -1 (F 1 -1) \u003d 1-F -1; F 1 (F -1 -1)=1-F 1 =1;
It characterizes the invariance of one mathematical monad to another by multiplying it by the reciprocal, i.e. we can say that the roots of the golden section equation themselves form
golden, self-normalized monad<Ф -1 ,Ф 1
>
.
Therefore, this equation can rightly be called
the golden ratio.
Additional properties of this equation can be learned by anyone using Newton's binomial and generating functions ( Continuity).
It is not difficult to understand that the process is increasingly complex
"golden monads"will be
"in the image and likeness"
, i.e. this process will be periodically repeated, and all the results are, as it were, closed in the framework of the golden section.
But perhaps the most remarkable properties of the golden ratio are connected, first of all, with the golden ratio equation given above. This equation is dual
X 2 + x - 1 = 0.
The roots of this equation are numerically equal:
X 1 = + 0.618033989..., x 2 = -1.618033989...,
This means that the golden section equations form a golden section cross with crossbars
Here he is, truly goldthe cross underlying the universe! It is directly visible in the right figure that the values of the expression at the poles of the vertical crossbar are equal to 1. From the cross in the left figure it is also seen that with each transition from one crossbar to the second, self-normalization is carried out. Self-normalization occurs in both addition and multiplication. The difference is only in the sign. And it's no coincidence . When moving along the crossbars, we get four more values · when added: 0 and0 , · when multiplying: -0,382 .., and-2,618 . It is easy to show that the following identities hold Ф -2 = 0.382...; Ф -1 =0.618...; F 1 =1,618...; F 2 =2,618...; Using a number of these values, and making a detour along the cross, we will get another golden cross.rice. 2
It is not difficult to show how to form a double cross from these crosses, which generates the law of the Cube.
Below we will show that the six obtained values fit perfectly into the framework of a complex relation - a unique regularity known from projective geometry. And now we will give another figure that directly speaks of the connection between the golden section and the Cube of the Law.rice. 3
rice. four
Compare this drawing, drawn by Leonardo da Vinci, with the previous one. Did you see?
Therefore, the hymn to the golden section can be continued indefinitely. So the Italian mathematician Luca Paciolli in his work "The Divine Proportion" gives 13 properties of the golden section, supplying each of them with epithets -
exceptional, inexpressible, wonderful, supernatural,
etc.
It is difficult to say whether these properties are related to the number 13 or not. But the chromatic scale is associated with both the number 13 and the number 8. Thus, the proportion 13/8 can be represented as 8/8 + 5/8. With these
many spiritual knowledge are also connected by proportions ( The path to oneself).
3. ROWS OF THE GOLDEN RATE
From the above properties of the golden section, it follows that the series
...; Ф -2 = 0.382...; Ф -1 =0.618...; F 0 ; F 1 \u003d 1.618 ...; F 2 \u003d 2.618 ...; ...;
can be continued either to the right or to the left.
Moreover, multiplying this series by
F
+
norF -ngenerates a new series, shifted to the right or left of the original, respectively. Odds
F
+
norF -ncan be considered as the similarity coefficients of golden series.
Golden section series can form a natural series of integers.
Look, these numbers are amazing properties. They form not only the Great Limits of the dual "golden monads". They form the Great Limits of the triads (numbers 5, 8, ..).
They also form a cross (number 9).
But there are other, more fundamental golden series. First of all, Newton's "golden" binomial formula should be given.
Newton's binomial already initially testifies to the existence of a monad (dual relation) and its properties underlie binomial series (arithmetic triangle, etc.). Now we can say that all binomial series can be expressed in terms of the golden ratio. The golden monad of Newton's binomial reflects another the most important property universe. She is
normalized(single).
4. RELATIONSHIP OF THE GOLDEN RATIO WITH THE FIBONACCI SERIES
Nature, as it were, solves the problem from two sides at once and adds up the results. As soon as it gets 1 in total, it moves to the next dimension, where it starts building everything from the beginning. But then she must build this golden ratio along certain rule. Nature does not use the golden ratio right away. She gets it through successive iterations. To generate the golden section, she uses another series, the Fibonacci series.
Fig.5
Rice. 6. Golden Ratio Spiral and Fibonacci Spiral
A remarkable property of this series is that as the numbers of the series increase, the ratio of two neighboring members of this series asymptotically approaches the exact proportion of the Golden Ratio (1: 1.618) the basis of beauty and harmony in the nature around us, including in human relations. Note that Fibonacci himself discovered his famous series, reflecting on the problem of the number of rabbits that should be born from one pair within one year. It turned out that in each subsequent month after the second, the number of pairs of rabbits exactly follows the digital series, which now bears his name. Therefore, it is no coincidence that man himself is arranged according to the Fibonacci series. Each organ is arranged according to internal or external duality. It should be said that the Fibonacci spiral can be double. There are numerous examples of these double helixes found all over the place. This is how sunflower spirals always correlate with the Fibonacci series. Even in an ordinary pinecone, you can see this double Fibonacci spiral. The first spiral goes in one direction, the second - in the other. If we count the number of scales in a spiral rotating in one direction and the number of scales in the other spiral, we can see that these are always two consecutive numbers of the Fibonacci series. It can be eight in one direction and 13 in the other, or 13 in one and 21 in the other. What is the difference between the golden ratio spirals and the Fibonacci spiral? The golden ratio spiral is perfect. It corresponds to the Primary source of harmony. This spiral has neither beginning nor end. She is endless. The Fibonacci spiral has a beginning, from which it starts “unwinding”. This is a very important property. It allows Nature, after the next closed cycle, to carry out the construction of a new spiral from “zero”. These facts once again confirm that the law of duality gives not only qualitative but also quantitative results. They make us think that the Macroworld and the Microworld around us evolve according to the same laws - the laws of hierarchy, and that these laws are the same for living and inanimate matter. The law of duality is responsible for the fact that the Hierarchy, having in its baggage only this algorithm for the formation of invariant shells, allows you to build the generating functions of these shells, build the Unified Periodic Law of the Evolution of Matter. Let we have the following generating function For n=1 we will have a generating function of the form etc. Now let's try to determine the next member of the generating function by recursive dependence, assuming that this member of the function will be obtained by summing its last two members. For example, if n=1, the value of the third term of the series will be equal to 2. As a result, we will get a series (1-1x+2x2). Then, multiplying the generating function by the operator (1-x) and using the recursive dependence to calculate the next member of the series, we will get the desired generating function. Denoting through the value of the n-th member of the series, and through the previous value of this series and assuming n = 1,2,3, .... the process of sequential formation of the members of the series can be depicted as follows (Table 1).
Table 1.
The table shows that after receiving the next resulting member of the series, this member is substituted into the original polynomial and addition is made to the previous one, then the new resulting member is substituted into the original series, etc. As a result, we get the Fibonacci series. It is directly seen from the table that the Fibonacci series has the property of invariance with respect to the operator (1-x) - it is formed as a series obtained by multiplying the Fibonacci series by the operator (1-x), i.e. the generating function of the Fibonacci series when multiplied by the operator (1 -x) generates itself. And this remarkable property is also a consequence of the manifestation of the pattern of duality. Indeed, in , , it was shown that the repeated application of an operator of the form (1 + x) leaves the structure of the polynomial unchanged, and the Fibonacci series has an additional, more more wonderful properties: each member of this series is the sum of its two last members. Therefore, Nature does not need to remember the Fibonacci series itself. It is only necessary to remember the last two terms of the series and the operator of the form P*(x)=(1-x) responsible for this doubling algorithm in order to obtain the Fibonacci series without error. But why does this series play a decisive role in Nature? The concept of triplicity, which determines the conditions for its self-preservation, can give an exhaustive answer to this question. If the "balance of interests" of the triad is violated by one of its "partners", the "opinions" of the other two "partners" must be corrected. The concept of trinity is especially evident in physics, where “almost” all elementary particles were built from quarks. If we recall that the ratios of fractional charges of quark particles make up a series, and these are the first members of the Fibonacci series, which are necessary for the formation of other elementary particles. It is possible that the Fibonacci spiral can also play a decisive role in the formation of the pattern of limitedness and closedness of hierarchical spaces. Indeed, imagine that at some stage of evolution, the Fibonacci spiral has reached perfection (it has become indistinguishable from the golden ratio spiral), and for this reason the particle must be transformed into the next “category”. The wonderful properties of the Fibonacci series are also manifested in the numbers themselves that are members of this series. Let's arrange the members of the Fibonacci series vertically., And then to the right, in descending order, we write integersEach line begins and ends with a Fibonacci number, i.e. there are only two such numbers in each line. The underlined numbers - 4, 7, 6, 11, 10, 18, 16, 29, 26, 47, 42 have special properties (the second level of the Fibonacci series hierarchy):1 2 32 543 8765 13 12 11 1 1 098 21 20 19 18 17 16 1514 13 34 33 32 31 30 29 28 27 26 25 24 23 22 21 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 ....
We have obtained a fractional Fibonacci series, which, perhaps, "profess" the collective spins of elementary particles and atoms of chemical elements. The next level of the hierarchy is formed as a result of splitting the intervals between the Fibonacci numbers and the selected numbers. For example, the numbers 52 and 50 from the interval 55-47 will rise to the third step of the hierarchy. The process of structuring a series of natural numbers can be continued, since the properties of periodicity and multi-level the structure of matter is reflected even in the properties of the Fibonacci series itself. But the Fibonacci series has another secret that reveals the essence of the periodicity of changes in the properties of a dual relation (monad). Above, the range of changes in the properties of the dual relation, which characterizes its norm of self-sufficiency, was determined U=<2/3, 1) Let's build a Fibonacci series for this range L=(5-4)/(4-3)= 1/1 (8-7)/(7-5) = 1/2 and (8-6)/(6-5)= 2/1 (13-11)/(11-8) = 2/3 and (13-10)/(10-8) = 3/2 (21-18)/(18-13) = 3/5 and (21-16)/(1b-13) = 5/3 (34-29)/(29-21) = 5/8 and (34-26)/(26-21) = 8/5 (55-47)/(47-34) = 8/13 and (55-42)/(42-34) = 13/8
We'll getL-tetrahedron, characterizing an increasing spiral of duality evolution. Let's continue this process. An attempt to go beyond this range of the norm of self-sufficiency will lead to its normalization, i.e. the first element in D-tetrahedron will be characterized by a norm of self-sufficiency equal to 1,0 . But, continuing this process further, we will be forced to constantly renormalize. Therefore, evolution cannot continue? But there is an answer in the question itself. After renormalization, the evolution should start from the beginning, but in the opposite direction, i.e. when forming a "parallel" D-tetrahedron, the sign of the number must change and the Fibonacci series begins to reverse.
D=Then the general series characterizing the norm of self-sufficiency of the "star tetrahedron" will be characterized by the relations
U=The stable state of the star tetrahedron will depend on the appropriate conjugation of the L- and D-tetrahedra. For U=1 we will have a cube. With U=2/3 we get self-sufficient star tetrahedron, with self-sufficient L- and D-tetrahedra. For smaller values, the stable state of the star tetrahedron will be achieved only by joint efforts of L- and D-tetrahedra. Obviously, in this case, the minimum value of the norm of self-sufficiency of the star tetrahedron will be equal to U=1/3, i.e. two n e self-sufficient tetrahedron jointly form self-sufficient star tetrahedron U. In the most general case, the stable states of the star tetrahedron U can be illustrated, for example, by the following diagram.
Rice. 7
The last drawing shows a figure resembling a Maltese cross, with eight peaks. i.e. this figure again evokes associations with the star tetrahedron.
The following information testifies to the miraculous properties of the Fibonacci series, its periodicity ( Mikhailov Vladimir Dmitrievich, "Live Information Universe", 2000, Russia, 656008, Barnaul, st. Partisan house. 242).
p.10."The laws of the "golden proportion", "golden section" are associated with the Fibonacci digital series, discovered in 1202, is a direction in the theory of information coding. Over the centuries-old history of the knowledge of Fibonacci numbers, the relations (numbers) formed by its members and their various invariants have been scrupulously studied and generalized, but have not been fully deciphered. Mathematical sequence of a series of Fibonacci numbers represents a a sequence of numbers, where each subsequent member of the series, starting from the third, is equal to the sum of the two previous ones: 1,1,2,3,5,8,13,21,34,55,89,144,233 ... to infinity. ... The digital code of civilization can be determined using various methods in numerology. For example, by converting complex numbers to single digits (for example: 13 is (1+3)=4, 21 is (2+3)=5, etc.) Carrying out a similar addition procedure with all the complex numbers of the Fibonacci series, we get the following series of 24 digits: 1 ,1 ,2 ,3 ,5 ,8 ,4 ,3 ,7 ,1 ,8 ,9 ,8 ,8 ,7 ,6 ,4 ,1 ,5 ,6 ,2 ,8 ,1 ,9 further, no matter how much you convert numbers into numbers, after 24 digits the cycle will consistently repeat an infinite number of times ... ...isn't a set of 24 digits a kind of digital code for the development of civilization? C.17 If the Pythagorean Four in the sequence of 24 Fibonacci digits is divided among themselves (as if broken) and superimposed on each other, then a picture of the relationship of 12 dualities of opposite digits arises, where each pair of digits in total gives 9 (duality , giving rise to trinity)....1 1 8 =9 2 1 8 =9 3 2 7 =9 4 3 6 =9 5 5 4 =9 6 8 1 =9 7 4 5 =9 8 3 6 =9 9 7 2 =9 10 1 8 =9 11 8 1 =9 12 9 9 = 18=1+8=9 (my edit)
1 1 1 1 75025
2 1 1 1 75025 3 2 2 2 150050 4 3 3 3 225075 5 5 5 5 375125 6 8 8 8 600200 7 4 1+3 13 4 975325 8 3 2+1 21 3 1575525 9 7 3+4 34 7 2550850 10 1 5+5=10=1 55 1 4126375 11 8 8+9=17=1+7 89 8 667722512 9 1+4+4 144 9 10803600
13 8 2+3+3 233 8 17480825 14 8 3+7+7=17=1+7=8 377 8 28284425 15 7 6+1+0=7 610 7 45765250 16 6 9+8+7=24=2+4=6 987 6 74049675 17 4 1+5+9+7=22=2+2=4 1597 4 119814925 18 1 2+5+8+4=19+1+9=10=1 2584 1 193864600 19 5 4+1+8+1=14=1+4=5 4181 5 313679525 20 6 6+7+6+5=24=2+4=6 6765 6 507544125 21 2 1+0+9+4+6=20=2 10946 2 821223650 22 8 1+7+7+1+1=17=1+7=8 17711 8 1328767775 23 1 2+8+6+5+7=28=2+8=10=1 28657 1 214999142524 9 4+6+3+6+8=27+2+7=9 46368 9 3478759200"
This information indicates that all "roads lead to Rome", i.e. a lot of periodically repeating accidents, coincidences. mystifications, etc., merging into a single stream, inevitably lead to the conclusion about the existence of a periodic pattern reflected in the Fibonacci series. And now consider one more, perhaps the most remarkable property of the Fibonacci series. On the Monad Forms page, we noted that there are only five unique forms that are of paramount importance. They are called Platanus bodies. Any Platonic solid has some special characteristics. Firstly, all faces of such a body are equal in size. Secondly, the edges of the Platonic solid are of the same length. Thirdly, the interior angles between its adjacent faces are equal. AND,fourth,being inscribed in a sphere, the Platonic solid touches the surface of this sphere with each of its vertices.
Rice. eight
There are only four shapes besides the cube (D) that have all of these characteristics. The second body (B) is a tetrahedron (tetra means "four"), having four faces in the form of equilateral triangles and four vertices. Another solid (C) is the octahedron (octa means "eight"), whose eight faces are equilateral triangles of the same size. The octahedron contains 6 vertices. A cube has 6 faces and eight vertices. The other two Platonic solids are somewhat more complicated. One (E) is called the icosahedron, which means "having 20 faces", represented by equilateral triangles. The icosahedron has 12 vertices. The other (F) is called the dodecahedron (dodecah is "twelve"). Its faces are 12 regular pentagons. The dodecahedron has twenty vertices.
These bodies have the remarkable properties of being inscribed all in just two figures - a sphere and a cube.
A similar relationship with the Platonic solids can be traced in all areas. So, for example, systems e The orbits of the planets of the solar system can be represented as nested Platonic solids inscribed in the corresponding spheres, which determine the radii of the orbits of the corresponding planets of the solar system.
Phase A (Fig. 8) characterizes the beginning of the evolution of the monadic form. Therefore, this form is, as it were, the simplest (sphere). Then a tetrahedron is born, and so on. The cube is located in this hexad opposite the sphere and therefore it has similar properties. Then the properties similar to the tetrahedron should have a monadic form located in the hexad opposite the tetrahedron. This is an icosahedron. The shapes of the dodecahedron must be "related" to the octahedron. And finally, the last shape becomes a sphere again. The last becomes the first! In addition, in the hexad, the continuity of the evolution of two neighboring Platonic solids should be observed. And, indeed, the octahedron and the cube, the icosahedron and the dodecahedron are mutual. If one of these polyhedra connects the centers of faces that have a common edge with line segments, then another polyhedron is obtained. In these properties lies their evolutionary origin from each other. In the Platonic hexad, two triads can be distinguished: “sphere-octahedron-icosahedron” and “tetrahedron-cube-dodecahedron”, endowing neighboring vertices of their own triads with reciprocity properties.
These figures have another remarkable quality. They are connected by strong ties with the Fibonacci series -<1:1:2:3:5:8:13:21:...>, in which each subsequent term is equal to the sum of the previous two. Let's calculate the differences between the members of the Fibbonacci series and the number of vertices in the Platonic solids:
· 2=2-A=2-2=0 (zero "charge"), · 3=3-B=3-4=-1 (negative "charge"), · 4=5-C=5-6=-1 (negative "charge"), · 5=8-D=8-8=0 (zero "charge"), · 6=13-E=13-12=1 (positive "charge"), · 7=21-F=21-20=1 (positive "charge"), Rice. 9At first glance, it may seem that the "monadic charges" of the Platonic solids reflect, as it were, the discrepancy between ideal forms from the Fibonacci series. However, considering that starting from the cube, the Platonic solids can form the GREAT LIMITS (Great Limit), it becomes clear that the dodecahedron and icosahedron, reflecting complementary the correspondence between the number of faces and the number of vertices, characterized by the numbers 12 and 20, actually expresses the ratios of 13 and 21 of the Fibonacci series. See how it goes rationingthe Fibonacci series. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,... 12, 20, ..... 1, 1, 2, 3, 5, 8, 13 The first line reflects the "normal" algorithm for the formation of the Fibonacci series. The second line begins with the icosahedron, in which the 13th vertex turned out to be the center of the structure, reflecting the properties of the GREAT LIMIT. A similar GREAT LIMIT exists for the dodecahedron. These two crystals give rise to a new dimension - the normalized icosahedron-dodecahedron monad, which begins to form a new round of the Fibonacci series (third line). The first Platonic solids, as it were, reflect the phase of analysis, when the GREAT LIMIT unfolds from the monad (1,1). The second phase is the synthesis of a new monad and its folding into the GREAT LIMIT. So the Fibonacci series generates the "golden proportion" responsible for the birth of the harmony of everything that exists, therefore the Platonic solids will also characterize the properties of all material structures. Thus, atoms are always associated with the five Platonic solids. Even if you take apart a very complex molecule, you can find simpler forms in it, and they can always be traced back to one of the five Platonic solids - no matter what its structure is. It doesn't matter if it's a metal, a crystal, or something else, the structure always goes back to one of the five original forms. Consequently, we come to the conclusion that the number of primordial monadic forms used by nature is limited and closed. The same conclusion was reached many centuries ago by Plato, who believed that complex particles of elements have the form of polyhedra; when crushed, these polyhedra give triangles, which are the true elements of the world. Having reached the most perfect form, nature takes this form as elementary and begins to build the following forms, using the latter as "single" elements. Therefore, all higher forms of inorganic, organic, biological and field forms of matter will necessarily be associated with simpler monadic crystals. From these forms, the most complex ones must be built - the highest forms of the Higher Mind. And these properties of monadic crystals should be manifested at all levels of the hierarchy: in the structure of elementary particles, in the structure of the Periodic system of elementary particles, in the structure of atoms, in the structure of the Periodic system of chemical elements, etc. So, in chemical elements, all subshells and shells can be represented in the form of monadic crystals. Naturally, the internal structure of atoms of chemical elements should be reflected in the structure of crystals and cells of living organisms. “Any form is a derivative of one of the five Platonic solids. With no exceptions. And it doesn't matter what the structure of the crystal is, it is always based on one of the Platonic solids..." . So the properties of the Platonic solids reflect the harmony of the golden section and the mechanisms of its generation by the Fibonacci series. And again we come to the most fundamental property of the UNIFIED LAW - PERIODICITY. The biblical "AND THE LAST BECOMES THE FIRST" is reflected in all the creations of the universe. The following figure shows a diagram of a chromatic scale, in which the 13th note is beyond the "boundary of the conscious world", and any neighboring pair can generate a new chromatic scale (The Laws of the Absolute).
rice. ten
This figure reflects the principles in accordance with which the SINGLE SELF-CONSISTENT FIELD OF HARMONY OF THE UNIVERSE is formed.
5. GOLDEN SECTION AND PRINCIPLES OF SELF-ORGANIZATION
5.1. SELF-SUFFICIENCY
Principlesself-organizations (self-sufficiency, self-regulation, self-reproduction, self-development and self-rationing) are very closely related to the golden ratio. Considering the principles of self-organization and the principles of new thinking (On new thinking, On globalistics), the conclusion was substantiated that the concept self-sufficiency definesshare the contribution of own objective functions to the general objective function of one or another object of the surrounding world. If the object's own share of the contribution to the overall objective function is not less than 2/3, then such an object will have a "controlling stake" in the objective function of the object and, therefore, will be self-sufficient, not a "puppet" object. But 2/3=0.66... and the golden ratio is 0.618... Very close match, or..? That's right OR! Therefore more accuratequantitative evaluationself-sufficiency can be considered the proportion of the golden section. However, for practical use measure of self-sufficiency determiningqualitythe state of the object, whether it lives in harmony with the surrounding world or not, a score of 2/3 is even preferable. The deep relationship of this principle with the golden ratio is shown in fig. 4, on which the most remarkable properties of the golden section and their relationship with the ONE LAW were given by the hand of the great master Leonardo da Vinci. And it is a pity that MANY SCIENTISTS DO NOT UNDERSTAND THIS EVEN TODAY. A SHAME!!!5.2. SELF-REPRODUCTION. SELF-DEVELOPMENT.
From the principles of construction of universal logic ( ) it follows that the infinite-dimensional logic, within the framework of the evolution of the same family, forms a binary spiral.
In this scheme, the nodal points characterize the downward spiral of the evolution of the logical family of the binary spiral (right screw). By induction, it can be determined that the left screw will determine the upward spiral of this family. This evolutionary binary spiral characterizes self-reproduction andself-developmentlogical family. Let we have the initial logic< - i ,-1 >. Then, depicting the axes integrated system reference in accordance with the rule of bypassing the tetrahedron along the cross, the evolution of logics can be reflected as shown in Fig. 12 rice. 12 It can be seen from the diagram that with each transition from one logic to another, in the direction of the arrows, a mirror effect occurs. self-copying logic. And when we complete the "circle of evolution", then the last and first logics will turn out to be opposite to each other. The next attempt leads already to the logic of binary doubling, since cell is occupied. As a result, a logic is born that differs from the first scale, instead of< -i, -1>a couple is born< -2 i ,-2 >. Note that successive mirroring of logics leads to their mirror inversion along the diagonals. Yes, diagonally. - i ,+1 we have logic <- i ,-1> <+1,+ i >. From the rules for traversing the vertices of the tetrahedron along the cross, we get that these logics form a cross in the tetrahedron if the corresponding edges are projected onto the plane. Pabout the diagonal-1,+ i we got complementary a couple of logics <-1,- i > <+ i ,+1> , also forming a cross. On fig. 11, the sides of the squares are oriented in the direction of the baptism. Therefore, the opposite sides of this square are the arms of the cross. Note that in the tetrahedron there is also a third cross formed by the edges <+ i ,- i > and<-1,+1> . But this cross has other functions, which will be discussed elsewhere. But the diagram in Fig. 6 justifies only the simple self-reproduction logician. It can generate a multidimensional world of "black and white" copies, which can only be characterized by different "shades". In accordance with the principles of self-organization, logics must have opportunity for self-development. And such an opportunity is realized (Fig. 13). rice. 13 Here in the square IIhappens first self-copying original logic, and in the third square, there is a process self-development. Here, first and second square are added with a shift, and then reproduced in a square III. Then the resulting chain is mirrored into a square IV, where the "closing" of the chain occurs. As a result, a tetrahedron is born, with four vertices, i.e. complex logic is born. So from a couple<1,1>a couple is born<2,2>. This is how the First period of the Periodic system of logical elements is born. Let us now take the second pair, consisting of two logical adjacent subshells -<1,2>. describing the evolution of this pair by squares in accordance with the above rules, we get a pair<3,3>. Attaching it to the initial chain<1,1,2>, we'll get<1,1,2,3>/ Then the evolution of the pair<2,3>will produce a couple<5,5>and, accordingly, the chain <1,1,3,5,>. It is easy to see that the Fibonacci series is born , which is the basis of the golden section. And this series is born in a natural way, it is based on the Unified Periodic Law of Evolution and the principles arising from it. self-organization (self-sufficiency, self-regulation, self-reproduction, self-development, self-rationing).
rice. eleven
5.3. FIBONACCCI SERIES AND BINARY SERIES
Let us now take, as logical pairs, an integral pair<2,2>. This pair will characterize the quantitative composition of the first logical shell. Then, in the process of its "baptism" we will produce the following binary pair<4,4>. This pair in its structure will characterize the star tetrahedron (or cube), which has eight vertices. We got the first subshell of the second period. Doubling these subshells will give a pair<8,8>, whose evolution will lead to the pair<16,16>, and then to the pair<32,32>. By connecting the resulting binary pairs into a single chain, we get a series <2, 8,16,32>. It is this sequence that characterizes the quantitative composition of the shells of the Periodic Table of chemical elements. In this way,unity of the Fibonacci series and the binary series is an indisputable fact. The periodic system of chemical elements, the binary series, the Fibonacci series and the golden section are closely related.
Rice. fourteen
It can be seen from the last scheme that the generating functions of these series are also closely related to the Newton binomial
(1-x) -n.
There is also a direct connection between the Fibonacci series and the binary series (Fig. 4)
Rice. fifteen
This figure shows how the entire Fibonacci series is built from the original ratio (1-1-2), using a binary series. This scheme is given in his book by D. Melchizedek (" ancient secret Flower of Life", vol. 2, p. 283). This drawing shows the family tree of the drone bee. Melchizedek emphasizes that the Fibonacci series (1-1-2-3-5-8-13-...) is a feminine series, while the binary series (1-2-4-8-16-32-.. .) is masculine. And rightly so (Gene memory, Information, About time). These pages provide a rationale for the fact that gene memory, reviving Past, or synthesizingFuture,forms precisely a binary series and precisely according to the law shown in Figure 4.6. OTHER PROPERTIES OF THE FIBONACCI SERIES
Everyone knows that rhythms (waves) permeate our entire life. Therefore, the universality of the proportion of the golden section must also be illustrated by the example of wave oscillations. Consider the harmonic process of string vibrations ( http://ftp.decsy.ru/nanoworld/index.htm). Standing waves of the fundamental and higher harmonics (overtones) can be created on the string. The half-wavelengths of the harmonic series correspond to the function 1/ n, wheren – natural number. The half-wavelengths can be expressed as a percentage of the half-wavelength of the main harmonic: 100%, 50%, 33%, 25%, 20%... In case of impact on an arbitrary section of the string, all harmonics will be excited with different amplitude coefficients, which depend on the coordinate area, on the width of the area and on the time-frequency characteristics of the impact. Considering different signs phases of even and odd harmonics, you can get an alternating function that looks something like this: If the fixing point is taken as the origin, and the middle of the string as 100%, then the maximum susceptibility for the 1st harmonic will correspond to 100%, for the 2nd - 50%, for the 3rd - 33%, etc. Let's see where our function will cross the x-axis. 62%, 38%, 23.6%, 14.6%, 9%, 5.6%, 3.44%, 2.13%,1.31%, 0.81%, 0.5%, 0.31%, 0.19%, 0.12%, ... This is the proportion of the golden wurf, which is understood as a sequential series of segments when adjacent segments are in relation to the golden ratio. Each next number is 0.618 times different from the previous one. It turned out the following: Excitation of a string at a point dividing it with respect to the golden section at a frequency close to the fundamental harmonic will not cause the string to vibrate, i.e. the point of the golden section is the point of compensation, damping. For damping at higher frequencies, for example at the 4th harmonic, the compensation point must be chosen at the 4th intersection of the function with the x-axis. Thus, the periodicity of changes in the properties of the dual relation turns out to be connected with the norm of self-sufficiency, the Fibonacci series, as well as with the properties of the star tetrahedron, which reflects the principle of an ascending and descending spiral. Therefore, it can be said that the secrets of the Golden Section, the secrets of the Fibonacci series, the secrets of their universality in the world of animate and inanimate Nature no longer exist. The golden ratio and the Fibonacci series reflect the most fundamental pattern of the Hierarchy - the pattern of duality, and the Fibonacci series itself reflects not only one of the main forms of manifestation of this pattern - the trinity, but also characterizes the norms of self-sufficiency of the dual relationship in the process of its evolution. 7. ABOUT COMPLEX ATTITUDE The properties of the golden section and the Fibonacci series considered above and their relationship allow us to make an assumption about the connection with the Unified Law of Evolution of the dual ratio of another remarkable ratio, which in projective geometry is known as complex relation of points ABCD. Rice. 16 This number has the property that it is exactly the same as. for both the image and the original. If you need to calculate x, then it doesn't matter if you are measuring the distance in the image or in the area itself. The camera can be deceiving. She cheats when she gives out equal lengths for unequal and right angles for indirect ones. The only thing it doesn't distort is the expression
ZnThe meaning of this expression can be found directly from the photograph. And everything that can be asserted with certainty, using the evidence of photography, can be expressed in terms of such quantities.
Usually, the symbol is used as an abbreviation for a complex relation.
ABCD.
Let us now redraw the scheme of a complex relation in a spatial form
Rice. 17
It is known that the golden ratio is expressed by the proportion
where the numerator is the smaller number, and denominator-large.
In relation to figure 17, the golden ratio will be reflected in the triangle
ABC,
for example,vector sum
AB=
BC+
CA.
If the angles between the legs are equal to zero, then we get a division of the segment in half. If the angle is
π
/
2,
then we get right triangle with the parties
1, F, F 0.5;
Therefore, we have the original equation
F 2 -F \u003d 1,written in vector form, the hypotenuse is a unit, and the legs are orthogonal to each other, which is reflected in the golden section equation.
For any other angle some closed spaces are described.
Comparison of Figures 16 and 17 also shows that a straight line (Fig. 16), which generates a complex relation, is transformed into a broken line, and a complex relation is generated by the process "
bypass on the cross
".
WhereinLast summit
broken linecloses on the first
.
As a result, we get the already known life-giving cross
the rule of leverage is "you win in strength, you lose in distance": - multiplying the crossbars and dividing by the length of the arms that determine transition from one crossbar to another. When constructing these more complex relationships, it must be taken into account that in the formation of a complex relationship, just like in the Fibonacci series, only two neighboring vertices of the polyline are involved. This rule of leverage, using the golden ratio, can be written in the following form . And now we can build a complex relationship on the tetrahedron, given that the distances from all the vertices of the pyramid to point O are the same.Rice. eighteen
From Figures 14-19, one can also understand the principles of constructing more complex relationships, for spaces with a higher dimension, i.e. it can be said that n-dimensionala complex relationship reflects the process of formation of a monadic crystal n -dimensions and that's why "exercises" on the formation of more complex relationships may be of independent interest ( Complex relationship). But all the meanings of the complex relation X, (1/X), (x-1)/ X, X/(x-1), 1/(1-x), (1-x), X,... are parts of the golden section equation x 2 - X - 1 =0 or X(X -1) =1. 7. THE LAW OF CONSERVATION OF THE GOLDEN RATIO The properties of the golden section considered above and, first of all, the properties of the complex ratio allow us to say that the golden section forms main law of the universe, reflecting the main law of conservation I- law of conservation of the golden ratio . Ratios x =0,618..., 1 / x =1,618, 1-1/ x =-0,618..., 1/(1-1/ x )=-1,618,.... form an infinite series in which the first four values form a cross of the golden ratio. Moreover, whenever a value is obtained that is greater than the value of the golden section, then normalization OBJECT. Isolates from it unit and the process of evolution continues! However, for the fifth and sixth values, we get the values " -2,616 " and " -0,382 ", after which the process starts from the beginning. The resulting infinite series of values of 0.618 and 1.618 is the reason why the golden ratio underlies the harmony of the world. The conservation law (Conservation laws) of the golden section can be demonstrate in a rotating cross (swastika).Rice. 19
Below, on the page that reveals the secrets of information (Information, About time), it will be shown that the golden ratio, gene memory underlies the very information concepts,
about the natural mechanisms of evolution of the monad "IMAGE-LIKELIHOOD" in TIME.
Thus, the essence of rationing is reduced to obtaining the proportions of the golden section, i.e. all the wonderful properties of the complex relationship of four points are determined by the properties of the life-giving cross, that the complex relationship is closely interconnected with the golden ratio, forming the law of conservation
golden ratio.
SUMMARY
1. No one doubts that the golden ratio underlies the harmony of the universe, and a number
Fibonacci generates this wonderful proportion.
Curious readers can get additional information about the properties of the golden section on the website
www .
goldenmuseum.
com
.
This truly golden proportion has so many wonderful properties that the discovery of new properties no longer surprises anyone.
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 acquainted with the golden rule, humanity no longer cheated on it.
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 acquainted with the golden rule, humanity no longer cheated on it.
Definition
The most capacious definition of the golden ratio says that the smaller part is related to the larger one, as the larger one is to the whole. Its approximate value is 1.6180339887. In a rounded percentage, the proportions of the parts of the whole will correlate as 62% by 38%. This ratio operates in the forms of space and time.
The ancients saw the golden section as a reflection of the cosmic order, and Johannes Kepler called it one of the treasures of geometry. Modern science considers the golden ratio as "asymmetric symmetry", calling it in a broad sense a universal rule that reflects the structure and order of our world order.
Story
The ancient Egyptians had the idea of golden proportions, they also knew about them in Russia, but for the first time the monk Luca Pacioli explained the golden ratio scientifically in the book The Divine Proportion (1509), which was supposedly illustrated by Leonardo da Vinci. Pacioli saw the divine trinity in the golden section: the small segment personified the Son, the large one - the Father, and the whole - the Holy Spirit.
The name of the Italian mathematician Leonardo Fibonacci is directly connected with the golden section 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 ratio: “It is arranged in such a way that the two lower terms of this infinite proportion add up to the third term, and any two last terms, if added together, give the next term, and the same proportion remains indefinitely. ". Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden section in all its manifestations.
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 golden division.
Over time, the rule of the golden ratio turned into an academic routine, and only the philosopher Adolf Zeising in 1855 brought it back to a second life. He brought the proportions of the golden section to the absolute, making them universal for all phenomena of the surrounding world. However, his "mathematical aestheticism" 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 the lizard, the distance between the leaves on the branch fall under it, there is a golden section and 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 proportions of the golden section. In his opinion, one of the most interesting forms is spiraling.
Even 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, such as the snail shell, the arrangement of sunflower seeds, 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 section. Man is a universal form for testing the laws of the golden section. Of course, by nature, not all people have ideal proportions, which creates certain difficulties with the selection of clothes.
In the diary of Leonardo da Vinci there is a drawing of a naked man inscribed in a circle, in two positions superimposed on each other. Based on the studies 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, exploring the proportionality of man, did a tremendous job. He measured about two thousand human bodies, as well as many ancient statues, and deduced that the golden ratio expresses the average law. In a person, almost all parts of the body are subordinate to him, but the main indicator of the golden section 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 “there is an immutable law in the composition, when nothing can be removed or added to the picture, even an extra point cannot be put, this is real mathematics.” For a long time, artists 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 Dürer used the proportional compass invented by him to determine the points of the golden section.
The art critic F. V. Kovalev, having studied in detail the painting by Nikolai Ge “Alexander Sergeevich Pushkin in the village of Mikhailovsky”, notes that every detail of the canvas, whether it be a fireplace, a bookcase, an armchair or the poet himself, is strictly inscribed in golden proportions.
Researchers of the golden ratio tirelessly study and measure the masterpieces of architecture, claiming that they have become 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, the Parthenon.
And today, in any art of spatial forms, they try to follow the proportions of the golden section, since, according to art historians, they facilitate the perception of the work and form an aesthetic sensation in the viewer.
Word, sound and film
The forms of temporal art in their own way demonstrate to us the principle of golden division. Literary critics, for example, noticed that the most popular number of lines in the 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. So the climax of The Queen of Spades is the dramatic scene of Herman and the Countess, ending with the death of the latter. There are 853 lines in the story, and the climax falls on line 535 (853:535=1.6) - this is the point of the golden ratio.
The Soviet musicologist E. K. Rozenov notes the amazing accuracy of the golden section ratios 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 golden ratio point usually accounts for the most striking or unexpected musical solution.
Film director Sergei Eisenstein deliberately coordinated the script for his film "Battleship Potemkin" with the rule of the golden section, dividing the tape into five parts. In the first three sections, the action takes place on a ship, and in the last two - in Odessa. The transition to the scenes in the city is the golden mean of the film.
Geometry is an exact and rather complex science, which, with all this, is a kind of art. Lines, planes, proportions - all this helps to create a lot of really beautiful things. And oddly enough, this is based on geometry in its most diverse forms. In this article, we will look at one very unusual thing which is directly related to this. The golden ratio is exactly the geometric approach that will be discussed.
The shape of the object and its perception
People most often focus on the shape of an object in order to recognize it among millions of others. It is by form that we determine what kind of thing lies in front of us or stands far away. We first of all recognize people by the shape of the body and face. Therefore, we can say with confidence that the form itself, its size and appearance is one of the most important things in human perception.
For people, the form of anything is of interest for two main reasons: either it is dictated by vital necessity, or it is caused by aesthetic pleasure from beauty. The best visual perception and a sense of harmony and beauty most often comes when a person observes a form in the construction of which symmetry and a special ratio were used, which is called the golden ratio.
The concept of the golden ratio
So, the golden ratio is the golden ratio, which is also a harmonic division. In order to explain this more clearly, consider some features of the form. Namely: the form is something whole, but the whole, in turn, always consists of some parts. These parts are most likely different characteristics, at least different sizes. Well, such dimensions are always in a certain ratio both among themselves and in relation to the whole.
So, in other words, we can say that the golden ratio is the ratio of two quantities, which has its own formula. Using this ratio when creating a form helps to make it as beautiful and harmonious as possible for the human eye.
From the ancient history of the golden section
The golden ratio is often used in most different areas life right today. But the history of this concept goes back to ancient times, when such sciences as mathematics and philosophy were just emerging. As a scientific concept, the golden ratio came into use during the time of Pythagoras, namely in the 6th century BC. But even before that, knowledge of such a ratio was used in practice in Ancient Egypt and Babylon. A striking evidence of this are the pyramids, for the construction of which they used just such a golden ratio.
new period
The Renaissance was a new breath for harmonic division, especially thanks to Leonardo da Vinci. This ratio has been increasingly used both in geometry and in art. Scientists and artists began to study the golden ratio more deeply and create books that deal with this issue.
One of the most important historical works related to the golden ratio is Luca Pancioli's book called The Divine Proportion. Historians suspect that the illustrations of this book were made by Leonardo pre-Vinci himself.
Mathematical expression of the golden ratio
Mathematics gives a very clear definition of proportion, which says that it is the equality of two ratios. Mathematically, this can be expressed by the following equality: a: b \u003d c: d, where a, b, c, d are some specific values.
If we consider the proportion of a segment divided into two parts, then we can meet only a few situations:
- The segment is divided into two absolutely even parts, which means that AB: AC \u003d AB: BC, if AB is the exact beginning and end of the segment, and C is the point that divides the segment into two equal parts.
- The segment is divided into two unequal parts, which can be in very different proportions to each other, which means that here they are absolutely disproportionate.
- The segment is divided so that AB:AC = AC:BC.
As for the golden section, this is such a proportional division of the segment into unequal parts, when the entire segment belongs to the larger part, like itself most of refers to the smaller one. There is another formulation: the smaller segment is related to the larger one, as well as the larger one to the entire segment. In mathematical terms, it looks like this: a:b = b:c or c:b = b:a. This is the form of the golden section formula.
Golden ratio in nature
The golden ratio, examples of which we will now consider, refers to the incredible phenomena in nature. This is very beautiful examples that mathematics is not just numbers and formulas, but a science that has more than a real reflection in nature and our life in general.
For living organisms, one of the main life tasks is growth. This desire to take its place in space, in fact, is carried out in several forms - upward growth, almost horizontal spreading on the ground, or spiraling on some kind of support. And as incredible as it is, many plants grow according to the golden ratio.
Another almost incredible fact- these are the ratios in the body of lizards. Their body looks pleasing enough to the human eye, and this is possible thanks to the same golden ratio. To be more precise, the length of their tail is related to the length of the whole body as 62:38.
Interesting facts about the rules of the golden ratio
The golden ratio is a truly incredible concept, which means that throughout history we can meet a lot of really interesting facts about this proportion. We present you some of them:
The golden ratio in the human body
In this section, it is necessary to mention a very significant person, namely, S. Zeising. This is a German researcher who has done a great job in the field of studying the golden ratio. He published a work entitled Aesthetic Research. In his work, he presented the golden ratio as an absolute concept, which is universal for all phenomena, both in nature and in art. Here we can recall the golden section of the pyramid, along with the harmonious proportion of the human body, and so on.
It was Zeising who was able to prove that the golden ratio, in fact, is the average statistical law for the human body. This was shown in practice, because during his work he had to measure a lot of human bodies. Historians believe that more than two thousand people took part in this experience. According to Zeising's research, the main indicator of the golden ratio is the division of the body by the navel point. Thus, a male body with an average ratio of 13:8 is slightly closer to the golden ratio than a female body, where the golden ratio is 8:5. Also, the golden ratio can be observed in other parts of the body, such as, for example, the hand.
On the construction of the golden section
In fact, the construction of the golden section is a simple matter. As we can see, even ancient people coped with this quite easily. What can we say about modern knowledge and technologies of mankind. In this article, we will not show how this can be done simply on a piece of paper and with a pencil in hand, but we will state with confidence that this is, in fact, possible. Moreover, this can be done in more than one way.
Since this is a fairly simple geometry, the golden ratio is quite simple to construct even in school. Therefore, information about this can be easily found in specialized books. By studying the golden ratio, grade 6 is fully able to understand the principles of its construction, which means that even children are smart enough to master such a task.
The Golden Ratio in Mathematics
The first acquaintance with the golden section in practice begins with a simple division of a straight line segment, all in the same proportions. Most often this is done with a ruler, a compass and, of course, a pencil.
Segments of the golden ratio are expressed as an infinite irrational fraction AE \u003d 0.618 ..., if AB is taken as a unit, BE \u003d 0.382 ... In order to make these calculations more practical, very often they use not exact, but approximate values, namely - 0 .62 and 0.38. If the segment AB is taken as 100 parts, then its larger part will be equal to 62, and the smaller one - to 38 parts, respectively.
The main property of the golden ratio can be expressed by the equation: x 2 -x-1=0. When solving, we get the following roots: x 1.2 =. Although mathematics is an exact and rigorous science, as well as its section - geometry, but it is precisely such properties as the laws of the golden section that bring mystery to this topic.
Harmony in art through the golden ratio
In order to sum up, let us briefly consider what has already been said.
Basically, many pieces of art fall under the rule of the golden ratio, where the ratio is close to 3/8 and 5/8. This is the rough formula for the golden ratio. The article has already mentioned a lot about examples of using the section, but we will look at it again through the prism of the ancient and contemporary art. So the most bright examples from ancient times:
As for the already conscious use of proportion, since the time of Leonardo da Vinci, it has come into use in almost all areas of life - from science to art. Even biology and medicine have proven that the golden ratio works even in living systems and organisms.
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