The golden ratio is the universal principle of harmony. Golden Ratio and Symmetry Golden Ratio in Architecture

Based on the described principle, a Golden (or harmonious) Rectangle is one in which the sides are related as 1: 1.618, i.e. the length of the longer side of the rectangle is equal to the length of the shorter side of the rectangle multiplied by ∳ (phi)=1.618:

Do you recognize? It's a harmonious table top! Or the facade of the cabinet and much more.

Similarly, the Golden (or harmonious) Parallelepiped is the one in which the sides are also related as 1: 1.618, i.e. the length of the longer side of the box is equal to the height of the box multiplied by ∳ (phi)=1.618, and the width of the box is equal to the height of the box divided by ∳ (phi)=1.618:

Do you recognize? This is a furniture cabinet, wall table (console), etc.

The Golden Ratio underlies many (if not all) natural relationships and even the construction of our universe. Examples abound at every level, from rabbit breeding, the arrangement of seeds in a sunflower and nuts in a cone, to astrophysics and quantum mechanics. Planetary orbits and even structure human figure are another confirmation of this remarkable proportion.

The ratio between adjacent phalanges of the fingers is ∳ (phi) = 1.618, The ratio between the elbow and the hand is ∳ (phi) = 1.618, the ratio of the distance from the crown to the eyes and the distance from the eyes to the chin is ∳ (phi) = 1.618, the ratio the distance from the crown to the navel and the distance from the navel to the heels is again ∳ (phi) = 1.618:

Distances between the sun and the first five planets in solar system also correlate (approximately) as ∳ (phi) = 1.618, therefore, as is certainly known, astrometry uses the golden ratio when determining the planets in their orbits:

Being so fundamental and so widespread in nature, this attitude simply calls us on a subconscious level as the absolutely correct one to follow. As such, this ratio has been used for centuries by designers and architects, from pyramids to furniture masterpieces.

The Great Pyramid at Giza, as is now clear, was also built in accordance with the Golden Section: the height of the side of the pyramid is equal to the length of the base of the side of the pyramid, multiplied by the same value ∳ (phi) = 1.618:

During the construction of the Parthenon (an ancient Greek temple located on the Athenian Acropolis, main temple in ancient Athens) used the ratio ∳ (phi) = 1.618 when determining external dimensions and the ratio of its parts:

It is not known for certain whether calculators or Fibonacci markers were used in the construction of the Parthenon, but the ratio was definitely applied. More details about the ratio ∳ (phi) = 1.618 in the construction of this architectural monument are given in the video, starting from the 48th second:

In the above video, finally, it came to a piece of furniture, albeit a simple one. The main thing is that the ratio is still the same - ∳ (phi) = 1.618.

One type of chest of drawers with many drawers, called in different publications as Highboy or Popadour ("Tall guy" or "Pompadour"), made in Philadelphia between 1762 and 1790, uses the Golden Ratio in the ratio of the sizes of many of its elements. The frame is a golden rectangle, the position of the narrowing ("waist" of the cabinet) is determined by dividing the overall height of the cabinet by ∳ (phi) = 1.618. The heights of the lower drawers are also related as ∳ (phi) = 1.618:

The Golden Section is used in the manufacture of furniture most often as a kind of rectangle, which is built using ∳ (phi) = 1.618 for its two dimensions, i.e. the already mentioned Golden Rectangle, where the length is 1.618 times the width (or vice versa). These proportions can be used to determine the overall dimensions of furniture, as well as interior details such as doors and drawers. One can apply calculations by dividing and multiplying by such a "round" and convenient number as 1.618, but one can simply use , simply taking the dimensions of the larger object and setting aside the size of the smaller object after that. Or vice versa. Fast, simple and convenient.

Furniture is three dimensional and the Golden Ratio can be applied to all three dimensions, i.e. a piece of furniture becomes a Golden Parallelepiped if it is made according to the rules of the Golden Ratio. For example, in simple case When looking at a piece of furniture from the side, its height may be the largest measurement in the Golden Rectangle. However, when looking at the same piece of furniture from the front, the same height can be a short measurement in the Golden Rectangle.

It should be noted, however, that the form of an object should follow its function. Even the perfect proportions of a piece of furniture can be meaningless if the item cannot be used, for example because it is too small or too large, or for other reasons cannot be used comfortably. Therefore, practical considerations should come first. In fact, most furniture projects require you to start designing with some given dimensions A: A table needs to be a certain height, a cabinet may need to be adjusted to a particular space, and a bookcase may need a certain number of shelves. But almost certainly you will be forced to define many other sizes in relation to which the correct proportions can be applied. But the end result will be worth the effort to see how the Golden Ratio can work for all of these elements. Deciding on the dimensions "by eye" or, even worse, based on the available blanks, will not allow you to get a perfectly balanced, with beautiful proportions of individual parts and a piece of furniture as a whole.

So, the dimensions of the individual pieces of furniture should be proportional in accordance with the Golden Ratio. Elements such as table legs, the relative dimensions of frame elements such as vertical and horizontal parts of facades, prolegs, drawers, etc., can be calculated using the Golden Ratio. golden ratio also offers one way to solve the problem of designing drawers in a chest of drawers with a stepped increase in the height of the drawers. With the help it is easy to carry out such marking - you just need to take the size of a larger box and set aside the dimensions of two adjacent boxes using the marker, etc. After that, taking the size of the box, use the marker to set aside the distance from the top of the box to the location of its handle.

This method of use as a tool for practical application The Golden Ratio will be effective for determining other dimensions as well, such as the position of shelves in a closet, dividers between drawers, etc. Any size of a piece of furniture is initially determined by functional and structural requirements, but many adjustments can be made by applying the Golden Ratio, which will certainly add harmony to the piece. Using the Golden Ratio when designing furniture will allow you to harmonize not only the object as a whole, but will also allow you to be sure that all the components - door panels, drawers, legs, sides, etc. fundamentally, harmoniously interconnected.

Designing something with absolutely perfect proportions is rarely possible in reality. Almost every piece of furniture or wood will have to be weighed against the constraints of functionality, joinery, or cost savings. But even an attempt to approach perfection, which can be defined as dimensions that exactly correspond to the Golden Ratio, guarantees you best result compared to developing without paying attention to these fundamental principles. Even if you are close to ideal proportions, the eye of the viewer will smooth out small flaws and consciousness will fill in some gaps in the design. It is desirable, but not necessary, that everything be perfect and according to the formula. But if your piece of furniture is absolutely out of proportion, no doubt it will be ugly. Therefore, it is necessary to strive for the correct proportions.

Finally, we often adjust things by eye to make the subjectlighter and better balanced, and we do this with the help of methodswhich are everyday in woodworking. These methods include taking into account changes in the dimensions of the workpiece, based on the direction of the wood fibers, taking into accountwood pattern, with which you can make a piece of furniture more attractive,finishing edges and corners that give the impression of greater or lesser thicknesselement of the product, the use of moldings to more closely match the product with the Golden Rectangle or Parallelepiped, the use of tapered legs to make the feelingbringing the piece of furniture closer to perfect proportion, and, in the end, mixing all these methods to achieve the perfect design. The use of the Golden Mean and the tool for its application, the Fibonacci Scatterer, is the beginning of this quest for perfection.

Materials used in the article chapters "A Guide to Good Design" from the book "Practical Furniture Design" by Graham Blackburn - recognized furniture maker, popularizer of woodworking and publisher

Often you have to deal with a situation when the element you have drawn "does not ring"? Something wrong? Wrong proportions?

It should not be argued that there is no ideal in nature, because it exists and was deduced long ago with the help of mathematics and geometry. The name of the person who first introduced the term "golden section" is unknown, although many are accustomed to believing that this is Leonardo Da Vinci. The earliest appearance of this term is in 1835 thanks to Martin Ohm, in a footnote to the second edition of his Pure Elementary Mathematics.

What does the golden section formula look like?

This harmonious ratio two quantities b and a, a > b, when a/b = (a+b)/a is true. A number equal to the ratio a/b is usually denoted by an uppercase Greek letter

(\displaystyle \phi )

In honor of ancient Greek sculptor and the architect Phidias.

For practical purposes, they are limited to an approximate value of = 1.618 or = 1.62. In a rounded percentage, the golden ratio is the division of a value in relation to 62% and 38%.

Sometimes the number is called the "golden number"

So that you and I do not bother with mathematics, smart people came up with such a circle. With it, you can already check finished projects on the ratio of parts, and build new ones, taking into account the principle of the "golden section"

Let your projects remain in the world cultural heritage!

The desire to give a fashionable shape to the nose or lips is rare, which cannot be said about the eyebrows, which are either plucked into a thin thread, or drawn daily or regularly tinted. Blindly following fashion trends is not always beneficial - thin eyebrows-threads often do not harmonize with the type of face at all, and those painted with a pencil look rather vulgar and almost always unnatural. But nature does not always take care of the harmony of facial features, therefore, if necessary, eyebrow correction has to be modeled. Since color and proportions are the basis of our visual perception, successful correction requires preliminary marking, for which Leonardo's eyebrow compasses are used.

What is Leonardo's compass

The Leonardo compass is a tool made of surgical steel that allows you to apply the principle of the Golden Ratio when modeling the shape of the eyebrows. Outwardly, in its upper part it resembles English letter W, as it has three legs. The design of the compass helps to measure the ratio between large and small distances (depending on the change in one of these distances, the other also changes) - the middle leg is involved in measuring both large and small distances.

The instrument owes its name to the great scientist and artist Leonardo da Vinci, who studied harmonious proportions and created his masterpieces using the principle of harmonic division.

The "golden ratio" is the proportion at which the ratio of one part to another is equal to the ratio of the whole to the first part.

Since the ideal shape of the eyebrows depends not so much on fashion as on the characteristics of a particular person (face shape, size and shape of the eyes), the master must take these features into account when “marking up”.

In order to give the eyebrows a shape that will not be a dissonant note in the overall harmony of the face, makeup artists have to make “marking”, based not on subjective aesthetic perception, but on precise geometric constructions.

Create a verified and correct form in the shortest possible time, the makeup artist is helped by an eyebrow compass.

What proportions does Leonardo's compass help determine?

Only those eyebrows that have a wide and narrow part look natural. However, in order to create a beautiful, harmonious shape, the makeup artist needs to determine:

• Where should the eyebrow start? They do not always start with the client where they are supposed to start according to harmonious proportions, so it is impossible to focus on the natural growth of hairs or intuitive perception.
• Where should the brow end. This point can be felt in the place where the frontal bone ends (a small depression is felt under the finger). Of course, during the correction procedure, it is inconvenient to probe this place every time, in addition, without accurate measurement, the eyebrows can turn out to be asymmetrical.

• Where should the wide end meet the narrow end (highest point). The location of this point depends on the school - in the Russian school it is located parallel to the pupil (you can see how such an eyebrow looks like in the photo of Lyubov Orlova), in the French school it is above the upper edge of the iris, and in Hollywood it goes to the outer edge of the eye.
• What should be the distance in the bridge of the nose.
• What should be the distance between the eye and the eyebrow (with a small vertical distance, the eyebrows seem to hang over).

Tips to help you in using the Leonardo eyebrow compass:

Why use Leonardo's compass

The location of the eyes visually changes depending on the slope of the base of the eyebrow - if this line is tilted towards the nose, the eyes become closer, and if this line is tilted away from the nose, the distance between the eyes seems wider. In this way, too wide or narrowly set eyes can be corrected.

The bridge of the nose will look more even in combination with a straight line at the base of the eyebrows.

The width of the eyebrows is adjusted depending on the proportions of the face (its widest part should correspond in width to half the iris and not exceed 1/3 of the length of the entire eyebrow).

There are a sufficient number of such recommendations, involving the removal of excess hairs or tattooing where there are not enough hairs. However, without the use of accurate measurements and the rule of the "golden section", one has to completely trust the experience and taste of the cosmetologist, and the taste of the client and the makeup artist may not coincide.

The use of Leonardo's compass allows you to create perfect shape eyebrows for a specific face and demonstrate to the client the advantage of the form chosen by the makeup artist.

How to work with Leonardo's compass

In order to build the correct lines as symmetrically as possible with the help of Leonardo's compass, it is important to know how to use the compass for marking. Marking with a compass is applied in a supine position.

• The construction of a sketch begins with the definition of a central point - a “reference point”. To do this, between the eyebrows, slightly above the bridge of the nose, it is necessary to determine the center of the forehead and mark this point. vertical line. The nose cannot serve as a guideline for a symmetrical construction, since so many people have a slight deformation of the nose, which, although not conspicuous, will affect the symmetry during correction.
• The second point necessary for construction is the point of the beginning of the eyebrow. In order to determine its location, Leonardo's compasses are taken, and the ends that determine long distances are placed on the lacrimal canals. The resulting small distance shows the distance between the eyebrows. Lines are drawn at the location of the points indicating the beginning.
• The third point is the end of the eyebrow, its "tail". To determine it, the compass is applied like a ruler - from the point of the edge of the nose (in the place where it touches the cheek) through the point of the edge of the eye to the end of the eyebrow. A vertical line is also drawn at the third point.

• The fourth important point is the highest point. It is necessary to determine this point regardless of the shape of the bend chosen by the client (this point can be either pronounced, “corner”, or smoothed, almost imperceptible). To determine this point, the extreme legs of the compass are placed at the end and beginning of the eyebrow. In this case, the middle leg of the compass should be directed towards the temple, and not towards the forehead. The location of the middle leg will be the highest point.
• After applying these points, the width of the eyebrows is determined and the upper and lower lines are adjusted. To do this, all the marked points are connected. The result should be a clear outline, with which the master will work in the future.

• In the process of work, points are applied simultaneously on each half of the face.
• How correctly the markings are applied should be checked in a sitting position. Symmetry is checked using a compass - the distance of each eyebrow from highest point before its beginning and end must match. It is also important to check whether the central point is correctly marked (the distance from this point to the beginning of the eyebrow should be the same on both sides).
• Eyebrows should lie on the same line. To check, the compass is used as a ruler, which is placed between the lower starting points. Similarly, the relationship between the upper starting points is checked.

All hairs that go beyond the intended lines are removed.

The use of Leonardo's eyebrow compass is recommended for beginners, since this method of marking is more convenient than using a flexible ruler.

Alexey Chulichkov

Why is beautiful, for example, a rose? Or a sunflower? Or a peacock tail? Your favorite dog and no less favorite cat? "Very simple!" - the mathematician will answer and begin to explain the law that was discovered in ancient times (perhaps it was noticed in nature) and was called the golden ratio. (See the article “Does God Know Math?” in the last issue.)

We suggest you make a "golden compass" - the simplest tool for measuring the golden ratio, known since antiquity. It will help to find mathematically verified harmony in the surrounding objects.

1. We need two strips of the same length - made of wood, cardboard or thick paper, as well as a bolt with a washer and a nut.

2. We drill a hole in both strips so that the middle of the hole divides the strip in the golden ratio, that is, the length of its greater part divided by the length of the entire strip should be equal. For example, if the length of the strip is 10 cm, then the hole must be drilled, stepping back from one of edges 10 x 0.618 \u003d 6.18 cm. If the length of the bar is 1 m, then we drill a hole, stepping back from the edge 100 x 0.618 \u003d 61.8 cm. It is convenient to have both large and small compasses on hand to measure objects of different scales.

3. We connect the planks with a bolt so that they can rotate around it with friction. The circle is ready. According to the laws of similarity of triangles, the distances between the ends of the smaller and larger legs of the compass are the same as the length of the smaller part of the bar to the larger one, that is, their ratio is φ.

4. Now you can start exploring! Let's check whether a person was created according to the laws of the golden ratio. Let's take in a larger compass solution the distance from the chin to the bridge of the nose. In a smaller solution fit the distance from the bridge of the nose to the roots of the hair. This means that the dot on the bridge of the nose divides our face in the golden ratio!

5. If you are fascinated by the laws of the golden ratio, we suggest making the “golden compass” a slightly more complex design. How? Try to think for yourself.

Look for golden proportions in things that seem beautiful to you - you will almost certainly find in them golden ratio and make sure that our world is beautiful and harmonious! Success in research!