Geometry is an exact and rather complicated science,which with all this is a kind of art. Lines, planes, proportions - all this helps to create a lot of really beautiful things. And oddly enough, the basis of this is precisely geometry in its most varied forms. In this article we will look at one very unusual thing that is directly related to this. The golden section is exactly the geometric approach that will be discussed.
The shape of the subject and its perception
People most often focus on the shape of the subject.in order to recognize it among millions of others. It is in form that we determine what a thing lies in front of us or is far away. We primarily recognize people by the shape of their bodies and faces. Therefore, we can confidently assert that the form itself, its dimensions and appearance are one of the most important things in human perception.
For people, the form of anything representsinterest for two main reasons: either it is dictated by vital necessity, or it is caused by aesthetic pleasure from beauty. The best visual perception and sensation of harmony and beauty most often comes when a person observes a form, in the construction of which we used symmetry and a special relationship, which is called the golden section.
The concept of the golden section
So the golden ratio is the golden proportionwhich is also a harmonic division. In order to explain it more clearly, consider some features of the form. Namely: the form is something whole, well, the whole, in turn, always consists of some parts. These parts are likely to have different characteristics, at least in different sizes. Well, such dimensions are always in a certain ratio, both among themselves and in relation to the whole.
So in other words, we can argue thatThe golden ratio is the ratio of two quantities, which has its own formula. Using this ratio when creating a form helps to make it as beautiful and harmonious as possible for the human eye.
From the ancient history of the golden section
The ratio of the golden section is often used inmost different areas of life today. But the history of this concept goes back to ancient times, when such sciences as mathematics and philosophy were born. As a scientific concept, the golden section came into use in the time of Pythagoras, namely in the VI century BC. But even before that, knowledge about a similar ratio was used in practice in ancient Egypt and Babylon. The bright evidence of this is the pyramids, for the construction of which they used just such a golden proportion.
New period
The Renaissance is a new breath forharmonic division, especially thanks to Leonardo da Vinci. This ratio has increasingly begun to be used both in the exact sciences, such as geometry, and in art. Scientists and artists began to study the golden section more deeply and create books that deal with this issue.
One of the most important historical worksassociated with the golden proportion, is a book by Luke Pancholi entitled Divine Proportion. Historians suspect that the illustrations of this book were made by Leonardo do Vinci himself.
Mathematical expression of the golden proportion
Math gives a very clear definition.proportions, which means that it is an equality of two relations. Mathematically, this can be expressed by the following equality: a: b = c: d, where a, b, c, d are some specific values.
If we consider the proportion of the segment divided into two parts, we can meet only a few situations:
- The segment is divided into two absolutely equal parts, which means AB: AC = AB: AF, if AB is the beginning and end of the segment, and C is the point that divides the segment into two equal parts.
- The segment is divided into two unequal parts, which can be in a very different ratio between each other, which means that here they are absolutely disproportionate.
- The segment is divided so that AB: AC = AC: BC.
As for the golden section, it isproportional division of the segment into unequal parts, when the whole segment belongs to the larger part, as the larger part belongs to the smaller part. There is another formulation: a smaller segment refers to a larger one, like a larger one to the whole segment. In the mathematical relationship, it looks like this: a: b = b: s or c: b = b: a. This is the form of the golden section formula.
Golden proportion in nature
The golden section, examples of which we are nowconsider, refers to the incredible phenomena in nature. These are very beautiful examples of the fact that mathematics is not just numbers and formulas, but a science that is more than a real reflection in nature and our life in general.
For living organisms, one of the main lifetasks is growth. Such a desire to take its place in space, in fact, is carried out in several forms - growth upward, almost horizontal spreading on the ground or twisting in a spiral on a certain support. And no matter how incredible it is, many plants grow in accordance with the golden proportion.
Another almost unbelievable fact is the ratios.in the body of lizards. Their body looks quite pleasant to the human eye, and this is possible thanks to the same golden ratio. To be more precise, the length of their tail refers to the length of the whole body as 62: 38.
Interesting facts about the rules of the golden section
The golden section is a truly incredible concept, which means that throughout history we can come across many really interesting facts about this proportion. We present you some of them:
- The golden section rule was actively applied inbuilding the pyramids. For example, the world famous tombs of Tutankhamen and Cheops were erected using this ratio. And the golden section of the pyramid still remains a mystery, because to this day it is not known, by chance or specifically such dimensions were chosen for their bases and heights.
- The golden section rule is clearly visible in the facade of the Parthenon - one of the most beautiful structures in the architecture of ancient Greece.
- The same applies to the building of Notre Dame de Paris, Notre Dame de Paris, here not only the facades, but also other parts of the structure were erected, relying on this incredible proportion.
- In Russian architecture you can find incredibly many examples of buildings that fully comply with the golden section.
- Harmonious division is also inherent in humanbody, and hence the sculpture, in particular, statues of people. For example, Apollo Belvedere - a statue where a person's height is divided by the umbilical line in the golden section.
- Painting is a separate story, especially considering the role of Leonard da Vinci in the history of the golden ratio. His famous Gioconda, of course, is subject to this law.
The golden ratio in the human body
В этом разделе нужно упомянуть очень значимую person, namely - S. Zeising. This is a German researcher who has done a tremendous amount of work in the field of studying the golden proportion. He published a work entitled "Aesthetic Research." In his work, he presented the golden section as an absolute concept that is universal for all phenomena, both in nature and in art. Here you can recall the golden section of the pyramid, along with the harmonious proportion of the human body and so on.
It was Zeizing who was able to prove that the goldensection, in fact, is the average statistical law for the human body. This was shown in practice, because during his work he had to measure a lot of human bodies. Historians believe that more than two thousand people took part in this experience. According to the study of Zeising, the main indicator of the golden ratio is the division of the body by the navel point. Thus, a male body with an average ratio of 13: 8 is slightly closer to the golden section than the female, where the number of the golden section is 8: 5. Also, the golden proportion can be observed in other parts of the body, such as, for example, the arm.
On the construction of the golden section
In fact, building the golden section is a matter ofsimple As we can see, even the ancient people coped with this quite easily. What to say about modern knowledge and technology of mankind. In this article we will not show how this can be done simply on a piece of paper and with a pencil in hand, but we can confidently state that this is actually possible. Moreover, this can be done in far more than one way.
Since this is a fairly simple geometry,The golden ratio is quite simple to build, even in school. Therefore, information about this can be easily found in specialized books. Studying the golden section of grade 6 is fully capable of understanding the principles of its construction, which means that even children are smart enough to master such a task.
The golden proportion in mathematics
The first acquaintance with the golden section in practice begins with a simple division of a straight line segment all in the same proportions. Most often this is done with a ruler, compass and, of course, a pencil.
The golden proportions are expressed asan infinite irrational fraction AE = 0.618 ... if AB is taken as one, BE = 0.382 ... In order to make these calculations more practical, very often not exact, but approximate values are used, namely, 0.62 and 0, 38 If the segment AB is taken as 100 parts, then the larger part will be equal to 62, and the smaller part - to 38 parts, respectively.
The main property of the golden ratio can be expressed by the equation: x2-x-1 = 0. When solving we get the following roots: x1,2=. Although mathematics is exact and rigorous science, like its section is geometry, it is precisely such properties as the laws of the golden section that induce mystery on this topic.
Harmony in art through the golden section
In order to summarize, consider briefly what has already been said.
Mainly under the rule of the golden ratioThere are many examples of art where the ratio is close to 3/8 and 5/8. This is the rough formula of the golden section. The article has already mentioned a lot of examples of using a section, but we will look at it again through the prism of ancient and modern art. So, the most vivid examples from ancient times:
- The golden section of the pyramid of Cheops and Tutankhamen is expressed literally in everything: temples, bas-reliefs, household items and, of course, decorations of the tombs themselves.
- The temple of Pharaoh Seti I in Abydos is famous for its reliefs with different images, and all this corresponds to the same law.
As for already surely consciousthe use of proportion, then, since the time of Leonardo da Vinci, she entered into use in almost all sectors of life - from science to art. Even biology and medicine have proven that the golden ratio works even in living systems and organisms.
