Introduction
The principle of the Golden Section was studied as early as the III century BC, and it is still one of the most famous pictorial canons. This principle is applied in many spheres of life: in nature, in painting, architecture, science, technology and even in the proportions of the human body. Objects surrounding a person vary in shape. The form, which is based on a combination of symmetry and the Golden Section, promotes visual perception and the appearance of a sense of beauty and harmony. The principle of the Golden Section is the highest manifestation of the structural and functional perfection of the whole and its parts. Scientists believe that the closer an object is to the golden ratio, the better the human brain perceives it.
1. The history of the origin of the Golden Section.
In the ancient literature that had come down to us, the Golden Section wa first found in Chapter II of Euclid's "Beginnings", in which a geometric construction of the Golden Section was given, equivalent to solving a quadratic equation of the form x(a+x)=a^2. Euclid used the Golden Ratio in the construction of regular 5- and 10-gons (IV and XIV books), as well as in stereometry in the construction of regular 12- and 20-gons. However, the Golden Section had been known before Euclid. It is very likely that the Golden Section problem was solved by the Pythagoreans, who are credited with constructing a regular 5-gon and geometric constructions equivalent to solving quadratic equations. After Euclid, Hypsicles (2nd century BC), Papp of Alexandria (3rd century BC) and others were engaged in the study of the Golden Section.
In medieval Europe, the Golden Ratio was introduced through Arabic translations of Euclid's "Beginnings". Translator and commentator of Euclid J. Campavo of Novara (13th century) added to the XIII book of the "Beginnings" a sentence containing an arithmetic proof of the incommensurability of the segment and both parts of its Golden Section.
In the 15th and 16th centuries, interest in the Golden Section increased among scientists and artists in connection with its applications both in geometry and in art, especially in architecture. L. Pacioli dedicated the treatise "On the Divine Proportion" (1509) to the Golden Section. By the "divine proportion" Pacioli means a continuous geometric proportion of three magnitudes, which Euclid calls "division in the middle and extreme ratio", and in the XIX century it was called the "golden section". In determining this proportion and describing its properties, Pacioli follows Euclid. This proportion occurs when the whole is divided into two parts, when the whole relates to the larger part as the larger part relates to the smaller one. In the language of equality of areas, the same proportion is given as follows: a square is mostly equal to a rectangle, the sides of which are the whole and the smaller part. Luca Pacioli outlines the various properties of the "divine proportion" known from the XIII and XIV book of the Principles of Euclid. In total, he considers thirteen such properties, associating this number with the number of participants in the last supper.
I. Kepler wrote about the Golden Section in one of his early works (1596).
The term "Golden Section" was introduced by Leonardo da Vinci (late 15th century), which has survived to our times. He made sections of a stereometric body formed by regular pentagons, and each time he received rectangles with the ratios of the sides in the golden division.
The Golden Section or proportional relations close to it formed the basis of the compositional construction of many works of world art.
2. The concept of the Golden Section.
The Golden Section (harmonic division, division in extreme and average ratio) is the division of a segment into two parts in such a way that most of it is the average proportional between the entire segment and its smaller part.
In the Middle Ages, Fibonacci discovered a numerical sequence, the numbers in which are called "Fibonacci numbers". It resembles the principle of the Golden Section. The sequence looks like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233... and on to infinity.
The Fibonacci sequence is a sequence of numbers where each subsequent number is equal to the sum of the previous two.
This sequence asymptotically tends to some constant relation, but it is impossible to express it exactly. If any member of this sequence is divided by the one preceding it, the result will be a value fluctuating around the irrational value of 1.61803398875... and every once in a while it surpasses, then does not reach it.
Golden Proportion:
a: b = b: c or с: b = b: а
This proportion is equal to:
In ancient Greek mathematics, the golden section was originally called the division of the segment AB by the point C into 2 parts, so that the larger part refers to the smaller one, as the entire segment refers to the larger one:
From equality, representing a as an independent variable, one can obtain a quadratic equation that describes the properties of the golden section:
Solving this equation, we get the roots:
3. The Golden Section in geometry.
If you approach Golden Section the geometrically, in order to find its supposed divine property, you need to build a rectangle, one side of which is 1,618 times longer than the other; you get a rectangle in which the aspect ratio is the golden ratio (more precisely, its approximate value).
4. The Golden Section in the architecture of Ancient World.
The Pyramid of Cheops
It was according to the rules of the Golden Section that the pyramid of Cheops was built. Looking at it, you can see a triangle with a right angle, one cathet of which is the height, and the second is half the length of the base. If we take the ratio of the hypotenuse to the smaller side, we get an ideal value of 1.61950 or 1.62
Temple of the Goddess Athena
The Greek Parthenon, erected in honor of the victory over the Persians, is the temple of the goddess Athena. The ratio of the length of this temple to the width gives the number of the Golden Section (with an error). If you subtract 14 cm from the length of the structure and add it to the width, you will get a complete match with the mathematical value. It should be assumed that the architects Iktin and Kallikrat deliberately laid the rule of the golden section in the project.
5. The Golden Section in the architecture of St. Petersburg.
St. Isaac 's Cathedral
In the drawing of St. Isaac's Cathedral, three rows of the Golden Section are visible.
First row: the width of the building is taken as 400 units and represents the following figures: 400, 247, 153, 94, 58…
If we divide the number 400 by ≈ 1,618, we get approximately 247
Repeat with the following number: 247: 1.618≈153.
Thus we find all the numbers. Based on the drawing, the main part of the cathedral with columns fits into a rectangle with sides 400 and 247. Since the sides are in a ratio of approximately 1.618, they form a Golden rectangle.
The next row is represented by the height of the building: 370, 228, 140, 87, 53, 33, 20, 12. Vertically, St. Isaac's Cathedral is divided by a Golden Section at the base of the dome, which makes the ratio of the main part and the dome harmonious.
The third row of dimensions starts from 113, and is the width of the base of the main dome: 113, 69, 42, 26, 16. The numbers of this series are found in the sizes of windows, in the heights of columns and other details of the cathedral.
As can be seen from the picture, the Golden rectangular and isosceles triangles take place in the building of St. Isaac's Cathedral.
Cabinet of curiosities
On the University embankment of Vasilievsky Island stands the building of the Kunstkammer, founded in 1718 under the direction of the German architect Georg Mattarnovi: Peter's Baroque, two 3-storey buildings and a complex multi-tiered dome tower.
The study begins with the main values: the height and length of the building, from which the golden row is built. Length — 450 units, further 277, 170, 105, 65, 40, 24. Such dimensions can be seen in the height and latitude of different levels of the tower, the length of the buildings. The tower part itself is inscribed in a golden isosceles triangle from the base to the top. The Golden Section is seen to a greater extent in this main element, which is correct from the point of view of architecture. Conclusion: the basis of the Kunstkamera obeys the golden rule and preserves compositional harmony.
New Golden row starts building height: 211, 130, 80, 49, 30. Looking at the dimensions of the drawing, it becomes clear that the choice of a three-storey type of buildings is due to the proportionality with the tower.
Smolny Cathedral
The subordination of the space and volumes of the Smolny Cathedral is based on the rule of the "Golden Section" and the shape of the Greek cross. The five-domed temple is perceived as the center of the four-domed monastery square inscribed in the star of the ten-towered fence.
6. The Golden Ratio in interior design.
1. The ratio of color in the room according to the Golden section rule.
The use of color according to the golden ratio rule also assumes a two-thirds ratio. The dominant color should occupy about 60% of the room, the main accompanying color should occupy about 30%, and the last additional color should occupy only 10% (it is usually used for accompanying decor).
Of course, these ratios are approximate, and additional colors may include several shades, but the basic principle should be preserved. You can choose a soft tonal transition within the same color or a bright contrasting chord — it all depends on the concept of the design of the room.
2. The correspondence of the height of the furniture to the line of the basement part of the wall.
If we divide the wall into the lower part and the upper frieze area, then in the classic version, the dividing border running along the perimeter of the room is usually located at a height of 75-100 cm from the floor (furniture factories are oriented to these dimensions). In our "golden" row there is a choice between the numbers 114,6 and 70.8. You can choose the number 70.8 by making it the bottom line of the curb.
The curb itself should not be massive, the values of 10.3 or 16.7 cm are quite suitable for its width. Thus, to the already existing lower curb line at a height of 70.8 cm from the floor, we will get the height of the upper curb line – either 81.1 or 87.5. These lines mark the boundary of the basement area of the wall. Ideally, the height of the chairs and chairs of the furniture set, as well as low furniture elements, such as dressers and bedside tables, should strive for it.
When choosing the width of the baseboard and the width of the cornice, we take numbers from the same "golden" row: 16.7, 10.3, 6.3 cm. The choice will depend on the width of the border and on other decorative elements.
3. The ratio of furniture and space.
Furniture should not occupy more than 60% of the composition, so as not to create a feeling of crowding and clutter. If we choose custom-made furniture, then it is convenient to use the proportions of the Golden Section calculated specifically for our apartment when making it. In this case, the furniture will look especially harmonious, and be perceived as a natural part of a single whole.
4. Rule 2/3.
The same golden rule of two-thirds (in each case, the numbers from our Fibonacci series will help to make a more accurate proportion) will allow you to correctly arrange all the elements of the interior decor. At a height of approximately 2/3 of the total height of the space, hanging lamps will look most harmoniously, the sofa should not occupy more than 2/3 of the space allotted to it, and the coffee table should not be more than 2/3 of the sofa.