How to draw a pentagon. Construction of a regular pentagon. Receiving with a strip of paper

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 inscribed in a circle regular pentagon equal to 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 cuboid with edges having lengths 1.618, 1 and 0.618.

Now consider the proof offered by Euclid in the Elements.

Let us now see how Euclid uses golden ratio in order to build an angle of 72 degrees - it is at this angle that the side of a regular pentagon is visible

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 information 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 presented in the book is explained in large numbers 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 "

Months	1	2	3	4	5	6	7	8	9	10	11	12
Pairs of rabbits	2	3	5	8	13	21	34	55	89	144	233	377

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 the ratios in this way, we obtain the main set of Fibonacci coefficients: 4.235,2.618,1.618,0.618,0.382,0.236. Let us 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 by 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 the numerical parameter S, which can take any values: 0, 1, 2, 3, 4, 5... Consider number series, S + 1 of which the first terms are units, and each of the subsequent ones is equal to the sum of two terms of the previous one and the one that is separated from the previous one by S steps. If nth member we denote 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 view the golden S-proportion is the positive root of the golden S-section 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 human body 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"

Bibliography

1. D. Pidow. Geometry and art. – M.: Mir, 1979.

2. Journal "Science and technology"

3. Magazine "Quantum", 1973, No. 8.

4. Journal "Mathematics at School", 1994, No. 2; No. 3.

5. Kovalev F.V. Golden section in painting. K .: Vyscha school, 1989.

6. Stakhov A. Codes of the golden ratio.

7. Vorobyov N.N. "Fibonacci numbers" - M.: Nauka 1964

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

9. Information from the Internet.

Fibonacci matrices and the so-called "golden" matrices, new computer arithmetic, new coding theory and new theory 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 geometric progression with 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...

It is impossible to do without studying the technology of this process. There are several ways to get the job done. How to draw a star with a ruler will help you understand the most famous methods of this process.

Varieties of stars

There are many options appearance such a figure as a star.

Since ancient times, its five-pointed variety has been used to draw pentagrams. This is due to its property, which allows you to make a drawing without lifting the pen from the paper.

There are also six-pointed, tailed comets.

The starfish traditionally has five peaks. Images of the Christmas version are often found in the same form.

In any case, to draw a five-pointed star in stages, you need to resort to the help of special tools, since a freehand image is unlikely to look symmetrical and beautiful.

Execution of the drawing

To understand how to draw an even star, you should understand the essence of this figure.

The basis for its outline is a broken line, the ends of which converge at the starting point. It forms a regular pentagon - a pentagon.

The distinctive properties of such a figure are the possibility of inscribing it in a circle, as well as a circle in this polygon.

All sides of the pentagon are equal. Understanding how to correctly draw a drawing, you can understand the essence of the process of building all the figures, as well as various schemes of parts and assemblies.

To achieve such a goal, how to draw a star using a ruler, you must have knowledge of the simplest mathematical formulas that are fundamental in geometry. You will also need to be able to count on a calculator. But the most important thing is logical thinking.

The work is not difficult, but it will require precision and scrupulousness. The effort spent will be rewarded with a good symmetrical, and therefore beautiful, image of a five-pointed star.

classical technique

The most famous way to draw a star with a compass, ruler and protractor is quite simple.

For this technique, you will need several tools: a compass or protractor, a ruler, a simple pencil, an eraser and a sheet of white paper.

To understand how to draw a star beautifully, you should act sequentially, stage by stage.

You can use special calculations in your work.

Figure calculation

At this stage of drawing the correct star, the contours of the finished figure appear.

If everything is done correctly, the resulting image will be smooth. This can be checked visually by rotating a sheet of paper and evaluating the shape. It will remain the same with every turn.

The main contours are drawn with a ruler and a simple pencil more clearly. All auxiliary lines are removed.

To understand how to draw a star in stages, you should carry out all the actions thoughtfully. In case of an error, you can correct the drawing with an eraser or carry out all the manipulations again.

Registration of work

The finished form can be decorated in a variety of ways. The main thing is not to be afraid to experiment. Fantasy will prompt an original and beautiful image.

You can decorate the drawn even star with a simple pencil or use a wide variety of colors and shades.

To figure out how to draw the right star, you need to stick to perfect lines in everything. Therefore, the most popular design option is to divide each ray of the figure into two equal parts with a line extending from the top to the center.

You can not separate the sides of the star with lines. It is allowed to simply paint over each ray of the figure with a darker shade from one side.

This option will also be the answer to the question of how to draw the correct star, because all its lines will be symmetrical.

If desired, with the aesthetic design of the figure, you can add an ornament or other various elements. By adding circles to the tops, you can get the sheriff's star. By applying a smooth shading of the shadow sides, you can get a starfish.

This technique is the most common, as it effortlessly allows you to understand how to draw a five-pointed star in stages. Without resorting to complex mathematical calculations, it is possible to obtain a correct, beautiful image.

Having considered all the ways of how to draw a star with a ruler, you can choose the most suitable one for yourself. The most popular is the geometric phased method. It is quite simple and effective. Applying fantasy and imagination, it is possible from the obtained correct, beautiful shape create an original composition. There are a lot of design options for drawing. But you can always come up with your own, the most unusual and memorable story. Most importantly, don't be afraid to experiment!

A regular pentagon can be constructed using a compass and straightedge, or by inscribing it within a given circle, or by building it from a given side. This process is described by Euclid in his Elements around 300 BC. e.

Here is one method for constructing a regular pentagon in a given circle:

1. Construct a circle in which the pentagon will be inscribed and designate its center as O. (This is the green circle in the diagram on the right).

Pick a point on the circle A, which will be one of the vertices of the pentagon. Draw a line through O And A.
Construct a line perpendicular to the line OA passing through the point O. Designate one of its intersections with the circle as a point B.
Build a point C midway between O And B.
C through a point A. Mark its intersection with the line OB(inside the original circle) as a point D.
Draw a circle centered at A through a point D. Designate its intersections with the original (green circle) as points E And F.
Draw a circle centered at E through a point A G.
Draw a circle centered at F through a point A. Designate its other intersection with the original circle as a point H.
Build a regular pentagon AEGHF.

icosahedron

icosahedron- one of the five Platonic solids, in simplicity following the tetrahedron and octahedron. They are united by the fact that the faces of each are equilateral triangles. When making a model of an icosahedron, you can choose any of two spectacular possibilities for the distribution of five colors.

First, the icosahedron can be colored so that each vertex has all five colors (however, in this case, the opposite faces will not be colored the same).

Another method provides the opposite faces with the same colors, but each vertex, with the exception of the two polar ones, will have one color repeated around the circle. Both colorings are very interesting and useful for our purposes, because many of the uniform polyhedra described below have icosahedral symmetry.

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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 construct, 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.

Instruction

Construct another diameter perpendicular to the MH diameter. To do this, use a compass to draw arcs from points M and H with the same radius. Choose a radius such that both arcs intersect with each other and with the given circle at one point. This will be the first point A of the second dimeter. Draw a straight line through it and point O. You get the diameter AB, perpendicular to the straight line MH.

Find the midpoint of the radius BO. To do this, use a compass with a circle radius to draw an arc from point B so that it intersects the circle at two points C and P. Draw a straight line through these points. This straight line will divide the radius of VO exactly in half. Put a point K at the intersection of SR and BO.

Connect points M and K with a line segment. Set a distance on the compass equal to the segment MK. From point M, draw an arc so that it intersects the radius AO. Place a point E at this intersection. The resulting distance ME corresponds to the length of one side of the inscribed pentagon.

Construct the remaining vertices of the pentagon. To do this, set the distance of the legs of the compass equal to the segment ME. From the first vertex of the pentagon M, draw an arc to the intersection with the circle. The intersection point will be the second vertex F. From the resulting point, in turn, also draw an arc of the same radius with the intersection of the circle. Get the third vertex of the pentagon G. Construct the other points S and L in the same way.