What is the golden ratio examples. Shkrudnev Fedor Dmitrievich - The Golden Ratio
"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 arrange 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, the merging of the various components of an 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 regular geometric shapes - a square, a circle, an isosceles triangle, a 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 connecting the Fibonacci series with 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 Tutankhamen, and other works of 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 seen 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. 4
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 have 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 important property of the 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 according to a 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 Section (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.
It can be seen from the table 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. The table directly shows 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, let us imagine that at some stage of evolution the Fibonacci spiral has reached perfection (it has become indistinguishable from the golden section 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 down the natural numbersEach 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. 10
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 of the complex reference system 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. thirteen 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. 14
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. 15
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 ("The Ancient Secret of the Flower of Life", vol. 2, p. 283). This drawing shows the drone bee family tree. 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. Given the different signs of the phases of even and odd harmonics, you can get a sign-changing function that looks 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. sixteen 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-big.
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 a right triangle with sides
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 peak
broken linecloses on the first
.
As a result, we get what is already known from the 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 above properties of the golden section and, first of all, the properties of a complex ratio allow us to say that the golden section forms the main law of the universe, reflecting the main law of conservation. I am- 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. nineteen
Below, on the page that reveals the secrets of information (Information, About time), it will be shown that the golden ratio, genetic memory underlie the very concept of information,
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.
Modern web design includes 2 features that must be clearly observed: aesthetics and the right scope. If you follow these concepts, web design can be considered successful.
As for aesthetics, it means that when drawing this or that image of an object, we use many different manipulations: creating a grid, layout, using typographic techniques in order to get a good structure of the object. It is important to maintain a sense of harmony, order and visual balance in any graphic processing. The Golden Ratio and the Rule of Three will help us with this.
You have probably heard of these terms before. And, perhaps, you have an idea in which specific projects they can be used. The "Golden Ratio" and the "Rule of Three" are used to change the image and present it in a better way than it actually is. Such technologies help to improve even the most primitive picture.
Let's take a closer look at these features and find out in which areas of web design they can be applied.
What is the "Golden Ratio" and how did it appear?
This term may not be clear at first glance. Why "Golden"? Why use this technology? To date, it still remains a mystery who invented the "Golden Section", where this name came from. However, it is known that the technology has been used for 2400 years. It is also worth noting that the golden ratio is used in various branches of science: in astronomy, mathematics, architecture, music, painting and many others.
The golden ratio is formed from a simple mathematical equation that shows a ratio. In its simplest mathematical form, this relationship looks like this:
As you can see, this is a unique equation that separates the relationship between two line sizes and proportions. In decimal, b divided by a equals 1.618033... if a>b. In the example below, let's say that b is 5. Then the equation will look like this:
You may have heard of the Fibonacci sequence before. How does it actually work? For example, there is a series of numbers in which any given number is created by adding the previous two. Starting from 0, the sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34… etc:
The written expression is represented as a formula: xn = xn-1 + xn-2 .
The sequence is closely related to the golden ratio, because if you take any two consecutive numbers and divide by the previous one, the fraction will be very close to the golden ratio. As the value of the number increases, the fraction gets even closer to the golden ratio. For example, 8/5 is 1.6, 34/21 is 1.619, and so on.
"Golden Spiral" Rectangle
So, you must have seen similar equations. But why do designers use geometry in their designs? Why is overlaying shapes necessary? The scheme is called the Fibonacci Spiral. It is actually quite simple and is the most optimal for many geometric shapes. The spiral is created using quarter circles that are drawn inside an array of squares based on the Fibonacci sequence.
The diagram below shows an example:
It turns out that each subsequent radius is greater than the previous one by a number close to the golden ratio. The resulting spiral is used in many areas, most often in drawing and architecture, but it can also be observed in natural phenomena.
"Rule of Three"
This rule is one of the variants of the golden spiral and is often used when cropping photos and videos. Used to trim frames and give them an aesthetic look. To apply the "Rule of Three", you need to divide the image into 9 equal parts. Draw 2 horizontal lines and 2 vertical lines. It is important to place them evenly. The point is to align focus with the leftmost vertical divider. The horizon or "vanishing point" must be level with the horizontal divider.
Application of the "Golden Spiral"
As already noted, the Fibonacci sequence is closely related to the golden ratio. The application of the golden section is performed using a traced spiral. The image shows an example of using this method. So, we see a rectangle, the base of which extends from the woman's right wrist to her left elbow.
The rectangle expands vertically until it reaches the crown. If we draw squares inside the golden rectangle, all the important parts of the woman are on the edges of the inner squares: her chin, eyes and lips. Leonardo da Vinci used the golden ratio many times in his work. Below are examples of the golden spiral in nature and space.
Application in web design
Many designers make the mistake of thinking that by simply dividing or multiplying by 1.61... you can get a harmonious proportion. This is far from true, it is just the basis of the process. You can't just use this or that number and get the magic proportion. However, there are certain ways that help to get the golden ratio. Some artists tend to think that the golden ratio theory is a myth. Here is another example of how the golden ratio works. Let's take a prototype site and consider the application of the golden ratio on it.
Looks pretty simple, right? Yes, in fact it is. The design is based on a 960 pixel grid. The design is represented using the golden ratio. If you use 1 golden spiral that spans 960px, you can see how the header, logo, etc. were positioned.
We move our spiral lower and rely on its dimensions
It turns out a kind of cascade of spirals in which the main design elements are inscribed in rectangles with a golden ratio
A grid based on the golden ratio has a number of proportional relationships within it that are clearly proportional within a rectangle. At the bottom of this article, I have attached a PSD file that contains my example, you can try using it in your project to experiment with the golden ratio.
It is said that the "divine proportion" is found in nature, and in many things around us. You can find it in flowers, beehives, sea shells, and even our bodies.
This divine proportion, also known as the golden ratio, divine ratio, or golden ratio, can be applied to a variety of arts and learning. Scientists argue that the closer an object is to the golden ratio, the better the human brain perceives it.
Since this ratio was discovered, many artists and architects have used it in their work. You can find the golden ratio in several Renaissance masterpieces, architecture, painting, and more. The result is a beautiful and aesthetically pleasing masterpiece.
Few people know what the secret of the golden ratio is, which is so pleasing to our eyes. Many believe that the fact that it appears everywhere and is a "universal" proportion makes us accept it as something logical, harmonious and organic. In other words, it just “feels” what we need.
So what is the golden ratio?
The golden ratio, also known as "phi" in Greek, is a mathematical constant. It can be expressed as a/b=a+b/a=1.618033987 where a is greater than b. This can also be explained by the Fibonacci sequence, another divine proportion. The Fibonacci sequence starts at 1 (some say 0) and adds the previous number to it to get the next one (i.e. 1, 1, 2, 3, 5, 8, 13, 21...)
If you try to find the quotient of the next two Fibonacci numbers (i.e. 8/5 or 5/3), the result is very close to the golden ratio of 1.6 or φ (phi).
The golden spiral is created using a golden rectangle. If you have a rectangle of squares 1, 1, 2, 3, 5 and 8 respectively, as shown in the picture above, you can start building a golden rectangle. Using the side of the square as the radius, you create an arc that touches the points of the square diagonally. Repeat this procedure with each square in the golden triangle and you will end up with a golden spiral.
Where can we see it in nature
The golden ratio and the Fibonacci sequence can be found in flower petals. In most flowers, the number of petals is reduced to two, three, five or more, which is like the golden ratio. For example, lilies have 3 petals, buttercups have 5, chicory flowers have 21, and daisies have 34. It is likely that flower seeds also follow the golden ratio. For example, sunflower seeds germinate from the center and grow towards the outside, filling the seed head. They are usually spiral and resemble a golden spiral. Moreover, the number of seeds tends to be reduced to Fibonacci numbers.
The hands and fingers are also an example of the golden ratio. Look closer! The base of the palm and the tip of the finger are divided into parts (bones). The ratio of one part to another is always 1.618! Even the forearms with the hands are in the same ratio. And fingers, and face, and the list goes on ...
Application in art and architecture
The Parthenon in Greece is said to have been built using golden proportions. It is believed that the dimensional ratios of height, width, columns, distance between pillars, and even the size of the portico are close to the golden section. This is possible because the building looks proportionately perfect, and it has been so since ancient times.
Leonardo Da Vinci was also a fan of the golden ratio (and many other curious items, in fact!). The marvelous beauty of the Mona Lisa may be due to the fact that her face and body represent the golden ratio, just like real human faces in life. In addition, the numbers in Leonardo Da Vinci's The Last Supper are arranged in the order that is used in the golden ratio. If you draw golden rectangles on canvas, Jesus will be right in the central lobe.
Application in logo design
Unsurprisingly, you can also find the use of the golden ratio in many modern projects, particularly design. For now, let's focus on how this can be used in logo design. First, let's take a look at some of the world's most famous brands that have used the golden ratio to perfect their logos.
Apparently, Apple used circles from Fibonacci numbers, connecting and cutting the shapes to get the Apple logo. It is unknown if this was done intentionally or not. However, the result is a perfect and visually aesthetic logo design.
The Toyota logo uses the ratio of a and b to form a grid that forms three rings. Notice how this logo uses rectangles instead of circles to create the golden ratio.
The Pepsi logo is created by two intersecting circles, one larger than the other. As shown in the picture above, the larger circle is proportional in relation to the smaller one - you guessed it! Their latest non-embossed logo is simple, effective and beautiful!
Apart from Toyota and Apple, the logos of several other companies such as BP, iCloud, Twitter, and Grupo Boticario are also believed to have used the golden ratio. And we all know how famous these logos are - all because the image immediately pops up in memory!
Here is how you can apply it in your projects
Sketch the golden rectangle as shown above in yellow. This can be achieved by constructing squares with height and width from numbers belonging to the golden ratio. Start with one block and place another next to it. And another square, whose area is equal to those two, place above them. You will automatically get a side of 3 blocks. After building this 3-block structure, you will end up with a side of 5 quads that can be used to make another (5-block area) box. This can go on as long as you like until you find the size you need!
The rectangle can move in any direction. Select small rectangles and use each one to put together a layout that will serve as the logo design grid.
If the logo is more rounded, then you will need a circular version of the golden rectangle. You can achieve this by drawing circles proportional to the Fibonacci numbers. Create a golden rectangle using only circles (this means the largest circle will have a diameter of 8, while the smaller circle will have a diameter of 5, and so on). Now separate these circles and place them so that you can form the main outline for your logo. Here is an example of a Twitter logo:
Note: You don't have to draw all the circles or rectangles of the golden ratio. You can also use the same size more than once.
How to apply it in text design
It's easier than designing a logo. A simple rule for applying the golden ratio in text is that subsequent larger or smaller text must match Phi. Let's take a look at this example:
If my font size is 11, then the subtitle should be written in a larger font. I multiply the font of the text by the number of the golden ratio to get a larger number (11 * 1.6 = 17). So the subtitle should be written in 17 font size. And now the title or title. I multiply the subtitle by the proportion and get 27 (1 * 1.6 = 27). Like this! Your text is now proportional to the golden ratio.
How to apply it in web design
And here it is a little more difficult. You can stay true to the golden ratio even in web design. If you are an experienced web designer, you have already guessed where and how it can be applied. Yes, we can make good use of the golden ratio and apply it to our web page grids and UI layouts.
Take the grid's total number of pixels as width or height and use that to build a golden rectangle. Divide the largest width or length to get smaller numbers. This can be the width or height of your main content. What's left could be the sidebar (or bottombar if you applied it to height). Now keep using the golden rectangle to further apply it to windows, buttons, panels, images and text. You can also build a complete mesh based on small versions of the golden rectangle both horizontally and vertically to create smaller UI objects that are proportional to the golden rectangle. You can use this calculator to get proportions.
Spiral
You can also use the golden spiral to determine where to place content on your site. If your homepage is loaded with graphic content, such as a website for an online store or photography blog, you can use the golden spiral method that many artists use in their work. The idea is to put the most valuable content in the center of the spiral.
Grouped content can also be placed using the golden rectangle. This means that the closer the spiral moves to the central squares (one square block), the “dense” the content is there.
You can use this technique to mark the location of your header, images, menus, toolbar, search box, and other elements. Not only is Twitter famous for its use of the golden rectangle in logo design, but it has been incorporated into web design as well. How? Through the use of the golden rectangle, or in other words the golden spiral concept, in the user profile page.
But it won't be easy to do this on CMS platforms where the author of the content defines the layout instead of the web designer. The golden ratio suits WordPress and other blog designs. This is probably because the sidebar is almost always present in the blog design, which fits nicely into the golden rectangle.
An easier way
Very often, designers omit complex math and apply the so-called “rule of thirds”. It can be achieved by dividing the area into three equal parts horizontally and vertically. The result is nine equal parts. The intersection line can be used as the focal point of the shape and design. You can place the key theme or main elements on one or all focal points. Photographers also use this concept for posters.
The closer the rectangles are to the ratio of 1:1.6, the more pleasant the picture is perceived by the human brain (since this is closer to the golden ratio).
Why do rituals work? This explains the Golden Ratio Method. Words in conspiracies are arranged in a certain order. The current structure is the golden ratio, but in speech design.
Such a conspiracy leads to results: the information located in the peak place of the golden section is a subconscious setting, the criticism of thinking is not involved here.
In ancient times, healers noted that it was forbidden to control the universe in a negative way. Therefore, the correct texts do not contain statements that denote destruction or eradication. These definitions are replaced by moving to a different space, where entities do not have the ability to harm. I give an example:
How are you, month, waning,
And so I'm going to decline
Bela fat from the body GO to the pig
In the name of the Father and the Son and the Holy Spirit. Amen.
This plot is a prime example of the golden ratio. His strength calls for good, given the reasonable distribution of energy in the universe. The goal is achieved due to the fact that it is used constantly. Even if a person does not believe in the result, he will come. This also applies to other conspiracies.
You can get acquainted with the effective training program "Super-profitable Feng Shui" by clicking on the link
To simplify the task, do the following:
- Write down the written text on the phone.
- Turn on repeat mode.
- Plug in your headphones so the text doesn't distract you.
- Listen to the plot for 30-45 minutes (15 days).
The first result can be seen after 5 days. In this case, the quantitative indicator of weight loss is not important, it is important to note that it is decreasing. Let your body decide how fast you want to reach your goal. Just watch the magic happen.
There is an extremely important point!
The principle of the golden section operates when the goal statement consists of 27 words-names, the keyword is at number 17. If there are 2-3 such definitions (no more), they can take positions 16-16 or 16-18.
Conjunctions, particles, prepositions are also words. The scheme is simple:
I ask my subconscious mind to help organize the events and circumstances of my life in such a way that I can (-la) (words No. 16, 17,18). I want to get it in a safe way for everything.
The technique will be the same as described above for weight reduction. Today, affirmations (conspiracy techniques) are actively used in psychotherapy.
This is due to the effectiveness in troubleshooting. After such procedures, people themselves begin to cope with possible failures, without seeking outside help.
It's time to try!!!
But first I want to tell you how to determine your own number. Find your year of birth in the table below, opposite is your number, your Palace of Birth.
But remember that the countdown is based on the Chinese solar calendar. In it, the new year begins on February 4th. If your date of birth is before February 4th, your year will be the previous year.
Listen to the directions of your own desires and needs. Here are the main parameters:
1 palace - career and professional growth, wedding, good income with little employment.
Palace 2 - change of image and the necessary costs for this in a specific amount of money. You must clearly understand how much you need to purchase a car, apartment, vacation, etc.
3 palace - status increase. Think about what projects you want to bring to life and how much you get for it. Clearly plan goals at this stage.
4 palace - create fame for your own brand in the new year, formulate romantic goals. Plan to spend money on your beauty and appearance.
5 palace - wish new acquaintances and travels, the development of female energy and sexuality, the exclusion of karmic lessons.
6 palace - think about the house, moving, repair. Let a team of like-minded people be in your desires. If you want a baby, plan for it.
Palace 7 - the coming year will bring a change of status and marriage, promising opportunities, business development. Think of your own growth in various areas.
8 palace - add creativity and romance, learning to your desires. Allow yourself to dream of self-improvement, and desires will come true.
9 palace - pay attention to the inner world and vitality, do not plan drastic changes. May your year become stable, improve what you already have.
“I ask my subconscious mind to help organize the events and circumstances of my life in such a way that from 2018 (it’s better to pronounce the EIGHTEENTH) (words No. 16, 17.18 - MONTHLY I GET TWO Hundred) thousand rubles only in a safer way for everything.”
As a result, we got 27 words. The golden ratio was the word "receive". Now the cash flow cannot get stuck anywhere, all payments will come on time.
The main word has fallen into its happy location, the goal will act on the subconscious in a direct way. We have generated momentum.
One more moment. Dangerous verbs should be avoided at the time of goal formation. Among them:
- Verbs with non-ending action (I'm looking for, I'm going, I'm selling). They make sense that the action will last and never end.
- Verbs with heavy vibrations that reduce the desire to move forward:
Earn - earn money by hard work (from the word "slave").
Manage - implies a lot of processes that repel our brain subconsciously.
Work - work for the "uncle" and suffer.
Achieve - ask for something.
To try is to act under torture.
Try to hiccup and not find a solution.
Your goals are important, so they need to be treated with care. They regulate life and lead to achievements. To speed up the solution of problems, you can use ready-made recipes:
- individual monetary figures;
- the use of noble helpers;
- ji fu energy is the big boss;
and many other features. And all these Feng Shui chips are collected in the training "Super-profitable Feng Shui". You can get acquainted with his program by clicking on the link.
And that's all. Tatyana Panyushkina was with you!
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Even in ancient Egypt it was known golden ratio, Leonardo da Vinci and Euclid studied its properties.The visual perception of a person is arranged in such a way that he distinguishes in form all the objects that surround him. His interest in an object or its form is sometimes dictated by necessity, or this interest could be caused by the beauty of the object. If in the very basis of the construction of the form, a combination is used golden ratio and the laws of symmetry, then this is the best combination for visual perception by a person who feels harmony and beauty. The whole whole consists of parts, large and small, and these different sizes of parts have a certain relationship, both to each other and to the whole. And the highest manifestation of functional and structural perfection in nature, science, art, architecture and technology is the Principle golden ratio. The concept of golden ratio introduced into scientific use the ancient Greek mathematician and philosopher (VI century BC) Pythagoras. But the very knowledge of golden ratio he borrowed from the ancient Egyptians. The proportions of all temple buildings, the pyramids of Cheops, bas-reliefs, household items and decorations from tombs show that the ratio golden ratio was actively used by ancient masters long before Pythagoras. As an example: the bas-relief from the temple of Seti I at Abydos and the bas-relief of Ramses use the principle golden ratio in the proportions of the figures. The architect Le Corbusier found this out. On a wooden board recovered from the tomb of the Architect Khesir, a relief drawing is depicted, on which the architect himself is visible, holding measuring instruments in his hands, which are depicted in a position fixing the principles golden ratio. Knew the principles golden ratio and Plato (427...347 BC). The Timaeus dialogue is proof of this, since it is devoted to questions golden division, aesthetic and mathematical views of the school of Pythagoras. Principles golden section used by ancient Greek architects in the facade of the Parthenon temple. The compasses that ancient architects and sculptors of the ancient world used in their work were discovered during excavations of the Parthenon temple.
Parthenon, Acropolis, Athens In Pompeii (museum in Naples) proportions golden division are also available.In ancient literature that has come down to us, the principle golden ratio first mentioned in Euclid's Elements. In the book "Beginnings" in the second part, a geometric principle is given golden ratio. Euclid's followers were Pappus (3rd century AD), Hypsicles (2nd century BC), and others. To medieval Europe with the principle golden ratio We met through translations from Arabic of Euclid's "Beginnings". Principles golden ratio were known only to a narrow circle of initiates, they were jealously guarded, kept in strict secrecy. A renaissance has come and an interest in the principles golden ratio increases among scientists and artists, since this principle is applicable in science, architecture, and art. And Leonardo Da Vinci began to use these principles in his works, even more than that, he began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, who got ahead of him and published the book "Divine Proportion" after which Leonardo left his the work is not finished. According to historians of science and contemporaries, Luca Pacioli was a real luminary, a brilliant Italian mathematician who lived between Galileo and Fibonacci. As a student of the painter Piero della Francesca, Luca Pacioli wrote two books, On Perspective in Painting, the title of one of them. He is considered by many to be the creator of descriptive geometry. Luca Pacioli, at the invitation of the Duke of Moreau, arrived in Milan in 1496 and lectured there on mathematics. Leonardo da Vinci at this time worked at the Moro court. Luca Pacioli's Divine Proportion, published in Venice in 1509, became an enthusiastic hymn golden ratio, with beautifully executed illustrations, there is every reason to believe that the illustrations were made by Leonardo da Vinci himself. Monk Luca Pacioli, as one of the virtues golden ratio emphasized its "divine essence". Understanding the scientific and artistic value of the golden ratio, Leonardo da Vinci devoted a lot of time to studying it. Performing a section of a stereometric body consisting of pentagons, he obtained rectangles with aspect ratios in accordance with golden ratio. And he gave it a name golden ratio". Which is still holding on. Albrecht Dürer, also studying golden ratio in Europe, meets with the monk Luca Pacioli. Johannes Kepler, the greatest astronomer of the time, was the first to draw attention to the importance golden ratio for botany calling it the treasure of geometry. He called the golden ratio self-continuing. “It is so arranged,” he said, “the sum of the two junior terms of an infinite proportion gives the third term, and any two last terms, if added together, give the next term, and the same proportion remains indefinitely.”
Golden Triangle:: Golden Ratio and Golden Ratio:: Golden Rectangle:: Golden Spiral
Golden Triangle
To find segments of the golden ratio of the descending and ascending rows, we will use the pentagram.
Rice. 5. Construction of a regular pentagon and pentagram
In order to build a pentagram, you need to draw a regular pentagon according to the construction method developed by the German painter and graphic artist Albrecht Dürer. If O is the center of the circle, A is a point on the circle, and E is the midpoint of segment OA. The perpendicular to the radius OA, raised at point O, intersects the circle at point D. Using a compass, mark a segment on the diameter CE = ED. Then the length of a side of a regular pentagon inscribed in a circle is equal to DC. We set aside segments DC on the circle and get five points for drawing a regular pentagon. Then, through one corner, we connect the corners of the pentagon with diagonals and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.
Each end of the pentagonal star is a golden triangle. Its sides form an angle of 36° at the top, and the base laid on the side divides it in proportion to the golden section. Draw straight line AB. From point A we lay off on it a segment O of arbitrary size three times, through the resulting point P we draw a perpendicular to the line AB, on the perpendicular to the right and left of point P we put off segments O. The resulting points d and d1 are connected by straight lines with point A. We put the segment dd1 on line Ad1, getting point C. She divided the line Ad1 in proportion to the golden ratio. The lines Ad1 and dd1 are used to build a "golden" rectangle.
Rice. 6. Building a golden
triangle
Golden Ratio and Golden Ratio
In mathematics and art, two quantities are in the golden ratio if the ratio between the sum of these quantities and the greater is the same as the ratio between the greater and the smaller. 
Expressed algebraically: 
The golden ratio is often denoted by the Greek letter phi (? or?). the figure of the golden ratio illustrates the geometric relationships that define this constant. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.
golden rectangle
The golden rectangle is a rectangle whose side lengths are in the golden ratio, 1:?
(one-to-fi), i.e. 1: or approximately 1:1.618. The golden rectangle can only be built with a ruler 
and a circle: 1. Construct a simple square 2. Draw a line from the middle of one side of the square to the opposite corner 3. Use this line as a radius to draw an arc that defines the height of the rectangle 4. Complete the golden rectangle
golden spiral
In geometry, the golden spiral is a logarithmic spiral whose growth factor b is related to?
, golden ratio. In particular, the golden spiral becomes wider (further away from where it started) by a factor ?
for every quarter turn it makes. 
The successive points of dividing the golden rectangle into squares lie on logarithmic spiral, sometimes known as the golden spiral.
Golden section in architecture and art.
Many architects and artists performed their work in accordance with the proportions of the golden section, especially in the form of a golden rectangle, in which the ratio of the larger side to the smaller one has the proportions of the golden section, believing that this ratio would be aesthetic. [Source: Wikipedia.org ]
Here are some examples:
Parthenon, Acropolis, Athens . This ancient temple fits almost exactly into the golden rectangle.
Vitruvian Man by Leonardo da Vinci you can draw many lines of rectangles in this figure. Then, there are three different sets of golden rectangles: Each set is for the head, torso, and legs area. Leonardo da Vinci's drawing Vitruvian Man is sometimes confused with the principles of the "golden rectangle", however, this is not the case. The construction of the Vitruvian Man is based on drawing a circle with a diameter equal to the diagonal of the square, moving it up so that it touches the base of the square and drawing the final circle between the base of the square and the midpoint between the area of the center of the square and the center of the circle: Detailed explanation about geometric construction >>
Golden ratio in nature.
Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio in the arrangement of branches along the stem of the plant and the veins in the leaves. He expanded his studies from plants to animals, studying the skeletons of animals and the branches of their veins and nerves, as well as the proportions of chemical compounds and the geometry of crystals, up to the use of the golden ratio in fine art. In these phenomena, he saw that the golden ratio was being used everywhere as a universal law, Zeising wrote in 1854: The golden ratio is a universal law, which contains the basic principle that forms the desire for beauty and completeness in such areas as nature and art, which permeates, as a paramount spiritual ideal, all structures, shapes and proportions, whether it be a cosmic or physical person, organic or inorganic, acoustic or optical, but the principle of the golden section finds its most complete realization, in human form.
Examples:
A cut of the Nautilus shell reveals the golden principle of spiral construction.
Mozart divided his sonatas into two parts, the lengths of which reflect golden ratio, although there is much debate as to whether he did it knowingly. In more modern times, the Hungarian composer Béla Bartók and the French architect Le Corbusier purposefully incorporated the golden ratio into their work. Even today golden ratio surrounds us everywhere in artificial objects. Look at almost any Christian cross, the ratio of vertical to horizontal is the golden ratio. To find the golden rectangle, look in your wallet and you will find credit cards there. Despite this much evidence given in works of art created over the centuries, there is currently a debate among psychologists about whether people really perceive golden proportions, in particular the golden rectangle, as more beautiful than other shapes. In a 1995 journal article, Professor Christopher Green, of York University in Toronto, discusses a number of experiments over the years that did not show any preference for the shape of the golden rectangle, but notes that several others have provided evidence that such a preference does not exist. . But regardless of the science, the golden ratio retains its mystique, in part because it applies so well to many unexpected places in nature. Spiral 
shells of the nautilus clam are surprisingly close to golden ratio, and the ratio of the length of the chest and abdomen in most bees is almost golden ratio. Even cross-sections of the most common forms of human DNA fit perfectly into the golden decagon. golden ratio and its relatives also appear in many unexpected contexts in mathematics, and they continue to arouse the interest of mathematical communities. Dr. Steven Marquardt, a former plastic surgeon, used this mysterious proportion golden ratio, in his work, which has long been responsible for beauty and harmony, to make a mask, which he considered the most beautiful form of the human face that can be. 
Mask perfect human face
Egyptian Queen Nefertiti (1400 BC)
The face of Jesus is a copy from the Shroud of Turin and corrected according to the mask of Dr. Stephen Marquardt.
An "averaged" (synthesized) celebrity face. With proportions of the golden section.
Site materials were used: http://blog.world-mysteries.com/
