How to draw a pentagon with a ruler. Dividing a circle into equal parts and inscribing regular polygons
Positive pentagon is a polygon in which all five sides and all five angles are equal. It is easy to describe a circle around it. Erect pentagon and this circle will help.
Instruction
1. First of all, you need to build a circle with a compass. Let the center of the circle coincide with point O. Draw axes of symmetry perpendicular to each other. At the intersection point of one of these axes with the circle, put a point V. This point will be the top of the future pentagon a. Place point D at the point of intersection of another axis with the circle.
2. On the segment OD, find the middle and mark point A in it. Later, you need to draw a circle with a compass centered at this point. In addition, it must pass through the point V, that is, with radius CV. Designate the point of intersection of the axis of symmetry and this circle as B.
3. Later, with the help compass draw a circle of the same radius, placing the needle at point V. Designate the intersection of this circle with the original one as point F. This point will become the 2nd vertex of the future true pentagon a.
4. Now it is necessary to draw the same circle through point E, but with the center at F. Designate the intersection of the circle just drawn with the original one as point G. This point will also become one of the vertices pentagon a. Similarly, you need to build another circle. Its center is in G. Let it intersect with the original circle H. This is the last vertex of a true polygon.
5. You should have five vertices. It remains easy to combine them along the line. As a result of all these operations, you will get a positive inscribed in a circle. pentagon .
Building positive pentagons allowed with the support of a compass and straightedge. True, the process is rather long, as, however, is the construction of any positive polygon with an odd number of sides. Modern computer programs allow you to do this in a few seconds.
You will need
- - A computer with AutoCAD software.
Instruction
1. Find the top menu in the AutoCAD program, and in it the "Basic" tab. Click on it with the left mouse button. The Draw panel appears. Various types of lines will appear. Select a closed polyline. It is a polygon, it remains only to enter the parameters. AutoCAD. Allows you to draw a variety of regular polygons. The number of sides can be up to 1024. You can also use the command line, depending on the version, by typing "_polygon" or "multi-angle".
2. Regardless of whether you use the command line or context menus, you will see a window on the screen in which you are prompted to enter the number of sides. Enter the number "5" there and press Enter. You will be prompted to determine the center of the pentagon. Enter the coordinates in the box that appears. It is allowed to denote them as (0,0), but there may be any other data.
3. Select the required construction method. . AutoCAD offers three options. A pentagon can be described around a circle or inscribed in it, but it is also allowed to build it according to a given side size. Select the desired option and press enter. If necessary, set the radius of the circle and also press enter.
4. A pentagon on a given side is first constructed correctly in the same way. Select Draw, a closed polyline, and enter the number of sides. Right-click to open the context menu. Press the command "edge" or "side". In the command line, type the coordinates of the initial and final points of one of the sides of the pentagon. Later this pentagon will appear on the screen.
5. All operations can be performed with command line support. Say, to build a pentagon along the side in the Russian version of the program, enter the letter "c". In the English version it will be "_e". In order to build an inscribed or circumscribed pentagon, enter later the number of sides of the letter "o" or "c" (or the English "_s" or "_i")
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Useful advice
With such a simple method, it is possible to build not only a pentagon. In order to construct a triangle, you need to spread the legs of the compass to a distance equal to the radius of the circle. After that, place the needle at any point. Draw a thin auxiliary circle. Two points of intersection of the circles, as well as the point where the leg of the compass was, form three vertices of a positive triangle.
The task of constructing a true pentagon is reduced to the task of dividing the circle into five equal parts. From the fact that a true pentagon is one of the figures that contains the proportions of the golden section, painters and mathematicians have long been interested in its construction. Several methods have now been discovered for constructing a true polygon inscribed in a given circle.
You will need
- - ruler
- - compasses
Instruction
1. Apparently, if we build a true decagon, and then combine its vertices through one, we get a pentagon. To construct a decagon, draw a circle with a given radius. Mark its center with the letter O. Draw two radii perpendicular to each other, in the figure they are designated as OA1 and OB. Divide the radius OB in half with the help of a ruler or by dividing the segment in half with the help of a compass. Construct a small circle with center C in the middle of segment OB with a radius equal to half OB. Unite point C with point A1 on the starting circle using a ruler. Segment CA1 intersects the auxiliary circle at point D. Segment DA1 is equal to the side of a regular decagon inscribed in this circle. With a compass, sweep this segment on a circle, then combine the intersection points through one and you will get a positive pentagon.
2. Another method was discovered by the German artist Albrecht Dürer. In order to construct a pentagon according to his method, start again by constructing a circle. Again sweep its center O and draw two perpendicular radii OA and OB. Divide the radius OA in half and mark the middle with the letter C. Place the needle of the compass at point C and open it to point B. Draw a circle of radius BC until it intersects with the diameter of the initial circle, where radius OA lies. Designate the point of intersection D. Segment BD is the side of the positive pentagon. Set aside this segment five times on the initial circle and unite the intersection points.
3. If you want to build a pentagon along its given side, then you need the 3rd method. Draw the side of the pentagon along the ruler, mark this segment with the letters A and B. Divide it into 6 equal parts. From the middle of segment AB, draw a ray perpendicular to the segment. Construct two circles with radius AB and centers at A and B, as if you were going to cut the segment in half. These circles intersect at point C. Point C lies on the ray emanating perpendicularly upward from the middle of AB. Set a distance from C up along this ray equal to 4/6 of the length of AB, designate this point D. Construct a circle of radius AB centered at point D. The intersection of this circle with the two auxiliary ones built earlier will give the last two vertices of the pentagon.
The topic of dividing a circle into equal parts in order to build correct inscribed polygons has long occupied the minds of ancient scientists. These theses of construction with the use of a compass and straightedge were expressed in the Euclidean Elements. However, only two millennia later this problem was completely solved not only graphically, but also mathematically.
Instruction
1. Approximate construction of a positive pentagon A. Dürer's method, with the help of a compass and ruler (through two circles with a common radius equal to the side pentagon).
2. Building the right pentagon based on a positive decagon inscribed in a circle (combining the vertices of the decagon through one).
3. Plotting via Calculated Internal Angle pentagon with the support of a protractor and a ruler (the sum of the angles of a convex n-gon is equal to Sn=180°(n – 2), since all angles of a positive polygon are equal). With n=5, S5=5400, then the angle value is 1080. (36005=720). Their intersection with the circle will give a segment equal to the side pentagon .
4. Another easy one graphic method: divide the diameter of the given circle AB into three parts (AC=CD=DE). From point D, lower the perpendicular to the intersection with the circle at points E, F. Drawing straight lines through the segments EC and FC until they intersect with the circle, we get points G, H. Points G, E, B, F, H are the vertices of the positive pentagon .
5. Construction with support for Bion's technique (which allows one to construct a true polygon inscribed in a circle with any number of sides n according to a given ratio). Let's say: for n=5. Let us construct a positive triangle ABC, where AB is the diameter of the given circle. Let's find the point D on AB, according to the further relation: AD: AB = 2: n. With n=5, AD=25*AB. Let us draw a straight line through CD until it intersects with the circle at point E. Segment AE is the side of the right inscribed pentagon.When n=5,7,9,10, the construction error does not exceed 1%. As n increases, the approximation error increases, but remains less than 10.3%.
6. Construction on a given side according to the method of L. Da Vinci (using the relationship between the side of the polygon (an) and the apothem (ha): an / 2: ha \u003d 3 / (n-1), which can be expressed as follows: tg180 ° / n \u003d 3 /(n-1)).
7. A general method for constructing positive polygons along a given side according to the method of F. Kovarzhik (1888), based on the rule of L. da Vinci. An integral method for constructing a positive n-gon based on the Thales theorem. primitive and beautiful.
There are two main methods for constructing a regular polygon with five sides. Both of them involve the use of a compass, ruler, and pencil. The 1st method is an inscription pentagon into a circle, and the 2nd method is based on the given side length of your future geometric figure.
You will need
- Compasses, ruler, pencil
Instruction
1. 1st construction method pentagon considered more "typical". First, build a circle and somehow designate its center (usually the letter O is used for this). After that, draw the diameter of this circle (let's call it AB) and divide one of the 2 radii obtained (say, OA) exactly in half. The middle of this radius is denoted by the letter C.
2. From the point O (the center of the initial circle), draw another radius (OD), one that will be strictly perpendicular to the previously drawn diameter (AB). After that, take a compass, put it at point C and measure the distance to the intersection of the new radius with the circle (CD). Set aside the same distance on the diameter AB. You will get a new point (let's call it E). Measure with a compass the distance from point D to point E - it will be equal to the length of the side of your future pentagon .
3. Put the compass at point D and set aside a distance on the circle equal to the segment DE. Repeat this procedure 3 more times, and after that, unite point D and 4 new points on the initial circle. The resulting figure will be a true pentagon.
4. To construct a pentagon using a different method, first draw a line segment. Let's say it will be a segment AB with a length of 9 cm. Next, divide your segment into 6 equal parts. In our case, the length of each part will be 1.5 cm. Now take a compass, put it at one of the ends of the segment and draw a circle or an arc with a radius, equal to the length segment (AB). After that, rearrange the compass to the other end and repeat the operation. The resulting circles (or arcs) will intersect at one point. Let's call her C.
5. Now take a ruler and draw a straight line through point C and the center of line segment AB. After that, starting from point C, set aside on this straight line a segment that is 4/6 of segment AB. The 2nd end of the segment will be denoted by the letter D. Point D will be one of the peaks of the future pentagon. From this point, draw a circle or an arc with a radius equal to AB. This circle (arc) will intersect the circles (arcs) you previously constructed at the points that are the two missing vertices pentagon. Unite these points with vertices D, A and B, and building a positive pentagon will be finished.
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Ray - it is a straight line drawn from a point and has no end. There are other definitions of a ray: say, "... it is a straight line bounded by a point on one side." How to draw a beam positively and what drawing supplies do you need?
You will need
- Sheet of paper, pencil and ruler.
Instruction
1. Take a sheet of paper and mark a dot in an arbitrary place. After that, attach a ruler and draw a line, starting from the indicated point and continuing to infinity. This drawn line is called a ray. Now mark another point on the beam, for example, with the letter C. The line from the original to point C will be called a segment. If you primitively draw a line and do not really notice one point, then this line will not be a ray.
2. It is not more difficult to draw a beam in any graphic editor or in the same MSOffice than manually. For example, take Microsoft program Office 2010. Go to the "Insert" section and select the "Shapes" element. Select the "Line" shape from the drop-down list. The cursor will then change to a cross. To draw a straight line, press the "Shift" key and draw a line of the desired length. Immediately after the style, the Format tab will open. Now you have drawn a primitively straight line and no fixed point, and based on the definition, the ray should be limited to a point on one side.
3. To make a point at the beginning of a line, do the following: select the drawn line and call the context menu by pressing the right mouse button.
4. Select Shape Format. Select "Line Type" from the menu on the left. Next, find the heading "Line Options" and select "Start Type" in the form of a circle. There you can also adjust the thickness of the start and end lines.
5. Remove the selection from 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 make a field where the inscription will be located. After writing the inscription, click on an empty space and it will be activated.
6. The beam is safely drawn and it took every few minutes. Drawing a beam in other editors is carried out according to the same thesis. When the Shift key is pressed, proportional figures will invariably be drawn. Nice use.
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Note!
The ratio of the diagonal of a true pentagon to its side is the golden ratio (irrational number (1+√5)/2). All of the five internal angles of the pentagon are 108°.
Useful advice
If you combine the vertices of a true pentagon with diagonals, you get a pentagram.
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.
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 based on the fact that if you build 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.
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 cuboid with edges having lengths 1.618, 1 and 0.618.
Now consider the proof offered by Euclid in the Elements.
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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 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.
Asymptotic behavior of the sequence, damped oscillations of its ratio about irrational numberФ can become more understandable if you show the relationship 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 back in ancient times called the golden ratio.
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 a 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.
AT general view 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 the aircraft with electromagnetic energy sources, rectangular cells are created with the proportion of the golden section.
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; Number 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 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...