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Observations of nature and attempts to reveal the secrets of her beautiful creatures have brought a lot of discoveries. One of them is the golden ratio. This is a certain regularity, which obeys everything that we call beautiful. People, animals, flowers, buildings, galaxies…
What is the golden ratio and how to understand it
Often we come across houses, objects, buildings, plants that fascinate us with something. People have long tried to understand why one thing seems beautiful to us, the other is not, they were looking for patterns. And they seem to have found it. This is some ratio of parts, which is called the golden ratio.
No one knows exactly who and when invented the golden ratio. Someone attributes the discovery to Pythagoras, but the first mention was found in the “Beginnings” of Euclid, and he lived in the 3rd century BC. So the find is obviously old. It is on this principle that ancient Greek and Roman temples were built. Of course, these may be coincidences, but they are very strange and there are a lot of them. So, most likely, they were aware of the ideal proportions.
It is absolutely certain that Leonardo da Vinci was looking for confirmation of this principle in the structure of the human body. And, what is most interesting, I found it. Those faces and bodies that seem beautiful to us have proportions that just obey the law of the golden section.
The formal definition sounds both simple and complex. It is associated with two segments of different sizes. This principle sounds something like this: if a segment is divided into two unequal parts, then this division will be proportional, if the larger part of the segment is related to the whole in the same way as the smaller part to the larger one. It will be clearer if you look at the illustration and the formula.
In the figure, the whole segment is divided so that if а split into b, we get 1,1618, the same figure is obtained if the whole segment is divided into a large part – a. This number is the embodiment of an ideal proportion. Now, if you look at the picture with the Parthenon, the proportions of this structure also obey the specified ratio.
The same pattern can be represented as a percentage. Maybe it’s easier for someone. In order for the division of the whole to be proportional, the parts must be 62% and 38%. It might be easier to remember that way.
This pattern was developed further by the mathematician Fibonacci. He developed a numerical sequence, the elements of which, starting from the ninth, obey the same law. The graphic representation of this sequence is a spiral. If you look closely, and in nature, and in architecture, and in the human body, the proportions of beauty are present.
How to build a rectangle with perfect proportions
In order to put into practice the information received, one must somehow learn how to divide the space or build it according to this law. First, let’s learn how to build a rectangle with perfect proportions. We take a square as a basis.
We divide the square in half, in one of the resulting rectangles we draw a line that connects the opposite corners. Then we take a compass, put the needle in the center of the bottom side of the square, set aside the length of the resulting diagonal and mark it on a line that will be a continuation of the bottom side of the square. The resulting rectangle has an aspect ratio of 1,62 (this is exactly the ratio that gives 62% and 38%).
What’s even more interesting is that if you start dividing a rectangle with an aspect ratio of 1,62 into a square and a rectangle, you will again get a rectangle with perfect proportions, but smaller. If you divide it again according to the same principle, there will be another pair of square + rectangle with sides whose ratio will correspond to the golden ratio. And so on until you can carry out the division. But what is even more interesting, the Fibonacci series, which looks like an unwinding spiral, fits perfectly into this division. The illustration is in the picture above.
How to divide a segment according to the rule of the golden section
This skill is useful, for example, when creating a house project, planning, designing an apartment, arranging furniture, etc. In the same way, it may be necessary when planning a site, flower beds, planting plants, etc. In general, it can be applied almost everywhere.
So, the order of division of the segment according to the rule of the golden section:
- Take a segment, divide it in half.
- From one of the ends we restore a perpendicular (a straight line at an angle of 90 °), which is equal to half the length of the segment. In the figure, this is a line BC.
- received point C connect the line with the other end of the segment (A).
- On the segment AC put an end D. It is at a distance equal to the length of the segment BС. The easiest way to do this is with a compass, but you can also use a ruler.
- We measure the length of the segment AD (again with a compass, or a ruler). We set aside the same length on the segment AB. We get a point E.
- Now, if we measure the lengths of the segments AE и EB and divide them, we get the same treasured number – 1,62.
By repeating the procedure a couple of times, you will learn how to do everything in just a matter of minutes. If you need, for example, to determine the height of the window, its shape, you can also use these proportions. By the same principle, you can determine the location of all architectural elements, their size. When planning existing objects, the division is easier to carry out using a percentage. Here you either count in your mind, or use a calculator.
Perfect triangle and pentagram
An isosceles triangle is called ideal, the base of which is related to the length of the side as 1/3. That is, again, the golden ratio is respected. It is not difficult to draw a triangle with an ideal aspect ratio. It is more convenient with a compass, but you can get by with a ruler.
The construction is like this. On a straight line from a point A three times we postpone a segment of arbitrary length. We denote this length O. We get a point B. Through it we draw a line perpendicular to the segment AB. On this line on both sides of the point B set aside the value O. We get two points d и d1. We connect them with a dot A. So we got a triangle, the sides of which are related as 1,62. You can check this if you use a compass to set aside the length of the base on the side (point C). The second check is that the opposite angle is 36°.
The construction of the pentagram is somewhat more complicated. We enter it in a circle, we cannot do without a compass.
- The center of the circle is denoted O, we draw a straight line through it until it intersects with a circle. One of the intersection points is denoted A. Line segment OA is the diameter of the circle.
- Finding the midpoint of a segment OD, put a dot E. From the center of the circle up to the intersection with the circle, we restore the perpendicular. This is the point D.
- Connect the dots E и D. Using a compass, set aside a point on the radius C. Line segment CD equal to the length of the segment ED. Measure the length of a line with a compass ED. We put the needle on the point E, we lead the lead to the intersection with the radius. So we got the point C.
- Length of cut DC side of the pentagram. We measure it, with the help of a compass we transfer it to a circle. To do this, with a compass with a deferred distance, we put four more points on the circle, connecting them one by one, we get a pentagram.
Here’s what’s interesting, if the vertices of the resulting pentagram are used to draw a star, it will consist of perfect triangles.
https://youtu.be/c3SVIQBXMnA
Application in construction
As already mentioned, no one knows who discovered the golden ratio, but everything that seems beautiful to us has exactly this aspect ratio. There are many examples in nature. If we consider famous buildings, then there is also the same pattern.
If you want your house to be attractive inside and out, to be remembered and liked, you can calculate at least the basic proportions when creating or choosing a project. Making adjustments to the proportions may not always be easy, often involving additional costs. But, if you immediately keep the golden ratio in mind when creating a project, the questions themselves disappear. Actually, it’s not that difficult.
For example, you want a house of about 100 square meters. The long side can be taken as 12 meters. Then the short one is located as 62% of the long one and will be 7,44 meters. You can make 7 meters or 7,5, you can increase it to 8. Accurate, up to a centimeter, compliance with dimensions is not at all necessary. The ratio is important. And “by eye” even in the approximation looks harmonious. The building area in this case is somewhat smaller – 90-96 squares. If you need more – take the long side equal to 13 meters and count again. It seems to be clear how to apply the golden ratio when creating a house plan.
The height of the floor in this case is taken as 32% of the long part. It will be 12 * 0,32 = 3,84 meters. In principle, this corresponds to the current ideas about the comfortable dimensions of the room, but if desired, you can make the height smaller. Approximately the same is calculated, all other fragments of the house are selected.
Do not forget that the house must also fit into the landscape. If there is some kind of dominant – a high hill, for example, then it is necessary to calculate the ratio with the hill, and with the proportions of the site. In general, to create a harmonious homestead, many factors must be taken into account.
According to the same principle, they develop an internal layout, trying, if possible, to comply with the required ratio. But once again we repeat: if possible. Do not get hung up on exact correspondence to the centimeter. The general trend is important.
Golden ratio in interior design
What else does the golden ratio give besides visual enjoyment? Psychologists say that in an interior created according to this rule, a person feels more comfortable. This is of course subjective, but you can try. So, here is how the golden ratio rule is interpreted in interior design:
- If you are going to divide the room into zones, use the rule. This means that one of the parts should be about 62%, the second – 38%.
- The area occupied by pieces of furniture should not be more than 2/3.
- When choosing furniture, we are guided by the rule: each medium-sized item in terms of dimensions belongs to large ones in the same way as small ones to medium ones.
- When choosing a color, follow approximately the same rules:
- The main color is about 2/3, all additional and accent – 1/3. Colors are chosen according to certain rules.
- The second option: 60% is the main color, 30% additional and 10% are accents.
An example of color matching according to the rules of correct proportionality
- When using horizontal division of the wall (panel), the height of the panel can be taken as 1/3 or 2/3 of the total height of the room. But at the same time, furniture is selected proportional in height, and not in length.
Regarding furniture, the rule seems incomprehensible, but this is only at first glance. For example, we select a group of rest. A large item in this case is a sofa or sofa. Medium – coffee or coffee table, chairs. Small – accessories. So, the dimensions of the coffee table should not be larger than the long side of the sofa, the armchairs should not be larger than its short side. Accessories are no larger than the size of a table or chairs. Ideally, they correlate with them as 62% and 38%.
Why is the exact ratio not specified? Because, firstly, to find such items is unrealistic. Secondly, the golden ratio is not only 62% and 38%. This is also the Fibonacci sequence, following which also makes the design harmonious. There are people for whom following this sequence is a “built-in feature”. They do not have to count, they choose based on their instinct and intuition. But if you analyze their choice, the proportions will be close to ideal. Like this.
The golden ratio in landscape design
When creating a landscape on a site, the principle of ideal proportions is applied, calling it the triangle rule. The composition should have one dominant, the rest of its components only emphasize, shade it. For example, there is a large tree on the site and you want to beat it. It will be the center of the composition – the dominant. Put it on the plan, draw a flower bed or rockery, rock garden – what you want to do.
From the dominant plant or stone, draw two lines at right angles. Lower plants will need to be planted on these lines. Moreover, the second in height should not be higher than 2/3 of the height of the main object. The third object is no higher than 1/3. Complement the composition with even shorter plantings. This is a short summary of how to apply the golden ratio in planting layouts.
But that’s not all. Plants must be selected by color – a combination of greens of different shades, interspersed with flowers and decorative leafy plants – everything obeys the same law. The dominant shade is about 60%, additional colors – 30%, accents – 10%. This is if we talk about the rules of selection in one group. But it is also necessary to coordinate the entire plan as a whole – in terms of size, height, colors.