Geometry is an exact and rather complex science, which, with all this, is a kind of art. Lines, planes, proportions - all this helps to create a lot of really beautiful things. And strangely enough, it is geometry that is at the heart of it in its most diverse forms. In this article, we will consider one very unusual thing that is directly related to this. The Golden Ratio is precisely the geometric approach that will be discussed.
The shape of the subject and its perception
People most often rely on the shape of an object in order to recognize it among millions of others. It is in form that we determine what kind of thing lies before us or stands in the distance. First of all, we recognize people by the shape of their bodies and faces. Therefore, we can confidently say that the form itself, its size and appearance is one of the most important things in the perception of man.
For people, the form of anything is of interest for two main reasons: either it is dictated by vital necessity, or it is caused by aesthetic pleasure from beauty. The best visual perception and a sense of harmony and beauty most often comes when a person observes a form in the construction of which symmetry and a special relation, which is called the golden ratio, were used.
Golden Ratio
So, the golden ratio is the golden proportion, which is also a harmonic division. In order to explain this more clearly, consider some features of the form. Namely: the form is something whole, but the whole, in turn, always consists of some parts. These parts most likely have different characteristics, at least different sizes. Well, such sizes are always in a certain ratio both among themselves and in relation to the whole.
So, in other words, we can argue that the golden ratio is a ratio of two quantities, which has its own formula. Using this ratio when creating a form helps to make it as beautiful and harmonious as possible for the human eye.
From the ancient history of the golden section
The Golden Ratio ratio is often used in a wide variety of areas of life today. But the history of this concept goes back to ancient times, when such sciences as mathematics and philosophy were just emerging. As a scientific concept, the golden ratio came into use during the time of Pythagoras, namely in the VI century BC. But even before that knowledge of such a ratio was used in practice in ancient Egypt and Babylon. Vivid evidence of this are the pyramids, for the construction of which they used just such a golden proportion.
New period
The Renaissance has become a new breath for harmonious division, especially thanks to Leonardo da Vinci. This ratio began to be used more and more both in the exact sciences, such as geometry, and in art. Scientists and artists began to study the Golden Ratio more deeply and create books addressing this issue.
One of the most important historical works related to the golden proportion is the book of Luke Pancholi entitled “The Divine Proportion”. Historians suspect that the illustrations in this book were made by Leonardo Vinci himself.
The mathematical expression of the golden ratio
Mathematics gives a very clear definition of proportion, which suggests that it is the equality of two ratios. Mathematically, this can be expressed by the following equation: a: b = c: d, where a, b, c, d are some certain values.
If we consider the proportion of the segment, divided into two parts, then we can meet only a few situations:
- A segment is divided into two absolutely even parts, which means AB: AC = AB: BC, if AB is exact at 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 to each other, which means that here they are absolutely disproportionate.
- The segment is divided so that AB: AC = AC: BC.
As for the golden ratio, this is such a proportional division of the segment into parts that are not equal to each other, when the entire segment refers to the majority, as the majority itself refers to the smaller. There is another formulation: a smaller segment refers to a larger one, as well as a larger one to the entire segment. In mathematical terms, this is as follows: a: b = b: c or c: b = b: a. This is exactly what the golden ratio formula has.
Golden proportion in nature
The Golden Ratio, examples of which we will now consider, refers to incredible phenomena in nature. These are very beautiful examples of the fact that mathematics is not just numbers and formulas, but a science that has more than a real reflection in nature and our life in general.
For living organisms, one of the main tasks in life is growth. Such a desire to take its place in space, in fact, is carried out in several forms - upward growth, almost horizontal spreading on the ground or spiraling on a certain support. And no matter how incredible it is, many plants grow in accordance with the golden ratio.
Another almost unbelievable fact is the ratio in the body of the lizards. Their body looks nice enough for the human eye, and this is possible due 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 Ratio
The golden ratio is a truly incredible concept, which means that throughout history we can meet many really interesting facts about this proportion. We present you some of them:
- The rule of the golden ratio was actively used in the construction of the pyramids. For example, the world famous tombs of Tutankhamun and Cheops were erected using this ratio. And the golden section of the pyramid is still a mystery, because to this day it is not known whether such sizes were randomly or specially chosen for their bases and heights.
- The rule of the golden ratio is clearly visible in the facade of the Parthenon - one of the most beautiful buildings in the architecture of ancient Greece.
- The same applies to the Notre Dame de Paris Cathedral building , 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 are fully consistent with the golden ratio.
- Harmonious division is also inherent in the human body, and therefore in sculpture, in particular, in the statues of people. For example, Apollo Belvedere is a statue where a person’s height is divided by the umbilical line in the golden ratio.
- 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.
Golden ratio in the human body
In this section, it is necessary to mention a very significant person, namely - S. Zeising. This is a German researcher who has done tremendous work in the study of the golden ratio. He published a work entitled Aesthetic Research. In his work, he presented the golden ratio as an absolute concept, which is universal for all phenomena both in nature and in art. Here you can recall the golden ratio of the pyramid along with the harmonious proportion of the human body and so on.
It was Zeising who was able to prove that the golden ratio, in fact, is the average statistical law for the human body. This was shown in practice, because during his work he had to measure a lot of human bodies. Historians believe that more than two thousand people took part in this experiment. According to Zeising research, the main indicator of the golden ratio is the division of the body by the navel point. So, the male body with an average ratio of 13: 8 is slightly closer to the golden ratio than the female, where the golden ratio is 8: 5. Also, the golden ratio can be observed in other parts of the body, such as, for example, the hand.
On the construction of the golden section
In fact, building a golden section is a simple matter. As we see, even 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 will confidently state that this is actually possible. Moreover, this can be done in more than one way.
Since this is a fairly simple geometry, the golden ratio is quite simple to build even at school. Therefore, information on this can easily be found in specialized books. Studying the golden ratio, Grade 6 is fully able to understand the principles of its construction, which means that even children are smart enough to master a similar task.
Golden ratio in mathematics
The first acquaintance with the golden ratio in practice begins with a simple division of a straight line segment in the same proportions. Most often this is done with a ruler, compass, and, of course, a pencil.
The segments of the golden ratio are expressed as an infinite irrational fraction AE = 0.618 ... if AB is taken as a unit, BE = 0.382 ... In order to make these calculations more practical, approximate values are often used, not exact, namely 0 , 62 and 0.38. If the segment AB is taken for 100 parts, then most of it will be 62, well, and the smaller - 38 parts, respectively.
The main property of the golden ratio can be expressed by the equation: x 2 -x-1 = 0. When solving, we get the following roots: x 1,2 =. Although mathematics is an exact and rigorous science, like its branch is geometry, it is precisely such properties as the laws of the golden section that lead to mystery on this subject.
Harmony in art through the golden ratio
In order to summarize, we will briefly consider what we have already talked about.
Basically, many art samples fall under the rule of the golden ratio, where a ratio close to 3/8 and 5/8 is observed. This is the crude golden ratio formula. The article has already mentioned a lot about examples of using a section, but we will look at it again through the prism of ancient and modern art. So, the most striking examples from ancient times:
- The golden section of the pyramid of Cheops and Tutankhamun is literally expressed in everything: temples, bas-reliefs, household items and, of course, the 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 the probably conscious use of proportion, then, since the time of Leonardo da Vinci, it came into use in almost all branches of life - from science to art. Even biology and medicine have proven that the golden ratio works even in living systems and organisms.