What is the golden ratio examples. Shkrudnev Fedor Dmitrievich - Golden section
"There is something more, which is composed of the combination and connection of three things (number, limitation and placement), something that miraculously illuminates the entire face of beauty. We call this harmony, which, without a doubt, is the source of all charm and beauty. After all, the purpose and purpose of harmony - to arrange the parts, generally speaking, different in nature, by some perfect ratio so that they correspond to one another, creating beauty ... It embraces all human life, permeates the entire 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, so that what it produces was perfect. This can never be achieved without harmony, for without it the higher harmony of the parts disintegrates. "The Great Soviet Encyclopedia gives the following definition of the concept of "harmony":
"Harmony is the proportionality of parts and the whole, the fusion of various components of an object into a single organic whole. In harmony, internal order and a measure of being are externalized."A lot of "beauty formulas" are already known. For a long time, in their creations, people have preferred regular geometric shapes - a square, a circle, an isosceles triangle, a pyramid, etc. In the proportions of structures, integer ratios are preferred. Of the many proportions that people have long used when creating harmonic works, there is one, the only one, which has unique properties. This proportion was called in different ways - "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 science 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 ratio. And this is not surprising. The harmony of Nature is associated with the concept of "golden ratio". At the same time, the principles of symmetry in living and inanimate Nature are usually associated with harmony. Therefore, the universality of the manifestation of the principle of the golden ratio today will surprise no one. And each new discovery in the field of revealing another golden ratio no longer amazes anyone, except perhaps the author of such a discovery. The universality of this principle is beyond doubt. Various reference books contain hundreds of formulas connecting the Fibonacci series with the golden ratio, including a number of formulas reflecting 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 the binomial of Newton, as is known, reflects the power function of the dual relation.
This formula binds the binomial of the golden ratio to the One.
Without this principle, in fact, no fundamental problem can be considered. In grace, this proportion is justified as a principle of self-sufficiency. And yet, despite its universality, the golden ratio is not always used in practice, and not everywhere.
2
... MONAD AND GOLDEN SECTION
The principles of symmetry underlie the theory of relativity, quantum mechanics, solid state physics, atomic and nuclear physics, and elementary particle physics. It was shown above that symmetry is one of the forms of manifestation of duality. Therefore, it is not surprising that these principles are most clearly expressed in the properties of 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 laws of duality.
The regularity of duality is one of the main mechanisms of the evolution of living and nonliving 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 limitation and isolation, to transformation from an organism with internal duality into an organism with an external duality, that is, doubling, which is carried out by dividing the original. Then the process is repeated. The duality pattern 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 known as far back as Babylonia and ancient Egypt. The proportions of the Cheops pyramid, objects from the tomb of Tutankhamun, and other works of ancient art eloquently testify to this, and the term "golden ratio" itself belongs to Leonardo da Vinci. Since then, many masterpieces of art, architecture and music have been performed in 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 in itself the property self-regulation.
Academician V.K. Tolkachev in his book "The Luxury of Systems Thinking" he writes about this important property of the golden ratio:
“Once, Claudius Ptolemy evenly divided the height of a person into 21 segments and identified two main parts: the large (major), consisting of 13 segments, and the smaller (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 greater part is equal to the ratio of the greater part to the smaller ...
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) refers to the major in the same way as the major refers to the 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 = 0, the numerical value of which is:
NS 1 = - 0.618033989 ..., x 2 = 1.618033989 ...,
The first root is denoted by the letter "
F", and second "- F
", but we will use other notation:
F
= 1.618033989 ..., and Ф -1 = 0.618033989 ...
This is the only number that has the property of being exactly one more than its inverse ratio. "
Note that another equation
NS 2
-
y- 1 =
xy
becomes an identity for the following values
NS 1
= + 0,618033989...,
y 1
=- 1,618033989...,
x 2
= -1,618033989...,
y 2
= 0,618033989...,
Maybe be in the aggregate, these roots give rise to the life-giving cross
- the cross of the golden ratio?
Golden ratio equation
Ф 2 -Ф = 1
whereF 1 = -F -1 = - 0.618033989 ...,
andФ 2 = Ф 1 = 1.618033989 ...,
satisfy the property self-regulation, which allows you to build more complex "structures" by
image and likeness
".
Substituting the roots in the equation
NS
(
x-1) = 1,we'll get
F 1 (Ф 1 -1) = 1.618 .. * 1.618 ..- 1.618 .. = 2.618 ..- 1.618 .. = 1
F -2 - (- F -1) = 0.382 ... + 0.6181 = 1.
Thus, this equation reflects not only the principle self-regulation arising from the Unified law of evolution of the dual relation (monad), but also the connection of the golden ratio with Newton's binomial (with the monad).
It is easy to show that the following identities will be valid
F -2 = 0.382 ...;
Ф -1 = 0.618 ...;
F 1
=1,618...;
F 2
=2,618...;
From where you can directly see that
roots of the equationФ 2 -Ф = 1also have other and remarkable properties
Ф 1 Ф -1 = Ф 0
=1
and
F -1 (F 1 -1) = 1-F -1; Ф 1 (Ф -1 -1) = 1-Ф 1 = 1;
It characterizes the invariance of one mathematical monad into another, by multiplying it by its reciprocal, i.e. we can say that the roots of the golden section equation themselves form
gold, self-normalized monad<Ф -1 ,Ф 1
>
.
Therefore, this equation can rightfully be called
the equation of the golden ratio.
Anyone can learn additional properties of this equation using Newton's binomial and generating functions ( Continuity).
It is easy to understand that the process is more and more complex
"golden monads"will be
"in the image and likeness"
, i.e. this process will be periodically repetitive, and all the results appear to be, as it were, closed within the framework of the golden section.
But, perhaps, the most remarkable properties of the golden ratio are associated, first of all, with the equation of the golden ratio, given above. This equation is dual
NS 2 + x - 1 = 0.
The roots of this equation are numerically equal:
NS 1 = + 0.618033989 ..., x 2 = -1.618033989 ...,
This means that the equations of the golden ratio form a cross of the golden ratio with crossbars
Here he is, truly goldthe cross underlying the universe! The right figure directly shows that the values of the expression at the poles of the vertical bar are equal to 1. It is also seen from the cross in the left figure that at each transition from one bar to the second, self-normalization is carried out. Self-normalization occurs both during addition and multiplication. The difference is only in the sign. And this is no coincidence . When moving along the crossbars, we get four more values · when adding: 0 and0 , · when multiplying: -0,382 .., and-2,618 . It is easy to show that the following identities will be valid F -2 = 0.382 ...; Ф -1 = 0.618 ...; F 1 =1,618...; F 2 =2,618...; Using a number of these meanings, and going around the cross, we get another gold-cut cross.rice. 2
It is not difficult to show how from these crosses, to form a double cross, giving rise to the law of Cuba.
Below we will show that the six obtained values fully fit into the framework of a complex relationship - a unique pattern known from projective geometry. And now we will give another figure that directly speaks about the connection between the golden ratio 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 ratio can be continued indefinitely. So the Italian mathematician Luca Pacholli in his work "Divine Proportion" gives 13 properties of the golden ratio, supplying each of them with epithets -
exceptional, unspeakable, wonderful, supernatural,
etc.
It is difficult to say if 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. So, 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 GOLDEN SECTION
From the above properties of the golden ratio, it follows that the series
...; F -2 = 0.382 ...; Ф -1 = 0.618 ...; Ф 0; Ф 1 = 1.618 ...; Ф 2 = 2.618 ...; ...;
can be continued both to the right and to the left.
Moreover, the multiplication of this series by
F
+
norF -ngenerates a new row, shifted respectively to the right or left of the original. Odds
F
+
norF -ncan be considered as coefficients of similarity of gold-cut rows.
Gold-cut 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, gold-cut series. First of all, the formula for the "golden" binomial of Newton should be given.
The binomial of Newton 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 binomial Newton reflects one more important property of the universe. She is
normalized(single).
4. ABOUT RELATIONSHIP OF THE GOLDEN SECTION WITH THE ROW OF FIBONACCI
Nature, as it were, solves the problem from two sides at once and adds up the results obtained. As soon as it gets 1 in total, it moves to the next dimension, where it starts to build everything all over again. But then she must build this golden ratio according to a certain rule. Nature does not immediately use the golden ratio. She gets it through successive iterations. She uses another row to generate the golden section, the Fibonacci row.
Fig. 5
Rice. 6.Golden Ratio and Fibonacci Spiral
A remarkable property of this series is that as the numbers of the series increase, the ratio of the 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 opened 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 that 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 an internal or external duality. It should be said that the Fibonacci spiral can be double. There are numerous examples of these double helixes found everywhere. So the spirals of sunflowers are always related to the Fibonacci series. Even in an ordinary pinecone, you can see this double Fibonacci spiral. The first spiral goes one way, the second the other. If you count the number of scales in a spiral rotating in one direction, and the number of scales in another spiral, you can see that these are always two consecutive numbers of the Fibonacci series. There may 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 spirals? The Golden Ratio spiral is perfect. It corresponds to the original source of harmony. This spiral has no beginning or end. It is endless. The Fibonacci spiral has a beginning, from which it starts "spinning". This is a very important property. It allows Nature, after another closed cycle, to build a new spiral from “scratch”. These facts confirm once again that the duality law gives not only qualitative, but also quantitative results. They make one think that the Macrocosm and Microcosm surrounding us evolve according to the same laws - the laws of hierarchy, and that these laws are the same for living and nonliving matter. The law of duality is the culprit for the fact that the Hierarchy, having in its luggage only this one algorithm for the formation of invariant shells, makes it possible to build the generating functions of these shells, to build the Unified Periodic Law of Evolution of Matter. Suppose 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 recurrent dependence, assuming that this member of the function will be obtained by summing its last two members. For example, for n = 1, the value of the third term of the series will be 2. As a result, we will receive the series (1-1x + 2x2). Then, multiplying the generating function by the operator (1-x) and using the recurrent dependence to calculate the next term of the series, we will get the desired generating function. Denoting through the value of the nth term 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 represented as follows (Table 1).
Table 1.
The table shows that after receiving the next resulting term of the series, this term is substituted into the original polynomial and addition with the previous one is performed, then the new resulting term is substituted into the original series, etc. As a result, we get the Fibonacci number. 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 as a result of 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 repeated application of an operator of the form (1 + x) leaves the structure of the polynomial unchanged, and the Fibonacci series has an additional, also more wonderful properties: each member of this series is the sum of its last two members. Therefore, Nature does not need to remember the Fibonacci series itself. You just need to remember the last two terms of the series and the operator of the form P * (x) = (1-x), which is responsible for this doubling algorithm, in order to get the Fibonacci series without error. But why in Nature this series plays a decisive role? This question can be answered with an exhaustive answer by the concept of triplicity, which determines the conditions for its self-preservation. If the “balance of interests” of the triad is violated by one of its “partners”, the “opinions” of the other two “partners” must be adjusted. The concept of trinity is especially clearly manifested 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 that are necessary for the formation of other elementary particles. It is possible that the Fibonacci spiral can play a decisive role in the formation of the pattern of limited and closed hierarchical spaces. Indeed, imagine that at some stage of evolution, the Fibonacci spiral has reached perfection (it has become indistinguishable from the golden ratio) and for this reason the particle must transform into the next "category". The wonderful properties of the Fibonacci series are manifested in the numbers themselves, which are members of this series.Let us arrange the members of the Fibonacci series vertically., And then to the right, in decreasing order, we write down the natural numbersEach line begins and ends with a Fibonacci number, that is, 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 hierarchy of the Fibonacci series):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 received a fractional Fibonacci series, which is, possibly, "professed" by the collective spins of elementary particles and atoms of chemical elements. The next level of the hierarchy is formed by splitting the intervals between the Fibonacci numbers and the highlighted numbers. For example, the third level of the hierarchy will be the numbers 52 and 50 from the interval 55-47. The process of structuring a series of natural numbers can be continued, since the properties of periodicity and multilevel 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 the dual relation (monad). Above, the range of change in the properties of the dual relation was determined, which characterizes its norm of self-sufficiency. U =<2/3, 1) Let's build a Fibonacci series for a given 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 the increasing spiral of the evolution of a dualistic attitude. Let's continue this process. An attempt to go beyond this range of the self-sufficiency rate will lead to its normalization, i.e. first element in D-tetrahedron will be characterized by a self-sufficiency rate equal to 1,0 ... But, continuing this process further, we will have to constantly perform renormalization. Therefore, evolution cannot continue? But, in the question itself, there is an answer. After renormalization, evolution should start over again, but in the opposite direction, i.e. during the formation of a "parallel" D-tetrahedron, the sign of the number should change and the Fibonacci series begins the opposite movement.
D =Then the general series characterizing the self-sufficiency rate of the "star tetrahedron" will be characterized by the relations
U =The stable state of a star tetrahedron will depend on the corresponding conjugation of the L- and D-tetrahedra. For U = 1 we will have a cube. For U = 2/3 we get self-sufficient star tetrahedron, with self-sufficient L- and D- tetrahedra. At lower values, the stable state of the star tetrahedron will be achieved only by joint efforts of the L- and D-tetrahedrons. Obviously, in this case, the minimum value of the self-sufficiency rate of the star tetrahedron will be equal to U = 1/3, i.e. two n e self-sufficient the tetrahedron jointly form self-sufficient star tetrahedron U. In the most general case, stable states of the star tetrahedron U can be illustrated, for example, by the following diagram.
Rice. 7
The last figure 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 wonderful properties of the Fibonacci series, its periodicity ( Mikhailov Vladimir Dmitrievich, "Living Information Universe", 2000, Russia, 656008, Barnaul, st. Partisan house. 242).
p.10."The laws of the" golden ratio "," golden section "are associated with the digital Fibonacci 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 ratios (numbers) formed by its members and their various invariants have been scrupulously studied and generalized, but never 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 a civilization can be determined using various methods in numerology. For example, by converting complex numbers to single-digit ones (for example: 13 is (1 + 3) = 4, 21 is (2 + 3) = 5, etc.) Carrying out a similar procedure of addition with all 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 the numbers into digits, after 24 digits the cycle will be sequentially repeated an infinite number of times ... ... isn't a set of 24 digits a kind of digital code for the development of civilization? P.17 If the Pythagorean Four in the sequence of 24 Fibonacci digits is divided among themselves (as if to break) and superimposed on each other, then a picture of the relationship of 12 dualities of opposite numbers appears, where each pair of numbers adds up to 9 (duality giving birth to a 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 edition)
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. m istifications, etc., merging into a single stream, inevitably lead to the conclusion about the existence of a periodic pattern, reflected in the Fibonacci series. Now let's look at 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 of primary importance. They are called Plane Bodies. Any Platonic solid has some special characteristics. Firstly, all the faces of such a body are equal in size. Secondly, the ribs of the Platonic solid are of the same length. Thirdly, the interior angles between its adjacent faces are equal. AND,fourthly,being inscribed in a sphere, the Platonic solid with each of its vertices touches the surface of this sphere.
Rice. eight
There are only four shapes other than the cube (D) that have all of these characteristics. The second body (B) is a tetrahedron (tetra means "four"), which has four sides in the form of equilateral triangles and four vertices. Another solid (C) is an octahedron (octa means "eight"), whose eight faces are equilateral triangles of the same size. The octahedron contains 6 vertices. The cube has 6 faces and eight vertices. The other two Platonic Solids are somewhat more complex. One (E) is called an icosahedron, which means "having 20 faces," represented by equilateral triangles. The icosahedron has 12 vertices. The other (F) is called the dodecahedron (dodeca is "twelve"). Its faces are 12 regular pentagons. The dodecahedron has twenty vertices.
These bodies have remarkable properties of being inscribed all in just two shapes - 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, etc. The cube is located in this hexad opposite the sphere and therefore it has similar properties. Then the properties similar to the tetrahedron should be possessed by the monad form located in the hexad opposite the tetrahedron. This is an icosahedron. The forms of the dodecahedron must be "related" to the octahedron. Finally, the last shape becomes a sphere again. The last one becomes the first! In addition, the continuity of the evolution of two neighboring Platonic solids should be observed in the hexad. And, indeed, the octahedron and the cube, the icosahedron and the dodecahedron are reciprocal. If one of these polyhedra is connected by straight line segments the centers of the faces that have a common edge, then we get another polyhedron. In these properties lies their evolutionary origin from each other. In the Plato hexad, two triads can be distinguished: "sphere-octahedron-icosahedron" and "tetrahedron-cube-dodecahedron", which endow the neighboring vertices of their own triads with reciprocity properties.
These figures have another remarkable quality. They are closely tied to 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 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. nineAt first glance, it may seem that the "monadic charges" of Platonic solids reflect, as it were, a discrepancy between ideal forms from the Fibonacci series. However, assuming that starting from the cube, Platonic solids can form 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 express the 13 and 21 ratios of the Fibonacci series. See how it goes rationingFibonacci 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" Fibonacci series formation algorithm. The second line begins with an icosahedron, in which the 13th vertex is the center of the structure, reflecting the properties of the GREAT LIMIT. The dodecahedron has a similar GREAT LIMIT. These two crystals give rise to a new dimension - the normalized "icosahedron-dodecahedron" monad, which begins to form a new turn of the Fibonacci series (third line). The first Platonic solids, as it were, reflect the phase of analysis, when the GREAT LIMIT is unfolded from the monad (1,1). The second phase is with the interests of the new monad and its folding into the GREAT LIMIT. So the Fibonacci series generates the "golden proportion", which is responsible for the birth of the harmony of all that exists, therefore the Platonic solids will also characterize the properties of all material structures. Thus, atoms always correspond to 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 is metal, crystal or anything else - the structure always goes back to one of the five original forms. Therefore, we come to the conclusion that the number of primordial monad 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 shape of polyhedrons; when split, these polyhedrons 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 must necessarily be associated with simpler monadic crystals. The most complex - the highest forms of the Higher Mind - should be built from these forms. And these properties of monadic crystals should manifest themselves 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 presented in the form of monad 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 ... " . Thus, the properties of 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 ONE 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 the chromatic scale, in which the 13th note is beyond the "border 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 according to which the UNIVERSE UNIVERSE HARMONY UNIVERSE HARMONY FIELD is formed.
5. GOLDEN RATIO AND THE PRINCIPLES OF SELF-ORGANIZATION
5.1. SELF-SUFFICIENCY
Principlesself-organization (self-sufficiency, self-regulation, self-reproduction, self-development and self-regulation) 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 justified that the concept self-sufficiency definesshare the contribution of own target functions to the overall target function of a particular object of the surrounding world. If the object's own share of the contribution to the total objective function is not less than 2/3, then such an object will have a "controlling stake" of the object's objective function and, therefore, will be self-sufficient, not a "puppet" object. But 2/3 = 0.66 ... and the golden ratio is 0.618 ... A very close match, or ..? That's exactly OR! Therefore, more accuratequantitative assessmentself-sufficiency can be considered the proportion of the golden ratio. However, for practical use a measure of self-sufficiency, definingqualitythe state of the object, whether he lives in harmony with the world around him 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 ratio and their relationship with the ONE LAW were given by the hand of the great master Leonardo da Vinci. And it's a pity that MANY STILL SCIENTISTS DO NOT UNDERSTAND THIS EVEN TODAY. A SHAME!!!5.2. SELF-PLAYBACK. SELF-DEVELOPMENT.
From the principles of constructing universal logic ( ) it follows that the infinite-dimensional logic within the evolution of the same family forms a binary spiral.
In this diagram, the nodal points characterize the downward spiral of evolution of the logical family of the binary spiral (right screw). It can be determined by induction that the left screw will define the upward spiral of this family. This evolutionary binary spiral characterizes self-reproduction andself-developmentlogical family. Suppose we have initial logic< - i ,-1 >. Then, depicting the axes of the complex frame of reference in accordance with the rule of traversing the tetrahedron along the cross, the evolution of logics can be reflected as shown in Fig. 12 rice. 12 The diagram shows that with each transition from one logic to another, in the direction of the arrows, a mirror image occurs. self-copying logic. And when we complete the "circle of evolution", then the last and the first logics will turn out to be opposite to each other. The next attempt leads to the logic of binary doubling, since the cell is occupied. As a result, a logic is born that differs from the first in scale, instead of< -i, -1> a couple is born< -2 i ,-2 >. Note that sequential mirroring of logics leads to their mirror inversion along the diagonals. So, diagonally - i ,+1 we have logic <- i ,-1> <+1,+ i >. From the rules for traversing the vertices of the tetrahedron along the cross, we find that these logics form a cross in the tetrahedron if the corresponding edges are projected onto the plane. NSabout the diagonal-1,+ i we got complementary a couple of logics <-1,- i > <+ i ,+1> , also forming a cross. In fig. 11, the sides of the squares are oriented in the direction of baptism. Therefore, the opposite sides of this square are the crossbeams 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 only justifies 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 should have opportunity for self-development... And this opportunity is realized (Fig. 13). rice. 13 Here squared IIfirst happens self-copying the original logic, and in the third square, the process takes place self-development... Here, at first, the first and second squares are added with a shift, and then reproduced in the square III... Then the resulting chain is mirrored into a square IVwhere the "closure" of the chain occurs. As a result, a tetrahedron is born, with four vertices, i.e. complex logic is born. So out of a pair<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 in squares in accordance with the above rules, we get a pair<3,3>... By 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 not hard to see that the Fibonacci series is born , which is the basis of the golden ratio. And this series is born in a natural way, it is based on the Unified Periodic Law of Evolution and the principles following from it self-organization (self-sufficiency, self-regulation, self-reproduction, self-development, self-regulation).
rice. eleven
5.3. FIBONACCI SERIES AND BINARY SERIES
Let us now take an integral pair as logical pairs<2,2>... This pair will characterize the quantitative composition of the first logical shell. Then, in the process of her "baptism", we will produce the following binary pair<4,4>... This pair in its structure will characterize a 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>, the evolution of which will lead to a pair<16,16>, and then to the pair<32,32>. 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. Thus,unity of the Fibonacci series and the binary series is an indisputable fact. The periodic table of chemical elements, the binary series, the Fibonacci series and the golden ratio are closely interrelated.
Rice. fourteen
It can be seen from the last diagram that the generating functions of these series are also closely interconnected with the Newton binomial
(1-x) -n.
There is also a direct relationship between the Fibonacci series and the binary series (Fig. 4)
Rice. fifteen
This figure shows how from the initial ratio (1-1-2), using a binary series, the entire Fibonacci series is built. This scheme is given in his book by D. Melchizedek ("The Ancient Secret of the Flower of Life", vol. 2, p. 283). This figure shows the bee drone 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 this is correct (Gene memory, Information, About time). These pages provide the rationale that gene memory, reviving Past, or synthesizingFuture,forms exactly a binary series and precisely according to the law shown in Figure 4.6. ABOUT OTHER PROPERTIES OF THE FIBONACCI SERIES
Everyone knows that rhythms (waves) permeate our entire life. Therefore, the universality of the proportion of the golden ratio must 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-wave lengths of the harmonic series correspond to the function 1 / n, wheren- natural number. The half-wave lengths can be expressed as a percentage of the half-wave length of the fundamental harmonic: 100%, 50%, 33%, 25%, 20% ... If an arbitrary section of the string is affected, all harmonics with different amplitude coefficients will be excited, which depend on the coordinate area, from the width of the area and from the time-frequency characteristics of the impact. Taking into account the different signs of the phases of even and odd harmonics, one can obtain an alternating function that looks approximately as follows: If the fixing point is taken as the origin and the middle of the string as 100%, then the maximum susceptibility at the 1st harmonic will correspond to 100%, at the 2nd - 50%, at the 3rd - 33%, etc. Let's see where our function will cross the abscissa 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 successive series of line segments when adjacent segments are in the ratio of 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 ratio 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 selected at the 4th intersection of the function with the abscissa. Thus, the periodicity of changes in the properties of the dual relation turns out to be associated with the self-sufficiency rate, the Fibonacci series, as well as with the properties of the star tetrahedron, which reflects the principle of an upward and downward spiral. Therefore, we can say that the secret of the Golden Section, the secret of the Fibonacci series, the secret of their universality in the world of animate and inanimate Nature no longer exists. The Golden Ratio and the Fibonacci series reflect the most fundamental regularity of the Hierarchy - the regularity of duality, and the Fibonacci series itself reflects not only one of the main forms of manifestation of this regularity - the three-unity, but also characterizes the norms of self-sufficiency of the dual attitude in the process of its evolution. 7. ABOUT A COMPLEX RELATIONSHIP The properties of the golden ratio 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 relation of another remarkable relationship, which in projective geometry is known as complex point ratio 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, it doesn't matter if you are measuring the distance on the image or on the plot itself. The camera can be deceiving. It deceives when it gives out equal lengths for unequal and right angles for indirect ones. The only thing it doesn't distort is the expression
ZnThe expression for 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 a shorthand for a complex relation
ABCD.
Let us now redraw the scheme of a complex relation in spatial form.
Rice. 17
It is known that the golden ratio is expressed by the proportion
where the numerator is the lower number and denominator-large.
In relation to Figure 17, the golden proportion 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 the division of the segment in half. If the angle is
π
/
2,
then we get a right-angled triangle with sides
1, F, F 0.5;
Therefore, we have the original equation
Ф 2 -Ф = 1,written in the vector form -g hypothenuse is a unit, and the legs are orthogonal to each other, which is reflected in the equation of the golden section.
At 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, transforms into a broken line, and a complex relation is generated by the process "
traversing the cross
".
WhereinThe last 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 crossbeams of the cross and dividing by the length of the shoulders that determine transition from one crossbar to another. When building these more complex relationships, it is necessary to take into account that only two adjacent vertices of the broken line participate in the formation of a complex relationship, just like in the Fibonacci series. This rule of leverage, using the golden ratio, can be written as follows . And now we can build a complex relation on the tetrahedron, given that the distances from all the vertices of the pyramid to the point O are the same.Rice. 18
From Figures 14-19, one can understand the principles of building more complex relationships, for spaces with a greater dimension, i.e. we can say that n-dimensionalcomplex relationship reflects the process of formation of a monad crystal n -dimensionality and that's why "exercises" for the formation of more complex relationships may be of independent interest ( Complicated attitude). But all the meanings of a complex relationship NS, (1/NS), (x-1) / NS, NS/ (x-1), 1 / (1-x), (1-x), NS,... are parts of the golden ratio equation x 2 - NS - 1 =0 or NS(NS -1) =1. 7. THE LAW OF PRESERVATION OF THE GOLDEN SECTION The properties of the golden ratio considered above and, first of all, the properties of a complex relationship, allow us to say that the golden ratio forms the main law of the universe, reflecting the main law of conservation I- conservation law 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 endless row, in which the first four values form the cross of the golden ratio. Moreover, whenever a value is obtained that is greater than the value of the golden ratio, then there is normalization OBJECT... Is isolated from him unit and the evolutionary process 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 0.618 and 1.618 is the reason why the golden ratio is at the heart of the harmony of the world. The conservation law (conservation laws) of the golden ratio can be to demonstrate in a rotating cross (swastika).Rice. nineteen
Below, on the page revealing the secrets of information (Information, About time), it will be shown that the golden ratio, gene memory, lie at the heart of the very concept of information,
about the natural mechanisms of evolution of the "IMAGE-LIKE" monad in TIME.
Thus, the essence of rationing is reduced to obtaining the proportions of the golden section, i.e. all the wonderful properties of a complex ratio of four points are determined by the properties of the life-giving cross, that a complex ratio is closely interconnected with the golden ratio, forming a conservation law
golden ratio.
SUMMARY
1. No one has any doubts that the golden ratio lies at the heart of the harmony of the universe, and a number
Fibonacci generates this wonderful proportion.
For more information on the properties of the golden ratio, curious readers can get on the site
www .
goldenmuseum.
com
.
This truly golden proportion has so many wonderful properties that the discovery of new properties does not surprise anyone.
Modern web design includes 2 features that must be strictly adhered to: aesthetics and the right scope. If you follow these concepts, web design can be considered successful.
As for aesthetics, here we mean that when drawing a particular image of an object, we use many different manipulations: creating a grid, a layout, using typographic techniques in order to get a good structure of an 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've probably heard of these concepts before. Or maybe 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 come about?
At first glance, this term may be incomprehensible. Why exactly "Golden"? Why use this technology? Today it is still a mystery who came up with the "Golden Section", where this name came from. However, the technology is known to have been in use for 2,400 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 derived 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 the two dimensions and aspect ratios of the lines. In decimal, b divided by a equals 1.618033 ... if a> b. In the example below, let's say b is 5. Then the equation looks 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 presented in the form of the formula: xn = xn-1 + xn-2.
Consistency is closely related to the golden ratio, because if you take any two consecutive numbers and divide by the previous one, the fraction will turn out to be very close to the golden ratio. As the value of the number increases, the fraction becomes even closer to the golden ratio. For example, 8/5 is 1.6, 34/21 is 1.619, and so on.
The Golden Spiral. Rectangle
So, you've probably come across similar equations. But why do designers use geometry in their designs? Why overlay shapes? The pattern 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 a sample:
It turns out that each subsequent radius is larger than the previous one by a number close to the golden ratio. The resulting spiral is used in many fields, more often in painting and architecture, but it can also be observed in natural phenomena.
The Rule of Three
This rule is one of the variants of the golden spiral and is often used when cropping photos and videos. Used for cropping frames and giving them an aesthetic look. To apply the Rule of Three, you need to divide the image into 9 equal parts. Draw 2 horizontal and 2 vertical lines. It is important to arrange them evenly. The point is to align the focus with the leftmost vertical separator. The horizon or vanishing point must be level with a horizontal divider.
Application of the "Golden Spiral"
As already noted, the Fibonacci sequence is closely related to the golden ratio. The Golden Ratio is applied 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 at the edges of the inner squares: her chin, eyes and lips. Leonardo Da Vinci used the Golden Ratio many times in his works. 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, this is just the basis of the process. It is impossible to just use this or that number and get a magic proportion. However, there are certain methods that can help you get the Golden Ratio. Some artists tend to think that the golden ratio theory is a myth. Here's another example of how the golden ratio works. Let's take a prototype site and look at applying the Golden Ratio to it.
Looks pretty straightforward, doesn't it? Yes, in fact it is. The design is based on a 960 pixel grid. The decoration is presented using the golden ratio. If you use 1 golden spiral that spans 960px, you can see how the heading, logo, etc. were positioned.
We move our spiral below and rely on its dimensions
It turns out a certain cascade of spirals in which the main design elements are inscribed in rectangles with a golden ratio
The grid based on the golden ratio has a number of proportional ratios within it, which are clearly proportional within the 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.
They say that "divine proportion" is inherent in nature, and in many things around us. You can find it in flowers, hives, seashells, and even our body.
This divine ratio, 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 relationship was discovered, many artists and architects have applied 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 know what the secret of the Golden Ratio lies, which pleases our eyes so much. 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 by the equation a / b = a + b / a = 1.618033987, where a is greater than b. It 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 two subsequent Fibonacci numbers (i.e. 8/5 or 5/3), the result is very close to the golden ratio of 1.6 or φ (phi).
A 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 image above, you can start building the 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 process for 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 Fibonacci sequence can be found in flower petals. For most flowers, the number of petals is reduced to two, three, five or more, which is similar to the golden ratio. For example, lilies have 3 petals, buttercups have 5, chicory flowers have 21, and daisies have 34. Probably, the flower seeds also follow the golden ratio. For example, sunflower seeds sprout from the center and grow outward, filling the head of the seed. They are usually spiral shaped and resemble a golden spiral. Moreover, the number of seeds is usually reduced to Fibonacci numbers.
Hands and fingers are also examples of the golden ratio. Look closer! The base of the palm and the tip of the finger are separated by parts (bones). The ratio of one part to the other is always 1.618! Even the forearms and 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 ratio. This is possible because the building looks proportionally perfect, and it has been that way since ancient times.
Leonardo Da Vinci was also a fan of the Golden Ratio (and many other curious subjects, as a matter of fact!). The wondrous beauty of the Mona Lisa can be attributed to the fact that her face and body represent the golden ratio, just like real human faces in life. In addition, the numbers in the painting "The Last Supper" by Leonardo Da Vinci are arranged in the order that is used in the golden ratio. If you draw golden rectangles on the 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 in 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 Fibonacci circles to connect and cut the shapes to get the Apple logo. It is unknown whether this was done on purpose 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 mesh in which three rings are formed. Notice how this logo uses rectangles instead of circles to create the golden ratio.
The Pepsi logo is created with two overlapping circles, one larger than the other. As shown in the picture above, the larger circle is proportional to the smaller - you guessed it! Their latest non-embossed logo is simple, eye-catching and beautiful!
In addition to Toyota and Apple, the logos of several other companies such as BP, iCloud, Twitter, and Grupo Boticario are believed to have used the golden ratio as well. And we all know how famous these logos are - all because the image immediately pops into memory!
This 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 of height and width from the numbers belonging to the golden ratio. Start with one block and place another next to it. And place another square, whose area is equal to those two, place above them. You will automatically receive a 3-block side. After building this three-block structure, you end up with a side of 5 quadrangles from which you can make another (5-block area) box. This can continue as long as you like until you find the size you want!
The rectangle can be moved in any direction. Select the smaller rectangles and use each one to build a layout that will serve as a grid for your logo design.
If the logo is more rounded, then you need a circular version of the golden rectangle. You can achieve this by tracing circles proportional to the Fibonacci numbers. Create a golden rectangle using only circles (this means that the largest circle will have a diameter of 8, and the smaller one will have a diameter of 5, and so on). Now split these circles and place them so that you can form the basic outline for your logo. Here's an example of a Twitter logo:
Note: You do not need to draw all the circles or rectangles of the Golden Ratio. You can also use the same size multiple times.
How to apply it in text design
It's easier than designing a logo. A simple rule of thumb for applying the golden ratio to 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 subheading 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). This means that the subheading should be written in 17 font size. And now the title or title. I multiply the subheading by the proportion to 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 complicated. 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 total number of grid pixels as the width or height and use that to draw the golden rectangle. Divide the largest width or length for smaller numbers. This can be the width or height of your main content. What is left could be a sidebar (or a bottom bar if you applied it to the height). Now keep using the golden rectangle to further apply it to windows, buttons, panels, images and text. You can also build a full mesh based on small versions of the golden rectangle, both horizontally and vertically to create smaller interface objects that are proportional to the golden rectangle. You can use this calculator to get the proportions.
Spiral
You can also use the golden spiral to determine where to place content on your site. If your home page is loaded with graphic content, such as on an e-commerce website or a photography blog, you can use the golden spiral method that many artists use in their work. The idea is to place the most valuable content at the center of the spiral.
Content with grouped material can also be placed using the golden rectangle. This means that the closer the spiral moves to the central squares (one square block), the denser the content there.
You can use this technique to indicate the location of your title, images, menu, toolbar, search box, and other elements. Twitter is famous not only for its use of the golden rectangle in logo design, but also for its web design. 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 material defines the location 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 blog design, which fits well with the golden rectangle.
An easier way
Very often, designers omit complex mathematics 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 a focal point for shape and design. You can place a key theme or main elements on one or all of the focal centers. Photographers also use this concept for posters.
The closer the rectangles are to the 1: 1.6 ratio, the more pleasant the picture is perceived by the human brain (since it is closer to the golden ratio).
Why do ceremonies work? This explains the Golden Section Method. The words in the 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 at the peak of the golden section is a subconscious mindset, criticism of thinking is not involved here.
In ancient times, healers noted that it is 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 another space, where entities have no opportunity to harm. Here's an example:
How are you, month, on the decline,
So I will begin to decline
From the body is white fat GO to the pig
In the name of the Father and the Son and the Holy Spirit. Amen.
This conspiracy is a prime example of the golden ratio. Its power 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 following the link
To simplify the task, do the following:
- Record your text on your phone.
- Turn on repeat mode.
- Plug in headphones to avoid distracting text.
- Listen to the conspiracy for 30-45 minutes (15 days).
The first result can be found already 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 choose the speed at which it will reach its goal. Just watch the magic happen.
There is an extremely important point!
The principle of the golden ratio is in effect when the formulation of the goal consists of 27 word 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 №№16, 17,18). I want to get it in a way that is safe 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 it !!!
But first, I want to tell you how to determine your own number. Find your year of birth in the table provided, opposite 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.
2 palace - a 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 upgrade. Think about what projects you want to bring to life and how much to get for it. Plan your goals clearly at this stage.
4 palace - create awareness of your own brand in the new year, formulate romantic goals. Make plans to spend money on your beauty and appearance.
5 palace - wish new acquaintances and travels, development of female energy and sexuality, exclusion of karmic lessons.
6 palace - think about home, moving, renovation. Let a team of like-minded people be in your desires. If you want a child, plan it.
7 palace - 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 your wishes 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 so that from 2018 (it is better to pronounce the Eighteenth) (words # 16, 17.18 - I GET TWO MONTHLY) thousand rubles only in a safer way for all that exists”.
As a result, we got 27 words. The word “I receive” has become the golden ratio. Now the cash flow will not be able to get stuck anywhere, all payments will arrive on time.
The main word got into its happy location, the target will act on the subconscious in a direct way. We have formed an impulse.
One more point. Dangerous verbs should be avoided when forming a goal. Among them:
- Verbs with non-ending action (search, go, sell). They make sense that the action will last and never end.
- Heavy vibrating verbs that reduce the urge to move forward:
To earn - to earn money by hard work (from the word "slave").
To control - implies many processes that subconsciously repulse our brain.
To work - to work for the "uncle" and suffer.
To seek - to beg for something.
To try is to act under torture.
Try - to hiccup and not find a solution.
Your goals are important, so you need to be careful about them. They regulate life and lead to achievements. To speed up solving problems, you can use ready-made recipes:
- individual monetary figures;
- the use of noble helpers;
- energy ji fu - big boss;
and many other features. And all these feng shui chips are collected in the "Super-profitable Feng Shui" training. You can familiarize yourself with its program by following the link
And that's all. Tatyana Panyushkina was with you!
Subscribe to Tatiana Panyushkina's closed Telegram channel - link to the channel https://t.me/tpanyushkina
There will be published high-quality video clips on health diagnostics, recommendations on longevity and health, disclosure of superpowers, and predicting the future. Practical recommendations, techniques and methods.
Community members every week on Sundays receive information for their development in a convenient form. These materials are not published in free sources or on the Internet.
In contact with
Even in ancient Egypt it was known Golden ratio, Leonardo da Vinci and Euclid studied its properties.A person's visual perception is arranged in such a way that he distinguishes in form all the objects that surround him. His interest in the subject or its form is sometimes dictated by necessity, or this interest could be aroused by the beauty of the subject. If in the very basis of constructing the form, the 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 parts of different sizes 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. Concept of golden ratio the ancient Greek mathematician and philosopher (VI century BC) Pythagoras introduced into scientific use. But the very knowledge of golden ratio he borrowed from the ancient Egyptians. The proportions of all the buildings of the temples, the pyramid of Cheops, bas-reliefs, household items and decorations from the 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 in Abydos and the bas-relief of Ramses used the principle golden ratio in the proportions of the figures. It was found out by the architect Le Corbusier. On a wooden board taken from the tomb of the Architect Khesir, there is a relief drawing, which shows the architect himself, holding in his hands the instruments for measurements, which are depicted in the position fixing the principles golden ratio. Knew the principles golden ratio and Plato (427 ... 347 BC). Dialogue "Timaeus" is proof of this, since it is devoted to questions gold division, aesthetic and mathematical views of the Pythagorean school. Principles Golden ratio used by ancient Greek architects in the facade of the Parthenon temple. The compasses that were used in their work by ancient architects and sculptors of the ancient world were discovered during excavations of the Parthenon temple.
Parthenon, Acropolis, Athens In Pompeii (Museum in Naples) proportions gold division are also available.In ancient literature that has come down to us, the principle golden ratio mentioned for the first time in Euclid's Elements. In the book "Beginnings" in the second part, the geometric principle is given golden ratio... The followers of Euclid were Pappus (III century AD) Hypsicles (II century BC), and others. To medieval Europe with the principle golden ratio we got acquainted by translations from Arabic of Euclidean "Elements". Principles golden ratio were known only to a narrow circle of initiates, they were jealously guarded, kept in strict secrecy. The era of renaissance and interest in principles has arrived golden ratio increases among scientists and artists, as this principle is applicable in science, and in architecture, and in art. And Leonardo Da Vinci began to use these principles in his works, even moreover, he began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, who was ahead of him and published the book "Divine Proportion" after which Leonardo left his labor 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 artist Piero della Franceschi, 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, came to Milan in 1496 and gave lectures on mathematics there. Leonardo da Vinci at this time worked at the court of Moreau. Luca Pacioli's book 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 singled out her "divine essence". Realizing the scientific and artistic value of the golden ratio, Leonardo da Vinci devoted a lot of time to studying it. Carrying out a section of a stereometric body consisting of pentagons, he obtained rectangles with aspect ratios in accordance with golden ratio... And he gave him the name “ golden ratio”. That holds up to this day. Albrecht Durer, also studies golden ratio in Europe, meets the monk Luca Pacioli. Johannes Kepler, the greatest astronomer of the time, was the first to notice the meaning golden ratio for botany, calling it the treasure of geometry. He called the golden proportion continuing to itself "It is so arranged," he said, "the sum of the two lower terms of the infinite proportion gives the third term, and any two last terms, if you add them, 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 the segments of the golden ratio of the descending and ascending series, 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 Durer. If O is the center of the circle, A is a point on the circle and E is the midpoint of the segment OA. The perpendicular to the radius OA, restored at point O, intersects with the circle at point D. Using a compass, mark the segment at the diameter CE = ED. Then the side length of a regular pentagon inscribed in a circle is DC. We put aside the 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 set aside on the side divides it in proportion to the golden ratio. We draw a straight line AB. From point A, we lay on it three times a segment O of an arbitrary size, through the resulting point P we draw a perpendicular to line AB, on the perpendicular to the right and left of point P we lay off segments O. We connect the obtained points d and d1 with straight lines to point A. line Ad1, getting point C. She divided line Ad1 in proportion to the golden ratio. Lines Ad1 and dd1 are used to draw a "golden" rectangle.
Rice. 6. Constructing gold
triangle
Golden ratio and Golden ratio
In mathematics and art, two quantities are in golden proportion if the ratio between the sum of these quantities and the greater is the same as the ratio between the greater and the lesser. 
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, the lengths of the sides are in the golden proportion, 1:?
(one-to-fi), that is, 1: or roughly 1: 1.618. The golden rectangle can only be built with a ruler 
and a compass: 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 the 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?
, the golden ratio. In particular, the golden spiral becomes wider (farther from the place of its beginning) by the coefficient ?
for every quarter of a turn she makes. 
Consecutive points of division of a golden rectangle into squares lie on logarithmic spiral, which is sometimes known as the golden spiral.
The golden ratio in architecture and art.
Many architects and artists performed their works in accordance with the proportions of the golden ratio, especially in the form of a golden rectangle, in which the ratio of the larger side to the smaller side has the proportions of the golden ratio, believing that this ratio would be aesthetically pleasing. [Source: Wikipedia.org ]
Here are some examples:
Parthenon, Acropolis, Athens ... This ancient temple fits almost exactly into a golden rectangle.
Vitruvian Man by Leonardo da Vinci you can draw many lines of rectangles into this figure. Then, there are three different sets of golden rectangles: Each set is for the head, torso, and legs area. Drawing by Leonardo Da Vinci 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 >>
The 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 research and moved from plants to animals, studying the skeletons of animals and the branches of their veins and nerves, as well as in the proportions of chemical compounds and the geometry of crystals, up to the use of the golden ratio in the visual arts. In these phenomena, he saw that the golden ratio is 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 primary spiritual ideal, all structures, forms and proportions, be it a cosmic person or an individual. organic or inorganic, acoustic or optical, but the principle of the golden section finds its most complete realization in human form.
Examples:
A slice 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 though there is much debate as to whether he deliberately did this. In more modern times, the Hungarian composer Bela Bartok 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 the vertical to the horizontal is the golden ratio. To find the golden rectangle, look in your wallet for credit cards. Despite this abundant evidence in artwork created over the centuries, there is currently debate among psychologists about whether people really perceive the golden proportions, in particular the golden rectangle, as more beautiful than other shapes. A 1995 journal article, Professor Christopher Green, of the University of York at Toronto, discusses a number of experiments over the years that have not shown any preference for the golden rectangle shape, but notes that several others have provided evidence that such a preference does not exist. ... But regardless of science, the Golden Ratio retains its mystery, in part because it is used so well in many unexpected places in nature. Spiral 
the shell of the nautilus clam is 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 from the most common forms of human DNA fit perfectly into the golden decagon. Golden ratio and his relatives also appear in many unexpected contexts, in mathematics, and they continue to generate interest in the mathematical communities. Dr. Stephen 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 a human face that can only be. 
Mask perfect human face
Egyptian Queen Nefertiti (1400 BC)
the face of Jesus is a copy from the Turin Shroud and corrected to match the mask of Dr. Stephen Marquardt.
The “average” (synthesized) face from among the celebrities. With proportions of the golden ratio.
The materials of the site were used: http://blog.world-mysteries.com/
