Geometry is an exact and rather complex science, which, with all this, is a kind of art. Lines, planes, proportions - all this helps to create a lot of really beautiful things. And oddly enough, this is based on geometry in its most diverse forms. In this article, we will look at one very unusual thing that is directly related to this. The golden ratio is exactly the geometric approach that will be discussed.
The shape of the object and its perception
People most often focus on the shape of an object in order to recognize it among millions of others. It is by form that we determine what kind of thing lies in front of us or stands far away. We first of all recognize people by the shape of the body and face. Therefore, we can say with confidence that the form itself, its size and appearance is one of the most important things in human perception.
For people the shape of somethingHowever, it is of interest for two main reasons: either it is dictated by vital necessity, or it is caused by aesthetic pleasure from beauty. The best visual perception and a sense of harmony and beauty most often comes when a person observes a form in the construction of which symmetry and a special ratio were used, which is called the golden ratio.
The concept of the golden ratio
So, the golden ratio is the golden ratio, which is also a harmonic division. In order to explain this more clearly, consider some features of the form. Namely: the form is something whole, but the whole, in turn, always consists of some parts. These parts most likely have different characteristics, at least different sizes. Well, such dimensions are always in a certain ratio both among themselves and in relation to the whole.
So, in other words, we can say that the golden ratio is the ratio of two quantities, which has its own formula. Using this ratio when creating a form helps to make it as beautiful and harmonious as possible for the human eye.
From the ancient history of the golden section
The golden ratio is often used in various areas of life right now. But the history of this concept goes back to ancient times, when such sciences as mathematics and philosophy were just emerging. As a scientific concept, the golden ratio came into use during the time of Pythagoras, namely in the 6th century BC. But even before that, knowledge of such a ratio was used in practice in ancient Egypt and Babylon. A striking evidence of this are the pyramids, for the construction of which they used exactly this golden ratio.
New Period
The Renaissance became a new breath for harmonic division, especially thanks to Leonardo da Vinci. This ratio has been increasingly used 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 that deal with this issue.
One of the most important historical works related to the golden ratio is a book by Luca Pancioli called "Divine Proportion". Historians suspect that the illustrations of this book were made by Leonardo pre-Vinci himself.
Mathematical expression of the golden ratio
Mathematics gives a very clear definition of proportion, which says that it is the equality of two ratios. Mathematically, this can be expressed as follows: a:b=c:d, where a, b, c, d are some specific values.
If we consider the proportion of a segment divided into two parts, we can meet only a few situations:
- The segment is divided into two absolutely even parts, which means that AB:AC=AB:BC, if AB is the exact beginning and end of the segment, and C is the point that divides the segment into two equal parts.
- The segment is divided into two unequal parts, which can be in very different proportions to each other, which means thathere they are completely disproportionate.
- The segment is divided so that AB:AC=AC:BC.
As for the golden section, this is such a proportional division of the segment into unequal parts, when the entire segment refers to the larger part, just as the larger part itself refers to the smaller one. There is another formulation: the smaller segment is related to the larger one, as well as the larger one to the entire segment. In mathematical terms, it looks like this: a:b=b:c or c:b=b:a. This is the form of the golden section formula.
Golden proportion in nature
The golden ratio, examples of which we will now consider, 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 has more than a real reflection in nature and our life in general.
For living organisms, one of the main tasks of 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 as incredible as it is, many plants grow according to the golden ratio.
Another almost unbelievable fact is the proportions in the body of lizards. Their body looks pleasing enough to the human eye, and this is possible thanks to the same golden ratio. To be more precise, the length of their tail is related to the length of the whole body as 62: 38.
Interesting facts about the rules of goldsections
The golden ratio is a truly incredible concept, which means that throughout history we can find a lot of really interesting facts about this proportion. Here are some of them:
- The rule of the golden section was actively used in the construction of the pyramids. For example, the world-famous tombs of Tutankhamen and Cheops were built using this ratio. And the golden section of the pyramid is still a mystery, because to this day it is not known whether such dimensions were chosen by chance or on purpose 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 building of Notre Dame Cathedral (Notre Dame de Paris), here not only the facades, but also other parts of the structure were erected based on this incredible proportion.
- In Russian architecture, you can find incredibly many examples of buildings that fully correspond to the golden ratio.
- Harmonious division is also inherent in the human body, and therefore in sculpture, in particular, 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 another story, especially considering the role of Leonard da Vinci in the history of the golden ratio. His famous Mona Lisa is, of course, subject to this law.
Golden ratio in the human body
In this section, a very significant person needs to be mentioned, namely -S. Zeising. This is a German researcher who has done a great job in the field of studying the golden ratio. He published a work en titled Aesthetic Research. In his work, he presented the golden ratio as an absolute concept, which is universal for all phenomena, both in nature and in art. Here we can recall the golden 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 experience. According to Zeising's research, the main indicator of the golden ratio is the division of the body by the navel point. Thus, a male body with an average ratio of 13:8 is slightly closer to the golden ratio than a female body, where the golden ratio is 8:5. Also, the golden ratio can be observed in other parts of the body, such as, for example, the hand.
On the construction of the golden ratio
In fact, the construction of the golden ratio is a simple matter. As we can see, even ancient people coped with this quite easily. What can we say about modern knowledge and technologies of mankind. In this article, we will not show how this can be done simply on a piece of paper and with a pencil in hand, but we will state with confidence that this is, in fact, possible. Moreover, there is more than one way to do this.
Because it's fairly simple geometry, the golden ratio is pretty easy to build even in school. Therefore, information about this can be easily found in specialized books. By studying the golden ratio, grade 6 is fully able to understand the principles of its construction, which means that even children are smart enough to master such a task.
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, a compass and, of course, a pencil.
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, 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 its larger part will be equal to 62, and the smaller one will be 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 an exact and rigorous science, as well as its section - geometry, but it is precisely such properties as the laws of the golden section that bring mystery to this topic.
Harmony in art through the golden ratio
To sum up, let's take a brief look at what has already been said.
Basically under the golden ratio rulemany examples of art fall under, where the ratio is close to 3/8 and 5/8. This is the rough formula for the golden ratio. The article has already mentioned a lot about examples of the use of the 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 ratio of the pyramids of Cheops and Tutankhamun is expressed literally in everything: temples, bas-reliefs, household items and, of course, decorations of the tombs themselves.
- Temple of Pharaoh Seti I in Abydos is famous for reliefs with different images, and all this corresponds to the same law.
As for the already conscious use of proportion, since the time of Leonardo da Vinci, it has come into use in almost all areas of life - from science to art. Even biology and medicine have proven that the golden ratio works even in living systems and organisms.
