Construction of a regular pentagon. Technical drawing. Construction of regular polygons Regular pentagon scheme

This figure is a polygon with the minimum number of corners that cannot be used to tile an area. Only a pentagon has the same number of diagonals as its sides. Using the formulas for an arbitrary regular polygon, you can determine all the necessary parameters that the pentagon has. For example, inscribe it in a circle with a given radius, or build it on the basis of a given lateral side.

How to draw a beam correctly and what drawing supplies will you need? Take a piece of paper and mark a dot anywhere. Then attach a ruler and draw a line from the indicated point to infinity. To draw a straight line, press the "Shift" key and draw a line of the desired length. Immediately after drawing, the "Format" tab will open. Deselect the line and you will see that a dot has appeared at the beginning of the line. To create an inscription, click the "Draw an inscription" button and create a field where the inscription will be located.

The first way to construct a pentagon is considered more "classical". The resulting figure will be a regular pentagon. The dodecagon is no exception, so its construction will be impossible without the use of a compass. The task of constructing a regular pentagon is reduced to the task of dividing a circle into five equal parts. You can draw a pentagram using the simplest tools.

I struggled for a long time trying to achieve this and independently find proportions and dependencies, but I did not succeed. It turned out that there are several different options for constructing a regular pentagon, developed by famous mathematicians. The interesting point is that arithmetically this problem can only be solved approximately exactly, since irrational numbers will have to be used. But it can be solved geometrically.

Division of circles. The intersection points of these lines with the circle are the vertices of the square. In a circle of radius R (Step 1) draw a vertical diameter. At the conjugation point N of a line and a circle, the line is tangent to the circle.

Receiving with a strip of paper

A regular hexagon can be constructed using a T-square and a 30X60° square. The vertices of such a triangle can be constructed using a compass and a square with angles of 30 and 60 °, or only one compass. To build side 2-3, set the T-square to the position shown by the dashed lines, and draw a straight line through point 2, which will define the third vertex of the triangle. We mark point 1 on the circle and take it as one of the vertices of the pentagon. We connect the found vertices in series with each other. The heptagon can be constructed by drawing rays from the F pole and through odd divisions of the vertical diameter.

And on the other end of the thread, the pencil is set and obsessed. If you know how to draw a star, but do not know how to draw a pentagon, draw a star with a pencil, then connect the adjacent ends of the star together, and then erase the star itself. Then put a sheet of paper (it is better to fix it on the table with four buttons or needles). Pin these 5 strips to a piece of paper with pins or needles so that they remain motionless. Then circle the resulting pentagon and remove these stripes from the sheet.

For example, we need to draw a five-pointed star (pentagram) for a picture about the Soviet past or about the present of China. True, for this you need to be able to create a drawing of a star in perspective. Similarly, you will be able to draw a figure with a pencil on paper. How to draw a star correctly, so that it looks even and beautiful, you won’t answer right away.

From the center, lower 2 rays onto the circle so that the angle between them is 72 degrees (protractor). The division of a circle into five parts is carried out using an ordinary compass or protractor. Since a regular pentagon is one of the figures that contains the proportions of the golden section, painters and mathematicians have long been interested in its construction. These principles of construction with the use of a compass and straightedge were set forth in the Euclidean Elements.

A regular pentagon is a geometric figure that is formed by the intersection of five straight lines that create five identical angles. This figure is called the Pentagon. The work of artists is closely related to the pentagon - their drawings are based on regular geometric shapes. To do this, you need to know how to quickly build a pentagon.

Why is this figure interesting? The building is shaped like a pentagon Department of Defense of the United States of America. This can be seen in the photos taken from the height of the flight. In nature, there are no crystals and stones, the shape of which would resemble a pentagon. Only in this figure the number of faces coincides with the number of diagonals.

Parameters of a regular pentagon

A rectangular pentagon, like every figure in geometry, has its own parameters. Knowing the necessary formulas, you can calculate these parameters, which will facilitate the process of building a pentagon. Calculation methods and formulas:

• the sum of all angles in polygons is 360 degrees. In a regular pentagon, all angles are equal, respectively, the central angle is found in this way: 360/5 \u003d 72 degrees;
• the inner corner is found in this way: 180*(n -2)/ n = 180*(5−2)/5 = 108 degrees. The sum of all interior angles: 108*5 = 540 degrees.

The side of the pentagon is found using the parameters that are already given in the problem statement:

• if a circle is circumscribed around the pentagon and its radius is known, the side is found according to the following formula: a \u003d 2 * R * sin (α / 2) \u003d 2 * R * sin (72/2) \u003d 1.1756 * R.
• If the radius of the circle inscribed in the pentagon is known, then the formula for calculating the side of the polygon is: 2*r*tg (α/2) = 2*r*tg (α/2) = 1.453*r.
• With a known diagonal of the pentagon, its side is calculated as follows: a \u003d D / 1.618.

The area of ​​the pentagon, like its side, depends on the parameters already found:

• using the known radius of the inscribed circle, the area is found as follows: S \u003d (n * a * r) / 2 \u003d 2.5 * a * r.
• the circumscribed circle around the pentagon allows you to find the area using the following formula: S \u003d (n * R2 * sin α) / 2 \u003d 2.3776 * R2.
• depending on the side of the pentagon: S = (5*a2*tg 54°)/4 = 1.7205* a2.

Building the Pentagon

You can build a regular pentagon using a ruler and a compass, based on a circle inscribed in it or one of the sides.

How to draw a pentagon based on an inscribed circle? To do this, stock up on a compass and a ruler and take the following steps:

1. First you need to draw a circle with center O, then select a point on it, A - the top of the pentagon. A line is drawn from the center to the top.
2. Then a segment perpendicular to the straight line OA is constructed, which also passes through O - the center of the circle. Its intersection with the circle is indicated by point B. The segment O.V. is bisected by point C.
3. Point C will become the center of a new circle passing through A. Point D is its intersection with the straight line OB within the boundaries of the first figure.
4. After that, a third circle is drawn through D, the center of which is point A. It intersects with the first figure at two points, they must be denoted by the letters E and F.
5. The next circle has its center at point E and passes through A, and its intersection with the original one is at the new point G.
6. The last circle in this figure is drawn through a point, A with a center F. Point H is placed at its intersection with the initial one.
7. On the first circle, after all the steps taken, five points appeared, which must be connected by segments. Thus, a regular pentagon AE G H F was obtained.

How to build a regular pentagon in a different way? With the help of a ruler and a compass, the pentagon can be built a little faster. For this you need:

1. First you need to use a compass to draw a circle, the center of which is point O.
2. The radius OA is drawn - a segment that is plotted on a circle. It is bisected by point B.
3. A segment OS is drawn perpendicular to the radius OA, points B and C are connected by a straight line.
4. The next step is to plot the length of segment BC with a compass on the diametral line. Point D appears perpendicular to segment OA. Points B and D are connected, forming a new segment.
5. In order to get the size of the side of the pentagon, you need to connect points C and D.
6. D with the help of a compass is transferred to a circle and is indicated by the point E. By connecting E and C, you can get the first side of a regular pentagon. Following this instruction, you can learn how to quickly build a pentagon with equal sides, continuing to build its other sides like the first one.

In a pentagon with the same sides, the diagonals are equal and form a five-pointed star, which is called a pentagram. The golden ratio is the ratio of the size of the diagonal to the side of the pentagon.

The Pentagon is not suitable for completely filling the plane. The use of any material in this form leaves gaps or forms overlaps. Although natural crystals of this form do not exist in nature, when ice forms on the surface of smooth copper products, molecules in the form of a pentagon appear, which are connected in chains.

The easiest way to get a regular pentagon from a strip of paper is to tie it in a knot and press down a little. This method is useful for parents of preschoolers who want to teach their toddlers to recognize geometric shapes.

Video

See how you can quickly draw a pentagon.

Construction of a regular hexagon inscribed in a circle.

The construction of a hexagon is based on the fact that its side is equal to the radius of the circumscribed circle. Therefore, to build, it is enough to divide the circle into six equal parts and connect the found points to each other.

A regular hexagon can be constructed using a T-square and a 30X60° square. To perform this construction, we take the horizontal diameter of the circle as the bisector of angles 1 and 4, build sides 1 - 6, 4 - 3, 4 - 5 and 7 - 2, after which we draw sides 5 - 6 and 3 - 2.

The vertices of such a triangle can be constructed using a compass and a square with angles of 30 and 60 °, or only one compass. Consider two ways to construct an equilateral triangle inscribed in a circle.

First way(Fig. 61, a) is based on the fact that all three angles of the triangle 7, 2, 3 each contain 60 °, and the vertical line drawn through the point 7 is both the height and the bisector of angle 1. Since the angle 0 - 1 - 2 is equal to 30°, then to find the side 1 - 2 it is enough to construct an angle of 30° from point 1 and side 0 - 1. To do this, set the T-square and square as shown in the figure, draw a line 1 - 2, which will be one of the sides of the desired triangle. To build side 2 - 3, set the T-square to the position shown by the dashed lines, and draw a straight line through point 2, which will define the third vertex of the triangle.

Second way is based on the fact that if you build a regular hexagon inscribed in a circle, and then connect its vertices through one, you get an equilateral triangle.

To build a triangle, we mark the vertex point 1 on the diameter and draw a diametrical line 1 - 4. Further, from point 4 with a radius equal to D / 2, we describe the arc until it intersects with the circle at points 3 and 2. The resulting points will be two other vertices of the desired triangle.

This construction can be done using a square and a compass.

First way is based on the fact that the diagonals of the square intersect at the center of the circumscribed circle and are inclined to its axes at an angle of 45°. Based on this, we install a T-square and a square with angles of 45 ° as shown in Fig. 62, a, and mark points 1 and 3. Further, through these points, we draw the horizontal sides of the square 4 - 1 and 3 -2 with the help of a T-square. Then, using a T-square along the leg of the square, we draw the vertical sides of the square 1 - 2 and 4 - 3.

Second way is based on the fact that the vertices of the square bisect the arcs of the circle enclosed between the ends of the diameter. We mark points A, B and C at the ends of two mutually perpendicular diameters, and from them with a radius y we describe the arcs until they intersect.

Further, through the points of intersection of the arcs, we draw auxiliary lines, marked on the figure with solid lines. Their points of intersection with the circle will define vertices 1 and 3; 4 and 2. The vertices of the desired square obtained in this way are connected in series with each other.

Construction of a regular pentagon inscribed in a circle.

To inscribe a regular pentagon in a circle, we make the following constructions. We mark point 1 on the circle and take it as one of the vertices of the pentagon. Divide segment AO in half. To do this, with the radius AO from point A, we describe the arc to the intersection with the circle at points M and B. Connecting these points with a straight line, we get the point K, which we then connect to point 1. With a radius equal to the segment A7, we describe the arc from point K to the intersection with the diametrical line AO ​​at point H. Connecting point 1 with point H, we get the side of the pentagon. Then, with a compass opening equal to the segment 1H, describing the arc from vertex 1 to the intersection with the circle, we find vertices 2 and 5. Having made notches from vertices 2 and 5 with the same compass opening, we obtain the remaining vertices 3 and 4. We connect the found points sequentially with each other.

Construction of a regular pentagon given its side.

To construct a regular pentagon along its given side (Fig. 64), we divide the segment AB into six equal parts. From points A and B with radius AB we describe arcs, the intersection of which will give point K. Through this point and division 3 on the line AB we draw a vertical line. Further from the point K on this straight line, we set aside a segment equal to 4/6 AB. We get point 1 - the vertex of the pentagon. Then, with a radius equal to AB, from point 1 we describe the arc to the intersection with the arcs previously drawn from points A and B. The intersection points of the arcs determine the vertices of the pentagon 2 and 5. We connect the found vertices in series with each other.

Construction of a regular heptagon inscribed in a circle.

Let a circle of diameter D be given; you need to inscribe a regular heptagon into it (Fig. 65). Divide the vertical diameter of the circle into seven equal parts. From point 7 with a radius equal to the diameter of the circle D, we describe the arc until it intersects with the continuation of the horizontal diameter at point F. Point F is called the pole of the polygon. Taking point VII as one of the vertices of the heptagon, we draw rays from the pole F through even divisions of the vertical diameter, the intersection of which with the circle will determine the vertices VI, V and IV of the heptagon. To obtain vertices / - // - /// from points IV, V and VI, we draw horizontal lines until they intersect with the circle. We connect the found vertices in series with each other. The heptagon can be constructed by drawing rays from the F pole and through odd divisions of the vertical diameter.

The above method is suitable for constructing regular polygons with any number of sides.

The division of a circle into any number of equal parts can also be done using the data in Table. 2, which shows the coefficients that make it possible to determine the dimensions of the sides of regular inscribed polygons.

Side lengths of regular inscribed polygons.

The first column of this table shows the number of sides of a regular inscribed polygon, and the second column shows the coefficients. The length of a side of a given polygon is obtained by multiplying the radius of a given circle by a factor corresponding to the number of sides of this polygon.

Ozhegov's explanatory dictionary says that a pentagon is a bounded by five intersecting straight lines forming five internal angles, as well as any object of a similar shape. If a given polygon has all the same sides and angles, then it is called a regular (pentagon).

What is interesting about a regular pentagon?

It was in this form that the well-known building of the United States Department of Defense was built. Of the voluminous regular polyhedra, only the dodecahedron has pentagon-shaped faces. And in nature, crystals are completely absent, the faces of which would resemble a regular pentagon. In addition, this figure is a polygon with a minimum number of corners that cannot be used to tile an area. Only a pentagon has the same number of diagonals as its sides. Agree, it's interesting!

Basic properties and formulas

Using the formulas for an arbitrary regular polygon, you can determine all the necessary parameters that the pentagon has.

• Central angle α = 360 / n = 360/5 = 72°.
• Internal angle β = 180° * (n-2)/n = 180° * 3/5 = 108°. Accordingly, the sum of the interior angles is 540°.
• The ratio of the diagonal to the side is (1+√5)/2, i.e. (approximately 1.618).
• The length of the side that a regular pentagon has can be calculated using one of three formulas, depending on which parameter is already known:

• if a circle is circumscribed around it and its radius R is known, then a = 2*R*sin (α/2) = 2*R*sin(72°/2) ≈1.1756*R;
• in the case when a circle with radius r is inscribed in a regular pentagon, a = 2*r*tg(α/2) = 2*r*tg(α/2) ≈ 1.453*r;
• it happens that instead of radii the value of the diagonal D is known, then the side is determined as follows: a ≈ D / 1.618.

• The area of ​​a regular pentagon is determined, again, depending on what parameter we know:
• if there is an inscribed or circumscribed circle, then one of two formulas is used:

S \u003d (n * a * r) / 2 \u003d 2.5 * a * r or S \u003d (n * R 2 * sin α) / 2 ≈ 2.3776 * R 2;

• the area can also be determined, knowing only the length of the side a:

S \u003d (5 * a 2 * tg54 °) / 4 ≈ 1.7205 * a 2.

Regular pentagon: construction

This geometric figure can be constructed in different ways. For example, inscribe it in a circle with a given radius, or build it on the basis of a given lateral side. The sequence of actions was described in Euclid's Elements around 300 BC. In any case, we need a compass and a ruler. Consider the method of construction using a given circle.

1. Select an arbitrary radius and draw a circle, marking its center with point O.

2. On the circle line, select a point that will serve as one of the vertices of our pentagon. Let this be point A. Connect points O and A with a straight line.

3. Draw a line through the point O perpendicular to the line OA. Mark the point where this line intersects with the circle line as point B.

4. In the middle of the distance between points O and B, build point C.

5. Now draw a circle, the center of which will be at point C and which will pass through point A. The place of its intersection with the line OB (it will be inside the very first circle) will be point D.

6. Construct a circle passing through D, the center of which will be at A. The places of its intersection with the original circle must be marked with points E and F.

7. Now build a circle, the center of which will be in E. You need to do this so that it passes through A. Its other intersection of the original circle must be indicated

8. Finally, draw a circle through A centered at point F. Mark another intersection of the original circle with point H.

9. Now it remains only to connect the vertices A, E, G, H, F. Our regular pentagon will be ready!

5.3. golden pentagon; construction of Euclid.

A wonderful example of the "golden section" is a regular pentagon - convex and star-shaped (Fig. 5).

To build a pentagram, you need to build a regular pentagon.

Let O be the center of the circle, A a point on the circle, and E the midpoint of segment OA. The perpendicular to the radius OA, restored at point O, intersects with the circle at point D. Using a compass, mark the segment CE = ED on the diameter. The length of a side of a regular pentagon inscribed in a circle is DC. We set aside segments DC on the circle and get five points for drawing a regular pentagon. We connect the corners of the pentagon through one diagonal and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.

Each end of the pentagonal star is a golden triangle. Its sides form an angle of 36° at the top, and the base laid on the side divides it in proportion to the golden section.

There is also a golden cuboid - this is a rectangular parallelepiped with edges having lengths of 1.618, 1 and 0.618.

Now consider the proof offered by Euclid in the Elements.

from the center of the circumscribed circle. Let's start with

segment ABE, divided in the middle and

So let AC = AE. Denote by a the equal angles EBC and CEB. Since AC=AE, the angle ACE is also equal to a. The theorem that the sum of the angles of a triangle is 180 degrees allows you to find the angle ALL: it is 180-2a, and the angle EAC is 3a - 180. But then the angle ABC is 180-a. Summing up the angles of triangle ABC, we get

180=(3a -180) + (3a-180) + (180 - a)

Whence 5a=360, so a=72.

So, each of the angles at the base of the triangle BEC is twice the angle at the top, equal to 36 degrees. Therefore, in order to construct a regular pentagon, it is only necessary to draw any circle centered at point E, intersecting EC at point X and side EB at point Y: the segment XY is one of the sides of the regular pentagon inscribed in the circle; Going around the entire circle, you can find all the other sides.

We now prove that AC=AE. Suppose that the vertex C is connected by a straight line segment to the midpoint N of the segment BE. Note that since CB = CE, then the angle CNE is a right angle. According to the Pythagorean theorem:

CN 2 \u003d a 2 - (a / 2j) 2 \u003d a 2 (1-4j 2)

Hence we have (AC/a) 2 = (1+1/2j) 2 + (1-1/4j 2) = 2+1/j = 1 + j =j 2

So, AC = ja = jAB = AE, which was to be proved

5.4. Spiral of Archimedes.

Sequentially cutting off squares from golden rectangles to infinity, each time connecting opposite points with a quarter of a circle, we get a rather elegant curve. The first attention was drawn to her by the ancient Greek scientist Archimedes, whose name she bears. He studied it and deduced the equation of this spiral.

Currently, the Archimedes spiral is widely used in technology.

6. Fibonacci numbers.

The name of the Italian mathematician Leonardo from Pisa, who is better known by his nickname Fibonacci (Fibonacci is an abbreviation of filius Bonacci, that is, the son of Bonacci), is indirectly associated with the golden ratio.

In 1202 he wrote the book "Liber abacci", that is, "The Book of the abacus". "Liber abacci" is a voluminous work containing almost all the arithmetic and algebraic knowledge of that time and played a significant role in the development of mathematics in Western Europe over the next few centuries. In particular, it was from this book that Europeans became acquainted with Hindu ("Arabic") numerals.

The material reported in the book is explained on a large number of problems that make up a significant part of this treatise.

Consider one such problem:

How many pairs of rabbits are born from one pair in one year?

Someone placed a pair of rabbits in a certain place, enclosed on all sides by a wall, in order to find out how many pairs of rabbits will be born during this year, if the nature of rabbits is such that in a month a pair of rabbits will reproduce another, and rabbits give birth from the second month after their birth "

Now let's move from rabbits to numbers and consider the following numerical sequence:

u 1 , u 2 … u n

in which each term is equal to the sum of the two previous ones, i.e. for any n>2

u n \u003d u n -1 + u n -2.

This sequence asymptotically (approaching more and more slowly) tends to some constant relation. However, this ratio is irrational, that is, it is a number with an infinite, unpredictable sequence of decimal digits in the fractional part. It cannot be expressed exactly.

If any member of the Fibonacci sequence is divided by the one preceding it (for example, 13:8), the result will be a value that fluctuates around the irrational value 1.61803398875... and sometimes exceeds it, sometimes not reaching it.

The asymptotic behavior of the sequence, the damped fluctuations of its ratio around an irrational number Φ can become more understandable if we show the ratios of several first terms of the sequence. This example shows the relationship of the second term to the first, the third to the second, the fourth to the third, and so on:

1:1 = 1.0000, which is less than phi by 0.6180

2:1 = 2.0000, which is 0.3820 more phi

3:2 = 1.5000, which is less than phi by 0.1180

5:3 = 1.6667, which is 0.0486 more phi

8:5 = 1.6000, which is less than phi by 0.0180

As you move along the Fibonacci summation sequence, each new term will divide the next with more and more approximation to the unattainable F.

A person subconsciously seeks the Divine proportion: it is needed to satisfy his need for comfort.

When dividing any member of the Fibonacci sequence by the next one, we get just the reciprocal of 1.618 (1: 1.618=0.618). But this is also a very unusual, even remarkable phenomenon. Since the original ratio is an infinite fraction, this ratio should also have no end.

When dividing each number by the next one after it, we get the number 0.382

Selecting ratios in this way, we obtain the main set of Fibonacci coefficients: 4.235, 2.618, 1.618, 0.618, 0.382, 0.236. We also mention 0.5. All of them play a special role in nature and in particular in technical analysis.

It should be noted here that Fibonacci only reminded mankind of his sequence, since it was known in ancient times under the name of the Golden Section.

The golden ratio, as we have seen, arises in connection with the regular pentagon, so the Fibonacci numbers play a role in everything that has to do with regular pentagons - convex and star-shaped.

The Fibonacci series could have remained only a mathematical incident if it were not for the fact that all researchers of the golden division in the plant and animal world, not to mention art, invariably came to this series as an arithmetic expression of the golden division law. Scientists continued to actively develop the theory of Fibonacci numbers and the golden ratio. Yu. Matiyasevich using Fibonacci numbers solves Hilbert's 10th problem (on the solution of Diophantine equations). There are elegant methods for solving a number of cybernetic problems (search theory, games, programming) using Fibonacci numbers and the golden section. In the USA, even the Mathematical Fibonacci Association is being created, which has been publishing a special journal since 1963.

One of the achievements in this area is the discovery of generalized Fibonacci numbers and generalized golden ratios. The Fibonacci series (1, 1, 2, 3, 5, 8) and the “binary” series of numbers discovered by him 1, 2, 4, 8, 16 ... (that is, a series of numbers up to n, where any natural number less than n can be represented as the sum of some numbers of this series) at first glance, they are completely different. But the algorithms for their construction are very similar to each other: in the first case, each number is the sum of the previous number with itself 2 = 1 + 1; 4 \u003d 2 + 2 ..., in the second - this is the sum of the two previous numbers 2 \u003d 1 + 1, 3 \u003d 2 + 1, 5 \u003d 3 + 2 .... Is it possible to find a general mathematical formula from which and " binary series, and the Fibonacci series?

Indeed, let's set a numerical parameter S, which can take any values: 0, 1, 2, 3, 4, 5... separated from the previous one by S steps. If we denote the nth member of this series by S (n), then we obtain the general formula S (n) = S (n - 1) + S (n - S - 1).

Obviously, with S = 0, from this formula we will get a “binary” series, with S = 1 - a Fibonacci series, with S = 2, 3, 4. new series of numbers, which are called S-Fibonacci numbers.

In general terms, the golden S-proportion is the positive root of the golden S-section equation x S+1 – x S – 1 = 0.

It is easy to show that at S = 0, the division of the segment in half is obtained, and at S = 1, the familiar classical golden ratio is obtained.

The ratios of neighboring Fibonacci S-numbers with absolute mathematical accuracy coincide in the limit with the golden S-proportions! That is, golden S-sections are numerical invariants of Fibonacci S-numbers.

7. Golden section in art.

7.1. Golden section in painting.

Turning to examples of the "golden section" in painting, one cannot but stop one's attention on the work of Leonardo da Vinci. His identity is one of the mysteries of history. Leonardo da Vinci himself said: "Let no one who is not a mathematician dare to read my works."

There is no doubt that Leonardo da Vinci was a great artist, his contemporaries already recognized this, but his personality and activities will remain shrouded in mystery, since he left to posterity not a coherent presentation of his ideas, but only numerous handwritten sketches, notes that say “both everyone in the world."

The portrait of Monna Lisa (Gioconda) has attracted the attention of researchers for many years, who found that the composition of the drawing is based on golden triangles that are parts of a regular star pentagon.

Also, the proportion of the golden section appears in Shishkin's painting. In this famous painting by I. I. Shishkin, the motifs of the golden section are clearly visible. The brightly lit pine tree (standing in the foreground) divides the length of the picture according to the golden ratio. To the right of the pine tree is a hillock illuminated by the sun. It divides the right side of the picture horizontally according to the golden ratio.

Raphael's painting "The Massacre of the Innocents" shows another element of the golden ratio - the golden spiral. On the preparatory sketch of Raphael, red lines are drawn running from the semantic center of the composition - the point where the warrior's fingers closed around the child's ankle - along the figures of the child, the woman clutching him to herself, the warrior with a raised sword and then along the figures of the same group on the right side of the sketch . It is not known whether Raphael built the golden spiral or felt it.

T. Cook used the golden section when analyzing the painting by Sandro Botticelli "The Birth of Venus".

7.2. Pyramids of the golden section.

The medical properties of the pyramids, especially the golden section, are widely known. According to some of the most common opinions, the room in which such a pyramid is located seems larger, and the air is more transparent. Dreams begin to be remembered better. It is also known that the golden ratio was widely used in architecture and sculpture. An example of this was: the Pantheon and Parthenon in Greece, the buildings of architects Bazhenov and Malevich

8. Conclusion.

It must be said that the golden ratio has a great application in our lives.

It has been proven that the human body is divided in proportion to the golden ratio by the belt line.

The shell of the nautilus is twisted like a golden spiral.

Thanks to the golden ratio, the asteroid belt between Mars and Jupiter was discovered - in proportion there should be another planet there.

The excitation of the string at the point dividing it in relation to the golden division will not cause the string to vibrate, that is, this is the point of compensation.

On aircraft with electromagnetic energy sources, rectangular cells with the proportion of the golden section are created.

Gioconda is built on golden triangles, the golden spiral is present in Raphael's painting "Massacre of the Innocents".

Proportion found in the painting by Sandro Botticelli "The Birth of Venus"

There are many architectural monuments built using the golden ratio, including the Pantheon and Parthenon in Athens, the buildings of architects Bazhenov and Malevich.

John Kepler, who lived five centuries ago, owns the statement: "Geometry has two great treasures. The first is the Pythagorean theorem, the second is the division of a segment in the extreme and average ratio"

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8. "Mathematics - Encyclopedia for children" M .: Avanta +, 1998

9. Information from the Internet.

Fibonacci matrices and the so-called "golden" matrices, new computer arithmetic, a new coding theory and a new theory of cryptography. The essence of the new science is the revision of all mathematics from the point of view of the golden section, starting with Pythagoras, which, of course, will entail new and certainly very interesting mathematical results in the theory. In practical terms - "golden" computerization. And because...

This result will not be affected. The basis of the golden ratio is an invariant of the recursive ratios 4 and 6. This shows the "stability" of the golden section, one of the principles of the organization of living matter. Also, the basis of the golden ratio is the solution of two exotic recursive sequences (Fig. 4.) Fig. 4 Recursive Fibonacci Sequences So...

The ear is j5 and the distance from ear to crown is j6. Thus, in this statue we see a geometric progression with the denominator j: 1, j, j2, j3, j4, j5, j6. (Fig. 9). Thus, the golden ratio is one of the fundamental principles in the art of ancient Greece. Rhythms of the heart and brain. The human heart beats evenly - about 60 beats per minute at rest. The heart compresses like a piston...