Friedrich V.A. 1

Dementyeva V.V. 1

1 Municipal budgetary educational institution"Average comprehensive school No. 6 ", Aleksandrovsk, Perm Territory

The text of the work is placed without images and formulas.
Full version work is available in the tab "Files of work" in PDF format

Introduction

“Standing in front of a black board and drawing on it

different shapes with chalk,

I was suddenly struck by the thought:

Why is symmetry pleasing to the eye?

What is symmetry?

This is an innate feeling, I answered myself "

L.N. Tolstoy

In the 6th grade mathematician textbook, author Nikolsky SM, on pages 132 - 133 section Additional tasks to chapter number 3, there are tasks for studying figures on a plane that are symmetric about a straight line. I was interested in this topic, I decided to complete the tasks and study this topic in more detail.

The object of research is symmetry.

The subject of research is symmetry as the fundamental law of the universe.

Which hypothesis will I test:

I believe that axial symmetry is not only a mathematical and geometric concept, and is used only for solving relevant problems, but also is the basis of harmony, beauty, balance and stability. The principle of symmetry is used in almost all sciences, in our Everyday life and is one of the "cornerstone" laws on which the universe as a whole is based.

Relevance of the topic

The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of its development. In our time, it is probably difficult to find a person who does not have any idea of ​​symmetry. The world in which we live is filled with the symmetry of houses, streets, creations of nature and man. We meet with symmetry literally at every step: in technology, art, science.

Therefore, knowledge and understanding about symmetry in the world around us is mandatory and necessary, which will be useful in the future for studying others. scientific disciplines... This is the relevance of the topic I have chosen.

Goal and tasks

Purpose of work: find out what role symmetry plays in human everyday life, in nature, architecture, everyday life, music and other sciences.

To achieve this goal, I need to complete the following tasks:

1. Find the information you need, literature and photographs. Establish the greatest amount of data necessary for my work, using the sources available to me: textbooks, encyclopedias or other media relevant to a given topic.

2. Give general concept about symmetry, types of symmetry and the history of the origin of the term.

3. To confirm your hypothesis, create crafts and conduct an experiment with these figures that have symmetry and are not asymmetrical.

4. Demonstrate and present the results of observations in your research.

For the practical part research work I need to do the following, for which I made a work plan:

1. Create with your own hands crafts with given properties - symmetrical and non-symmetrical models, composition, using colored paper, cardboard, scissors, felt-tip pens, glue, etc .;

2. Conduct an experiment with my crafts, with two symmetry options.

3. Research, analyze and systematize the results obtained by compiling a table.

4. For a visual and interesting consolidation of the knowledge gained, using the "Paint 3 D" application, create drawings for clarity, as well as draw pictures, with tasks - draw a symmetrical half (starting with simple drawings and ending with complex ones) and combine them, creating an electronic book.

Research methods:

1. Analysis of articles and all information about symmetry.

2. Computer modeling (photo processing by means of a graphic editor).

3. Generalization and systematization of the data obtained.

Main part.

Axial symmetry and the concept of perfection

Since ancient times, man has developed an idea of ​​beauty and tried to comprehend the meaning of perfection. All the creations of nature are beautiful. People are beautiful in their own way, animals and plants are delightful. The sight pleases the eye precious stone or a crystal of salt, it's hard not to admire a snowflake or a butterfly. But why is this happening? It seems to us that the appearance of the objects is correct and complete, the right and left half of which looks the same.

Apparently, people of art were the first to think about the essence of beauty.

For the first time, this concept was substantiated by artists, philosophers and mathematicians. Ancient Greece... Ancient sculptors who studied the structure of the human body as far back as the 5th century BC. began to use the concept of "symmetry". This word has Greek origin and means harmony, proportionality and similarity of the arrangement of the constituent parts. The ancient Greek thinker and philosopher Plato argued that only that which is symmetrical and proportionate can be beautiful.

Indeed, those phenomena and forms that have proportionality and completeness are “pleasing to the eye”. We call them correct.

Symmetry types

In geometry and mathematics, three types of symmetry are considered: axial symmetry (relative to a straight line), central (relative to a point) and mirror (relative to a plane).

Axial symmetry as a mathematical concept

Points are symmetrical about a certain straight line (axis of symmetry) if they lie on a straight line perpendicular to this straight line and at the same distance from the axis of symmetry.

A figure is considered symmetric with respect to a straight line if, for each point of the figure in question, a point symmetric for it with respect to a given straight line is also located on this figure. In this case, the straight line is the axis of symmetry of the figure.

Shapes that are symmetrical about a straight line are equal. If a geometric figure is characterized by axial symmetry, the definition of mirror points can be visualized by simply bending it along the axis and folding equal halves "face to face". The sought-for points will touch in this case.

Examples of the axis of symmetry: the bisector of an undeveloped angle of an isosceles triangle, any straight line drawn through the center of a circle, etc. If a geometric figure is characterized by axial symmetry, the definition of mirror points can be visualized by simply bending it along the axis and folding equal halves "face to face". The sought-for points will touch in this case.

Shapes can have several axes of symmetry:

· The axis of symmetry of the angle is the straight line on which its bisector lies;

· The axis of symmetry of a circle and a circle is any straight line passing through their diameter;

An isosceles triangle has one axis of symmetry, an equilateral triangle has three axes of symmetry;

· A rectangle has 2 axes of symmetry, a square - 4, a rhombus - 2 axes of symmetry.

An axis of symmetry is an imaginary line dividing an object into symmetrical parts. It is shown in my picture for clarity.

There are figures that do not have any axis of symmetry. Such figures include a parallelogram, different from a rectangle and a rhombus, a versatile triangle.

Axial symmetry in nature

Nature is wise and rational; therefore, almost all of her creations have a harmonious structure. This applies to both living beings and inanimate objects.

Close observation shows that symmetry is the basis of the beauty of many forms created by nature. Leaves, flowers, fruits have a pronounced symmetry. Their mirror, radial, central, axial symmetry are obvious. It is largely due to the phenomenon of gravity.

The geometric shapes of crystals with their flat surfaces are an amazing natural phenomenon. However, the true physical symmetry of the crystal is manifested not so much in its appearance how much is in the internal structure of the crystalline substance.

Axial symmetry in the animal kingdom

Symmetry in the world of living things is manifested in the regular arrangement of identical parts of the body relative to the center or axis. Axial symmetry is more common in nature. It conditions not only general structure organism, but also the possibility of its subsequent development. Each type of animal has a characteristic color. If a drawing appears in the color, then, as a rule, it is duplicated on both sides.

Axial symmetry and man

If you look at any Living being, the symmetry of the body structure is immediately evident. Human: two arms, two legs, two eyes, two ears, and so on.

This means that there is a certain line along which animals and people can be visually "divided" into two identical halves, that is, axial symmetry is the basis of their geometric structure.

As can be seen from the above examples, nature creates any living organism not chaotically and senselessly, but according to the general laws of the world order, because in the Universe nothing has a purely aesthetic, decorative purpose. This is due to a natural necessity.

Of course, mathematical accuracy is rarely inherent in nature, but the similarity of the elements of the organism is still striking.

Symmetry in architecture

Since ancient times, architects have been well aware of mathematical proportion and symmetry, and have used them in the construction of architectural structures. For example, the architecture of the Russians Orthodox churches and cathedrals of Russia: the Kremlin, Cathedral of Christ the Savior, Moscow, Kazan and Isaac Cathedral St. Petersburg, etc.

And also other world famous sights, many of which are in all countries of the world, we can see now: Egyptian pyramids, Louvre, Taj Mahal, Cologne Cathedral, etc. All of them, as we can see, have symmetry.

Symmetry in music

I study at a music school, it was interesting for me to find examples of symmetry in this area. Not only musical instruments have obvious symmetry, but parts of musical works sound in a certain order, in accordance with the score and the composer's intention.

For example, a reprise - (French reprise, from reprendre - to renew). Repetition of a topic or a group of topics after the stage of its (their) development or presentation of a new thematic material.

Also in one-dimensional repetition in time through equal intervals is the musical principle of rhythm.

Symmetry in technology

We live in a rapidly changing high-tech, information society, and do not think why some objects and phenomena around us awaken a sense of beauty, while others do not. We do not notice them, we don’t even think about their properties.

But besides this, these technical and mechanical devices, parts, mechanisms, aggregates will not be able to work correctly and generally function if symmetry is not observed, or rather, a certain axis, in mechanics this is the center of gravity.

Center balance, in this case, is a must technical requirement, the observance of which is strictly regulated by GOST or TU and must be observed.

Symmetry and space objects

But, perhaps, the most mysterious, exciting the minds of many, since ancient times, are space objects. Which also have symmetry - the sun, moon, planets.

This chain can be continued, but we are now talking about something unified: that axial symmetry is the fundamental law of the universe, is the basis of beauty, harmony and proportionality, and in relation to this with mathematics.

Practical part

Having found the necessary information, having studied the literature, I was convinced of the correctness of my hypothesis and concluded that in the eyes of a person, asymmetry is most often associated with incorrectness or inferiority. Therefore, in most of the creations of human hands, symmetry and harmony can be traced as a necessary and obligatory requirement.

This can be clearly seen in my drawing, where a piglet is depicted, with disproportionate body parts, which immediately catches the eye!

And only after you take a closer look at him, do you find him cute?

Despite the fact that this topic is well-known, well-studied, but all these data are considered separately in each discipline. Generalized data that the principle of symmetry is used, and it is on it that many other sciences are based, and I have not met their relationship with mathematics.

Therefore, I decided to prove my statement using the simplest and most accessible way for me. That solution, I believe, would be to conduct an experiment with tests.

For clear evidence that asymmetric models are not stable, do not have the necessary requirements and vital skills, and to confirm my hypothesis, I need to create crafts, drawings and composition:

Option 1 - symmetrical about the axis;

Option 2 - with a clear violation of symmetry.

Since I believe that such an imbalance will be clearly visible in the following examples, for which I created origami crafts (an airplane and a frog) from colored paper. For the purity of the experiment, they are made of the same colored paper and tested under the same conditions. And the composition "Lighthouse", where the lighthouse is made of empty plastic bottle, pasted over with colored paper. To decorate the composition, toy figures of a person, models of a sailing ship and a boat were used, decorative stones, and to simulate light, I used a battery-powered element.

I conducted tests with these crafts, recorded all the indicators and entered them into the table (all indicators can be viewed in Appendix No. 1, pp. 18 - 21).

All crafts were done in compliance with safety precautions (Appendix No. 2, p. 21)

I analyzed all the data obtained, that's what I got.

Analysis of the received data

Experiment # 1

Trial- Long jump of frogs, measuring this distance.

The Green (symmetrical) frog jumps evenly, a greater distance, while the Red (not symmetrical) frog has never jumped exactly, always with a turn or flip to the side, at a distance of 2 - 3 times less.

Thus, we can conclude that such an animal will not be able to hunt quickly or, on the contrary, run away, effectively get food, which reduces the chances of survival, this proves that everything in nature is balanced, proportional, correct - symmetric.

Experiment # 2

Test type- launching aircraft in flight and measuring the distance of the flight length.

Plane No. 1 "Pink" (symmetrical) flies 10 times, 8 times evenly and directly, on maximum length, (that is, the entire length of my room), and the flight trajectory of aircraft No. 2 "Orange" (not symmetrical) out of 10 times - never flew smoothly, always with a turn or overturn, at a shorter distance. That is, if it were a real plane, then it would not be able to fly smoothly, in the right direction... Such a flight would be very inconvenient or even dangerous for humans (as well as for birds), and cars and other vehicles would not be able to drive, swim, etc. in the necessary direction.

Experiment # 3

Test type - checking the stability of the building "Mayak", with a decrease in the angle of inclination of the structure, relative to the surface.

1. Having created the composition "Lighthouse", I installed it directly, i.e. perpendicular (at an angle of 90 0) relative to the walls of the structure to the surface. This construction stands upright, withstands the installed light element and a human figurine.

2. To carry out the experiment further, I needed to outline the base of the tower at angles equal to 10 0.

Then I cut off an angle equal to 10 0 from the base.

At an angle of 80 0, the building stands crooked, staggers, but withstands additional load.

3. Cutting off another 10 0, I got an angle of inclination of 70 0, at which my whole structure collapses.

This experience proves that the historically established tradition of building at right angles and maintaining the symmetry of the building itself is necessary condition for sustainable, reliable construction and operation of architectural buildings and structures.

For illustrative example axial symmetry and evidence of the statement that a person perceives any objects around him, images of animals, etc. only symmetrically, that is, when both sides, "halves" are the same, equal, I created an electronic coloring book that can be printed by compiling a children's coloring book. This manual will help everyone to better understand the topic, spend their free time interesting and happy. (Title page shown in this figure, the rest of the figures are located in Appendix No. 3, pp. 21-24).

The experiments I have carried out prove that symmetry is not only a mathematical and geometric concept, but is a sphere, our living environment, a kind of technical requirement, as well as a necessary condition for survival in general, both for humans and animals. Symmetry brings it all together, and goes far beyond conventional science!

Conclusion

Conclusions:

I found out that symmetry is one of the main components in a person's daily life, in household items, in architecture, technology, in nature, music, science, etc.

Result:

I found the necessary information, proved my hypothesis, verified and confirmed it empirically. I created crafts, composition, drawings and electronic coloring for a visual experiment.

I found out that all the laws of nature - biological, chemical, genetic, astronomical - are associated with symmetry. Practically, everything that surrounds us, that is created by man, is subordinated to the principles of symmetry that are common for all of us, since they have an enviable consistency. Thus, balance, identity as a principle has a universal scale.

Can we say that symmetry is the fundamental law on which the basic laws of science are based? Maybe yes.

This secret was tried to comprehend the great thinkers of mankind. Today we also plunged into the solution to this mystery.

One of the famous mathematicians, Hermann Weil, wrote that "symmetry is the idea through which man has been trying for centuries to comprehend and create order, beauty and perfection."

Maybe we found the secret to creating beauty, perfection, or even creating the basic laws of the universe? Is it symmetry?

Applications

Appendix No. 1 Test table:

Experiment # 1
Attempt #	Test type	"Green Frog" (symmetrical)	Test result and characteristics "Red Frog" (not symmetrical)
	Long jump of a frog (measurement in cm.)		6.0 to the left
		14.4 with a slight turn to the right	9.0 flip back
		10.5 almost exactly	2.0 coup
		9.5 with a slight turn to the right	5.0 flip to the left
		10.6 with a slight turn to the right	3.0 to the left
		9.0 coup	9.0 turn left
		13.5 almost exactly	1.5 backward, with a turn to the left
			9.5 left with a coup
		21.2 almost exactly	4.5 left with a coup
Experiment # 2
		Plane "Pink" (Symmetric)	Airplane "Orange" (Not symmetrical)
	Launching an airplane in length Maximum (5.1 meters)	5.1 with 2 flips	3.04 with flips to the right
			2.78 with flips to the right
		5.1 tilted to the right	3, 65 with flips to the right
		5.1 tilted to the right	1.51 almost exactly
		5.1 almost exactly	4.73 with flips to the right
		5.1 tilted to the left	3.82 turn right
		5.1 almost exactly	3.41 with coups
		5.1 almost exactly	3.37 turn left
		5.1 with flip	3.51 with flips to the left
		5.1 almost exactly	3.19 with flips to the right

Experiment # 3
Attempt #	Characteristics of properties object	Type and characteristics of the test	Result
	The structure is perpendicular to the surface (i.e. at an angle of 90 0)	Additional load setting: luminous element and human toy figure	The lighthouse stands level, securely
	Angle 80 0	From the base of the lighthouse, I sketched and cut off an angle of 10 0	The lighthouse can withstand the load, but it is unreliable, staggers
	At an angle of 70 0	From the base of the lighthouse, I once again cut 10 0	The structure falls and collapses

Appendix No. 2

When making my crafts, safety precautions were followed, namely:

The scissors or knife should be well sharpened and adjusted.

It is necessary to store in a certain and safe place or box.

When using scissors (knife), you should not be distracted, you need to be as attentive and disciplined as possible.

Passing the scissors (knife), hold them by the closed blades (point).

Put the scissors (knife) on the right with closed blades (tip) directed away from you.

When cutting, the narrow blade of the scissors (knife edge) should be down.

Wash hands after using the glue.

Appendix No. 3

Coloring e-book

Symmetry-

This means that one part of the object is similar to another.

Axial symmetry is symmetry about a straight line (line).

An axis of symmetry is an imaginary line dividing an object into symmetrical parts. It is shown in the figures for clarity.

In this book, you need to finish the drawings by connecting the dots.

Then you can paint what happened.

Try to finish these drawings:

Heart

Triangle Small house

Asterisk Leaf

Mouse Christmas tree

DogLock

TO Besides axial symmetry, there is also symmetry about a point.

This ball is symmetrical

And another kind of symmetry is mirror symmetry.

Mirror symmetry

it is symmetry about the plane. For example, relative to the mirror.

Symmetry is -

Used Books

2. Hermann Weil "Symmetry" (Publishing house "Nauka" main edition of physical and mathematical literature, Moscow 1968)

4. My drawings and photographs.

5. Handbook of mechanical engineering, volume 1, (State Scientific and Technical Publishing House of Mechanical Engineering Literature, Moscow 1960)

6. Photos and drawings from the Internet.

Goals:

• educational:
  • give an idea of ​​symmetry;
  • to acquaint with the basic types of symmetry on the plane and in space;
  • develop strong skills in building symmetrical figures;
  • expand the understanding of known figures, introducing the properties associated with symmetry;
  • show the possibilities of using symmetry when solving different tasks;
  • consolidate the knowledge gained;
• general educational:
  • teach yourself to set yourself up for work;
  • teach to control yourself and your neighbor on your desk;
  • teach how to evaluate yourself and your deskmate;
• developing:
  • to intensify independent activity;
  • develop cognitive activity;
  • teach to generalize and systematize the information received;
• educational:
  • to instill in students a "sense of the shoulder";
  • educate communication;
  • instill a culture of communication.

DURING THE CLASSES

In front of each are scissors and a sheet of paper.

Exercise 1(3 min).

“Let's take a sheet of paper, fold it into pieces and cut out some figurine. Now expand the sheet and look at the fold line.

Question: What is the function of this line?

Supposed answer: This line divides the shape in half.

Question: How are all the points of the figure located on the two resulting halves?

Supposed answer: All points of the halves are at the same distance from the fold line and at the same level.

- This means that the fold line divides the figure in half so that 1 half is a copy of 2 halves, i.e. this line is not simple, it has a remarkable property (all points are at the same distance relative to it), this line is the axis of symmetry.

Assignment 2 (2 minutes).

- Cut out a snowflake, find the axis of symmetry, characterize it.

Assignment 3 (5 minutes).

- Draw a circle in a notebook.

Question: Determine how the axis of symmetry runs?

Supposed answer: Differently.

Question: So how many axes of symmetry does a circle have?

Supposed answer: Many.

- That's right, a circle has many axes of symmetry. The same remarkable figure is the ball (spatial figure)

Question: What other figures have more than one axis of symmetry?

Supposed answer: Square, rectangle, isosceles and equilateral triangles.

- Consider volumetric figures: cube, pyramid, cone, cylinder, etc. These figures also have an axis of symmetry. Determine how many axes of symmetry a square, rectangle, equilateral triangle and the proposed volumetric figures have?

I distribute to students the halves of plasticine figures.

Assignment 4 (3 min).

- Using the information received, fill in the missing part of the figure.

Note: the figure can be both planar and volumetric. It is important that the students determine how the axis of symmetry goes and complete the missing piece. The correctness of the execution is determined by the neighbor on the desk, assesses how correctly the work has been done.

A line is laid out of a lace of the same color on the desktop (closed, open, with self-intersection, without self-intersection).

Assignment 5 (group work 5 min).

- Determine visually the axis of symmetry and build the second part from a lace of a different color relative to it.

The correctness of the work performed is determined by the students themselves.

The elements of the drawings are presented to the students

Assignment 6 (2 minutes).

Find the symmetrical parts of these patterns.

To consolidate the material covered, I propose the following tasks, provided for 15 minutes:

Name all equal elements of the triangle KOR and KOM. What is the appearance of these triangles?

2. Draw in a notebook several isosceles triangles with a common base equal to 6 cm.

3. Draw line segment AB. Construct a straight line perpendicular to line segment AB and passing through its middle. Mark points C and D on it so that the quadrangle ACBD is symmetrical with respect to line AB.

- Our initial ideas about the form date back to a very distant era of the ancient Stone Age - the Paleolithic. For hundreds of millennia of this period, people lived in caves, in conditions that did not differ much from the life of animals. Humans made tools for hunting and fishing, developed languages ​​to communicate with each other, and in the late Paleolithic era adorned their existence, creating works of art, figurines and drawings that reveal a wonderful sense of form.
When there was a transition from simple gathering of food to active production, from hunting and fishing to agriculture, humanity enters a new stone Age, in the Neolithic.
Neolithic man had a keen sense of geometric shape. The burning and painting of earthen vessels, the making of reed mats, baskets, fabrics, and later - the processing of metals developed ideas about planar and spatial figures. Neolithic ornaments were pleasing to the eye, revealing equality and symmetry.
- Where does symmetry occur in nature?

Supposed answer: wings of butterflies, beetles, tree leaves ...

“Symmetry can be seen in architecture as well. When constructing buildings, builders adhere to symmetry.

That is why the buildings are so beautiful. Also, an example of symmetry is a person, animals.

Home assignment:

1. Come up with your own ornament, depict it on an A4 sheet (you can draw it in the form of a carpet).
2. Draw butterflies, mark where the elements of symmetry are present.

In a broad sense, symmetry refers to the preservation of something unchanged during some transformations. Some geometric shapes also have this property.

Geometric symmetry

When applied to a geometric shape, it means that if the given shape is transformed - for example, rotated - some of its properties will remain the same.

The ability to do these transformations differs from shape to shape. For example, a circle can be rotated as much as you like around a point located in its center, it will remain a circle, nothing will change for it.

Symmetry can be explained without resorting to rotation. It is enough to draw a straight line through the center of the circle and build a segment perpendicular to it at any place of the figure, connecting two points on the circle. The point of intersection with a straight line will be divided into two parts, which will be equal to each other.

In other words, the straight line divided the figure into two equal parts. The points of the parts of the figure, located on the lines perpendicular to the given one, are at an equal distance from it. This line will be called the axis of symmetry. Symmetry of this kind - - is called axial symmetry.

Number of axes of symmetry

The number will be different. For example, a circle and a ball have many such axes. An equilateral triangle will have a perpendicular axis of symmetry, lowered to each side, therefore, it has three axes. A square and a rectangle can have four axes of symmetry. Two of them are perpendicular to the sides of the quadrangles, and the other two are diagonals. But the isosceles triangle has only one axis of symmetry, located on the honey's equal sides.

Axial symmetry is also found in nature. It can be seen in two ways.

The first type is radial symmetry, which implies the presence of several axes. It is typical for starfish, for example. Bilateral or bilateral symmetry is inherent in more highly developed organisms with a single axis dividing the body into two parts.

Bilateral symmetry is also inherent in the human body, but it cannot be called ideal. The legs, arms, eyes, lungs, but not the heart, liver or spleen are located symmetrically. Deviations from bilateral symmetry are noticeable even externally. For example, it is extremely rare that a person has the same moles on both cheeks.

You will need

• - properties of symmetric points;
• - properties of symmetrical figures;
• - ruler;
• - square;
• - compasses;
• - pencil;
• - paper;
• - a computer with a graphic editor.

Instructions

Draw a straight line a, which will be the axis of symmetry. If its coordinates are not specified, draw it at random. On one side of this straight line, put an arbitrary point A. You need to find a symmetrical point.

Helpful advice

Symmetry properties are constantly used in AutoCAD. For this, the Mirror option is used. To build an isosceles triangle or isosceles trapezoid, it is enough to draw the lower base and the angle between it and the side. Flip them with the command indicated and extend the sides as needed. In the case of a triangle, this will be the point of their intersection, and for a trapezoid, it will be set value.

You constantly encounter symmetry in graphic editors when you use the "flip vertically / horizontally" option. In this case, the line corresponding to one of the vertical or horizontal sides of the picture frame is taken as the axis of symmetry.

Sources:

• how to draw central symmetry

Cone sectioning is not so difficult task... The main thing is to follow a strict sequence of actions. Then this task will be easily accomplished and will not require much labor from you.

You will need

• - paper;
• - a pen;
• - circus;
• - ruler.

Instructions

When answering this question, you first need to decide what parameters the section is given.
Let it be the line of intersection of the plane l with the plane and the point O, which is the point of intersection with its section.

The construction is illustrated in Fig. 1. The first step in constructing a section is through the center of the section of its diameter, extended to l perpendicular to this line. As a result, point L is obtained. Then, through point O draw a straight line LW, and construct two guide cones lying in the main section O2M and O2C. At the intersection of these guides lie the point Q, as well as the already shown point W. These are the first two points of the desired section.

Now draw at the base of the cone BB1 ​​perpendicular to the MC and construct the generators of the perpendicular section О2В and О2В1. In this section, through T.O, draw a straight line RG parallel to BB1. T.R and T.G - two more points of the desired section. If the cross-section of the ball is known, then it could be built already at this stage. However, this is not an ellipse at all, but something elliptical, having symmetry about the segment QW. Therefore, you should build as many points of the section as possible in order to connect them in the future with a smooth curve to obtain the most reliable sketch.

Draw an arbitrary section point. To do this, draw an arbitrary diameter AN at the base of the cone and draw the corresponding guides O2A and O2N. Through so, draw a straight line passing through PQ and WG, until it intersects with the just drawn guides at points P and E. These are two more points of the desired section. Continuing the same way and further, you can arbitrarily desired points.

True, the procedure for obtaining them can be slightly simplified using the symmetry with respect to QW. To do this, you can draw straight lines SS 'in the plane of the desired section, parallel to RG until they intersect with the surface of the cone. The construction is completed by rounding the constructed polyline from chords. It suffices to construct half of the sought-for section due to the already mentioned symmetry with respect to QW.

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Tip 3: How to build a graph trigonometric function

You need to draw schedule trigonometric functions? Master the algorithm of actions using the example of building a sinusoid. To solve the problem, use the research method.

You will need

• - ruler;
• - pencil;
• - knowledge of the basics of trigonometry.

Instructions

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note

If two semiaxes of a single-strip hyperboloid are equal, then the figure can be obtained by rotating a hyperbola with semiaxes, one of which is the above, and the other, different from two equal, around the imaginary axis.

Helpful advice

When considering this figure relative to the Oxz and Oyz axes, it can be seen that its main sections are hyperbolas. And when a given spatial figure of rotation is cut by the Oxy plane, its section is an ellipse. The throat ellipse of a single-strip hyperboloid passes through the origin, since z = 0.

The throat ellipse is x² / a² + y² / b² = 1, and the other ellipses are x² / a² + y² / b² = 1 + h² / c².

Sources:

• Ellipsoids, paraboloids, hyperboloids. Straight generators

The shape of a five-pointed star has been widely used by humans since ancient times. We consider its form to be beautiful, since we unconsciously distinguish the ratio of the golden section in it, i.e. the beauty of the five-pointed star is mathematically based. Euclid was the first to describe the construction of the five-pointed star in his "Elements". Let's share his experience.

You will need

• ruler;
• pencil;
• compass;
• protractor.

Instructions

The construction of a star is reduced to the construction with the subsequent connection of its vertices with each other sequentially through one. In order to build the correct one, you need to break the circle into five.
Construct an arbitrary circle using a compass. Mark its center with O.

Mark point A and use the ruler to draw line segment OA. Now you need to divide segment OA in half, for this, draw an arc from point A with radius OA until it intersects with the circle at two points M and N. Construct segment MN. Point E, at which MN intersects OA, will bisect OA.

Restore OD perpendicular to radius OA and connect point D and E. Resection B at OA from point E with radius ED.

Now use line segment DB to mark the circle into five equal parts. Designate the vertices of the regular pentagon sequentially with numbers from 1 to 5. Connect the dots in the following sequence: 1 with 3, 2 with 4, 3 with 5, 4 with 1, 5 with 2. Here is the correct five pointed star, v regular pentagon... It was in this way that he built

May 20, 2014

Human life is filled with symmetry. It is convenient, beautiful, there is no need to invent new standards. But what is she really and is it so beautiful in nature, as is commonly believed?

Symmetry

Since ancient times, people have sought to organize the world around them. Therefore, something is considered beautiful, but something is not very. From an aesthetic point of view, gold and silver ratios are considered attractive, as well as, of course, symmetry. This term is of Greek origin and literally means "proportionality". Of course it comes not only about the coincidence on this basis, but also on some others. In a general sense, symmetry is a property of an object when, as a result of certain formations, the result is equal to the initial data. This is found in both living and inanimate nature, as well as in objects made by man.

First of all, the term "symmetry" is used in geometry, but finds application in many scientific fields, and its meaning remains generally unchanged. This phenomenon is quite common and is considered interesting, since several of its types, as well as elements, are distinguished. The use of symmetry is also interesting, because it is found not only in nature, but also in ornaments on fabrics, borders of buildings and many other man-made objects. It is worth considering this phenomenon in more detail, since it is extremely exciting.

Use of the term in other scientific fields

In what follows, symmetry will be considered from the point of view of geometry, but it is worth mentioning that given word used not only here. Biology, virology, chemistry, physics, crystallography - all this is an incomplete list of areas in which this phenomenon studied from various angles and in different conditions... For example, the classification depends on which science this term refers to. So, the division into types varies greatly, although some of the basic ones, perhaps, remain the same everywhere.

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Classification

There are several main types of symmetry, of which three are most common:

In addition, the following types are also distinguished in geometry, they are much less common, but no less curious:

• sliding;
• rotational;
• point;
• translational;
• screw;
• fractal;
• etc.

In biology, all species are called somewhat differently, although in essence they can be the same. Subdivision into certain groups occurs based on the presence or absence, as well as the number of certain elements, such as centers, planes and axes of symmetry. They should be considered separately and in more detail.

Basic elements

Some features are distinguished in the phenomenon, one of which is necessarily present. The so-called reference elements include planes, centers and axes of symmetry. It is in accordance with their presence, absence and quantity that the type is determined.

The center of symmetry is the point inside a figure or crystal, at which lines converge, connecting in pairs all sides parallel to each other. Of course, it does not always exist. If there are sides to which there is no parallel pair, then such a point cannot be found, since it does not exist. By definition, it is obvious that the center of symmetry is that through which a figure can be reflected back onto itself. An example is, for example, a circle and a point in its middle. This element is usually referred to as C.

The plane of symmetry is, of course, imaginary, but it is this plane that divides the figure into two equal parts to each other. It can pass through one or more sides, be parallel to it, or it can divide them. Several planes can exist for the same figure. These elements are commonly referred to as P.

But perhaps the most common is what is called the "axis of symmetry." This common phenomenon can be seen both in geometry and in nature. And it is worthy of separate consideration.

Axles

Often an element with respect to which a figure can be called symmetrical is

a straight line or segment protrudes. In any case, we are not talking about a point or a plane. Then the axes of symmetry of the figures are considered. There can be a lot of them, and they can be located as you like: divide the sides or be parallel to them, and also intersect the corners or not. Symmetry axes are usually denoted as L.

Examples include isosceles and equilateral triangles. In the first case, there will be a vertical axis of symmetry, on both sides of which there are equal faces, and in the second, the lines will intersect each angle and coincide with all bisectors, medians and heights. Ordinary triangles do not have it.

By the way, the totality of all the above-mentioned elements in crystallography and stereometry is called the degree of symmetry. This indicator depends on the number of axes, planes and centers.

Examples in geometry

Conventionally, it is possible to divide the entire set of objects of study of mathematicians into figures that have an axis of symmetry, and those that do not. The first category automatically includes all regular polygons, circles, ovals, as well as some special cases, while the rest fall into the second group.

As in the case when it was said about the axis of symmetry of a triangle, this element for a quadrilateral does not always exist. For a square, rectangle, rhombus or parallelogram, it is, but for an irregular figure, accordingly, it is not. For a circle, the axis of symmetry is the set of straight lines that pass through its center.

In addition, it is interesting to consider volumetric figures from this point of view. At least one axis of symmetry in addition to all regular polygons and the ball will possess some cones, as well as pyramids, parallelograms and some others. Each case must be considered separately.

Examples in nature

Mirror symmetry in life is called bilateral, it is most common
often. Any person and many animals are an example of this. The axial is called radial and is much less common, as a rule, in flora... And yet they are. For example, it is worth considering how many axes of symmetry a star has, and does it have them at all? Of course, we are talking about marine life, and not about the subject of study by astronomers. And the correct answer would be this: it depends on the number of rays of the star, for example, five, if it is five-pointed.

In addition, radial symmetry is observed in many flowers: chamomile, cornflowers, sunflowers, etc. There are a lot of examples, they are literally everywhere around.

Arrhythmia

This term, first of all, reminds the majority of medicine and cardiology, however, it initially has a slightly different meaning. In this case, the synonym will be "asymmetry", that is, the absence or violation of regularity in one form or another. It can be seen as an accident, and sometimes it can be a wonderful technique, for example, in clothing or architecture. After all, there are a lot of symmetrical buildings, but the famous Leaning Tower of Pisa is slightly inclined, and although it is not the only one, this is the most famous example. It is known that this happened by accident, but this has its own charm.

In addition, it is clear that the faces and bodies of humans and animals are also not completely symmetrical. There have even been studies that have judged the "right" faces as inanimate or simply unattractive. Still, the perception of symmetry and this phenomenon in itself is amazing and has not yet been fully studied, and therefore extremely interesting.