The golden section is ... The golden section of the pyramid. Golden Section Formula

Geometry is an exact and rather complex science, which, for all this, is a kind of art. Lines, planes, proportions - all this helps to create many really beautiful things. And strangely enough, at the heart of this lies precisely the geometry in its various forms. In this article we will consider one very unusual thing, which is directly related to this. The golden section is precisely the geometric approach that will be discussed.

The form of the object and its perception

People often focus on the form of the 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. We first of all recognize people in the form of body and face. Therefore, we can confidently say that the form itself, its size and appearance is one of the most important things in human perception.

For people, the form of anything is of interest for two main reasons: either it is dictated by a vital necessity, or it is caused by aesthetic pleasure from beauty. The best visual perception and the sense of harmony and beauty most often comes when a person observes a form in the construction of which symmetry and a special relationship, which is called the golden section, were used.

The concept of the golden section

So, the golden section is a golden proportion, which is also a harmonic division. In order to explain this more clearly, let us 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 in size. Well, such dimensions are always in a certain ratio both among themselves and with respect to the whole.

So, in other words, we can say that the golden ratio is a ratio of two quantities, which has its own formula. The use of such a ratio when creating a form helps 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 in the most diverse spheres of life right now. But the history of this concept goes back to ancient times, when only such sciences as mathematics and philosophy were born. As a scientific concept, the golden section became common during the time of Pythagoras, namely, in the 6th century BC. But even before that knowledge about a similar relationship in practice was used in ancient Egypt and Babylon. A clear evidence of this are the pyramids, for the construction of which they used exactly such a golden proportion.

New period

The Renaissance became a new breath for harmonic division, especially thanks to Leonardo da Vinci. This ratio has been increasingly used in both 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 works related to the golden proportion is the book of Luke Pancholi entitled "The Divine Proportion". Historians suspect that the illustrations of this book were executed by Leonardo himself before Vinci.

The mathematical expression of the golden ratio

Mathematics gives a very clear definition of the proportion, which indicates that it is the equality of the two ratios. Mathematically, this can be expressed by the following equality: a: b = c: d, where a, b, c, d are certain definite values.

If we consider the proportion of a segment divided into two parts, then we can meet only a few situations:

• The segment is divided into two absolutely equal parts, and therefore AB: AC = AB: BC, if AB is the exact start 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 among themselves, and hence, here they are absolutely disproportionate.
• The segment is divided so that AB: AC = AC: BC.

As for the golden section, it is such a proportional division of a segment into unequal parts, when the whole segment refers to the most part, as the most part refers to the smaller. There is another formulation: the smaller segment refers to the larger, as well as the larger to the whole segment. In the mathematical relation, it looks like this: a: b = b: c or c: b = b: a. It is this kind of formula has a golden section.

The golden proportion in nature

The golden section, examples of which we will now consider, refers to incredible phenomena in nature. These are very beautiful examples of that mathematics are 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 life tasks is growth. This desire to take its place in space, in fact, is carried out in several forms - upward growth, almost horizontal spreading on the ground or twisting in a spiral on a certain support. And no matter how incredible, many plants grow in accordance with the golden proportion.

Another almost incredible fact is the relationship in the body of lizards. Their body looks nice enough for 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 meet many 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 Tutankhamun and Cheops were built using such a ratio. And the golden section of the pyramid is still a mystery, because to this day it is not known whether such dimensions were accidentally or specially chosen for their bases and heights.
• The rule of the golden section 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 the Notre Dame de Paris Cathedral, here, not only facades, but also other parts of the construction were erected, relying on this incredible proportion.
• In Russian architecture, you can find incredibly many examples of buildings that fully correspond to the golden section.
• 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 an umbilical line in a golden section.
• Painting - a separate story, especially if you consider the role of Leonardo da Vinci in the history of the golden ratio. His famous Gioconda, of course, is subject to this law.

The golden section in the human body

In this section, it is necessary to mention a very significant person, namely S. Zeising. He is a German researcher who has done a tremendous job in the study of the golden proportion. He published a paper entitled "Aesthetic Studies." In his work he presented the golden section as an absolute concept, which 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 Zeasing 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, the main indicator of the golden ratio is the division of the body by the navel point. Thus, the 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 hand.

About the construction of the golden section

In fact, the construction of the golden section is a simple matter. As we see, even ancient people coped with this quite easily. What can we say about the modern knowledge and technologies of humanity. 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 with certainty we will say that this, in fact, is possible. Moreover, this can be done in more than one way.

Since this is a fairly simple geometry, the golden section is quite simple to build, even in school. Therefore, information about this can easily be found in specialized books. Studying the golden section of the 6th grade is fully capable of understanding the principles of its construction, and therefore even children are clever enough to master this task.

The golden proportion in mathematics

The first acquaintance with the golden section in practice begins with a simple division of a line segment all in the same proportions. Most often this is realized using a ruler, a compass and, of course, a pencil.

The segments of the golden proportion are expressed as an infinite irrational fraction AE = 0.618 ... if AB is taken as unity, 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 most of it will be 62, and the smaller will be 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 section - geometry, but just such properties as the laws of the golden section, suggest mystery on this topic.

Harmony in art through the golden section

In order to sum up, we will briefly consider what has already been said.

Basically, under the rule of the golden ratio there are many art samples, where the ratio is close to 3/8 and 5/8. This is the rough formula of the golden section. The article already mentioned a lot about the use of cross-sections, 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, decorating the most tombs.
• The 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 proportions, then, since the time of Leonardo da Vinci, it has come into use in almost all branches of life - from science to art. Even biology and medicine have proved that the golden ratio works even in living systems and organisms.