You will need

• - properties of symmetrical points;
• - properties of symmetrical figures;
• - ruler;
• - square;
• - compass;
• - pencil;
• - a sheet of paper;
• - a computer with a graphics editor.

Instructions

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

Useful advice

Symmetry properties are used constantly in AutoCAD. To do this, use the Mirror option. To construct an isosceles triangle or isosceles trapezoid, it is enough to draw the lower base and the angle between it and the side. Reflect them using the specified command and extend the sides to the required size. In the case of a triangle, this will be the point of their intersection, and for a trapezoid, this will be a given value.

You constantly come across symmetry in graphic editors when you use the “flip vertically/horizontally” option. In this case, the axis of symmetry is taken to be a straight line corresponding to one of the vertical or horizontal sides of the picture frame.

Sources:

• how to draw central symmetry

Constructing a cross section of a cone is not such a 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;
• - pen;
• - circle;
• - ruler.

Instructions

When answering this question, you first need to decide what parameters define the section.
Let this be the straight line of intersection of the plane l with the plane and the point O, which is the 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. The result is point L. Next, draw a straight line LW through point O, and construct two guide cones lying in the main section O2M and O2C. At the intersection of these guides lie point Q, as well as the already shown point W. These are the first two points of the desired section.

Now draw a perpendicular MS at the base of the cone BB1 ​​and construct generatrices of the perpendicular section O2B and O2B1. In this section, through point O, draw a straight line RG parallel to BB1. Т.R and Т.G are two more points of the desired section. If the cross section of the ball were known, then it could be built already at this stage. However, this is not an ellipse at all, but something elliptical that has symmetry with respect to the segment QW. Therefore, you should build as many section points as possible in order to connect them later with a smooth curve to obtain the most reliable sketch.

Construct an arbitrary section point. To do this, draw an arbitrary diameter AN at the base of the cone and construct the corresponding guides O2A and O2N. Through t.O, draw a straight line passing through PQ and WG until it intersects with the newly constructed guides at points P and E. These are two more points of the desired section. Continuing in the same way, you can find as many points as you want.

True, the procedure for obtaining them can be slightly simplified using 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 is enough to construct half of the desired section due to the already mentioned symmetry with respect to QW.

Video on the topic

Tip 3: How to make a graph trigonometric function

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

You will need

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

Instructions

Video on the topic

Please note

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

Useful advice

When examining this figure relative to the Oxz and Oyz axes, it is clear that its main sections are hyperbolas. And when this spatial figure of rotation is cut by the Oxy plane, its section is an ellipse. The neck ellipse of a single-strip hyperboloid passes through the origin of coordinates, because z=0.

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

Sources:

• Ellipsoids, paraboloids, hyperboloids. Rectilinear generators

The shape of a five-pointed star has been widely used by man since ancient times. We consider its shape beautiful because we unconsciously recognize in it the relationships of the golden section, i.e. the beauty of the five-pointed star is mathematically justified. Euclid was the first to describe the construction of a five-pointed star in his Elements. Let's join in with his experience.

You will need

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

Instructions

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

Mark point A and use a ruler to draw line segment OA. Now you need to divide the segment OA in half; to do this, from point A, draw an arc of radius OA until it intersects the circle at two points M and N. Construct the segment MN. The point E where MN intersects OA will bisect segment OA.

Restore the perpendicular OD to the radius OA and connect points D and E. Make a notch B on OA from point E with radius ED.

Now, using line segment DB, mark the circle into five equal parts. Label 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 regular five-pointed star, into a regular pentagon. This is exactly the way I built it

So, as for geometry: there are three main types of symmetry.

Firstly, central symmetry (or symmetry about a point) - this is a transformation of the plane (or space), in which a single point (point O - the center of symmetry) remains in place, while the remaining points change their position: instead of point A, we get point A1 such that point O is the middle of the segment AA1. To construct a figure Ф1, symmetrical to the figure Ф relative to point O, you need to draw a ray through each point of the figure Ф, passing through point O (center of symmetry), and on this ray lay a point symmetrical to the chosen one relative to point O. The set of points constructed in this way will give the figure F1.

Of great interest are figures that have a center of symmetry: with symmetry about the point O, any point in the figure Φ is again transformed into a certain point in the figure Φ. There are many such figures in geometry. For example: a segment (the middle of the segment is the center of symmetry), a straight line (any point of it is the center of its symmetry), a circle (the center of the circle is the center of symmetry), a rectangle (the point of intersection of its diagonals is the center of symmetry). Many centrally symmetrical objects in living and inanimate nature(student message). Often people themselves create objects that have a center symmetries (examples from handicrafts, examples from mechanical engineering, examples from architecture and many other examples).

Secondly, axial symmetry (or symmetry about a straight line) - this is a transformation of a plane (or space), in which only the points of the straight line p remain in place (this straight line is the axis of symmetry), while the remaining points change their position: instead of point B we obtain a point B1 such that the straight line p is the perpendicular bisector to the segment BB1 . To construct a figure Ф1, symmetrical to the figure Ф, relative to the straight line р, it is necessary for each point of the figure Ф to construct a point symmetrical to it relative to the straight line р. The set of all these constructed points gives the desired figure F1. There are many geometric shapes having an axis of symmetry.

A rectangle has two, a square has four, a circle has any straight line passing through its center. If you look closely at the letters of the alphabet, you can find among them those that have horizontal or vertical, and sometimes both, axes of symmetry. Objects with axes of symmetry are quite often found in living and inanimate nature (student reports). In his activity, a person creates many objects (for example, ornaments) that have several axes of symmetry.

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Thirdly, plane (mirror) symmetry (or symmetry about a plane) - this is a transformation of space in which only points of one plane retain their location (α-symmetry plane), the remaining points of space change their position: instead of point C, a point C1 is obtained such that the plane α passes through the middle of the segment CC1, perpendicular to it.

To construct a figure Ф1, symmetrical to the figure Ф relative to the plane α, it is necessary for each point of the figure Ф to build points symmetrical relative to α; they, in their set, form the figure Ф1.

Most often, in the world of things and objects around us, we encounter three-dimensional bodies. And some of these bodies have planes of symmetry, sometimes even several. And man himself, in his activities (construction, handicrafts, modeling, ...) creates objects with planes of symmetry.

It is worth noting that, along with the three listed types of symmetry, there are (in architecture)portable and rotating, which in geometry are compositions of several movements.

Axial symmetry and the concept of perfection

Axial symmetry is inherent in all forms in nature and is one of fundamental principles beauty. Since ancient times, man has tried

to comprehend the meaning of perfection. This concept was first substantiated by artists, philosophers and mathematicians Ancient Greece. And the word “symmetry” itself was invented by them. It denotes proportionality, harmony and identity of the parts of the whole. The ancient Greek thinker Plato argued that only an object that is symmetrical and proportionate can be beautiful. Indeed, those phenomena and forms that are proportional and complete “please the eye.” We call them correct.

Axial symmetry as a concept

Symmetry in the world of living beings is manifested in the regular arrangement of identical parts of the body relative to the center or axis. More often in

Axial symmetry occurs in nature. It determines not only general structure organism, but also the possibilities of its subsequent development. The geometric shapes and proportions of living beings are formed by “axial symmetry”. Its definition is formulated as follows: this is the property of objects to be combined under various transformations. The ancients believed that the sphere possesses the principle of symmetry to the fullest extent. They considered this form harmonious and perfect.

Axial symmetry in living nature

If you look at any living creature, the symmetry of the body’s structure immediately catches the eye. Human: two arms, two legs, two eyes, two ears and so on. Each animal species has a characteristic color. If a pattern appears in the coloring, then, as a rule, it is mirrored on both sides. This means that there is a certain line along which animals and people can be visually divided into two identical halves, that is, their geometric structure is based on axial symmetry. Nature creates any living organism not chaotically and senselessly, but according to general laws world order, because nothing in the Universe has a purely aesthetic, decorative purpose. Availability various forms also due to natural necessity.

Axial symmetry in inanimate nature

In the world, we are surrounded everywhere by such phenomena and objects as: typhoon, rainbow, drop, leaves, flowers, etc. Their mirror, radial, central, axial symmetry is obvious. It is largely due to the phenomenon of gravity. Often the concept of symmetry refers to the regularity of changes in certain phenomena: day and night, winter, spring, summer and autumn, and so on. In practice, this property exists wherever order is observed. And the laws of nature themselves - biological, chemical, genetic, astronomical - are subject to the principles of symmetry common to us all, since they have an enviable systematicity. Thus, balance, identity as a principle has a universal scope. Axial symmetry in nature is one of the “cornerstone” laws on which the universe as a whole is based.

Axial symmetry. With axial symmetry, each point of the figure goes to a point that is symmetrical to it relative to a fixed straight line.

Picture 35 from the presentation “Ornament” for geometry lessons on the topic “Symmetry”

Dimensions: 360 x 260 pixels, format: jpg. To download a free image for a geometry lesson, right-click on the image and click “Save image as...”. To display pictures in the lesson, you can also download the entire presentation “Ornament.ppt” with all the pictures in a zip archive for free. The archive size is 3324 KB.

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Symmetry

"Point of symmetry" - Central symmetry. A a A1. Axial and central symmetry. Point C is called the center of symmetry. Symmetry in everyday life. A circular cone has axial symmetry; the axis of symmetry is the axis of the cone. Figures that have more than two axes of symmetry. A parallelogram has only central symmetry.

“Mathematical symmetry” - What is symmetry? Physical symmetry. Symmetry in biology. History of symmetry. However, complex molecules generally lack symmetry. Palindromes. Symmetry. In x and m and i. HAS A LOT IN COMMON WITH PROGRESSAL SYMMETRY IN MATHEMATICS. But actually, how would we live without symmetry? Axial symmetry.

“Ornament” - b) On the strip. Parallel translation Central symmetry Axial symmetry Rotation. Linear (arrangement options): Creating a pattern using central symmetry and parallel translation. Planar. One of the varieties of ornament is a mesh ornament. Transformations used to create an ornament:

“Symmetry in Nature” - One of the main properties of geometric shapes is symmetry. The topic was not chosen by chance, because next year we will have to start studying a new subject - geometry. The phenomenon of symmetry in living nature was noticed back in Ancient Greece. We study in the school scientific society because we love to learn something new and unknown.

“Movement in Geometry” - Mathematics is beautiful and harmonious! Give examples of movement. Movement in geometry. What is movement? What sciences does motion apply to? How movement is used in various fields human activity? A group of theorists. The concept of movement Axial symmetry Central symmetry. Can we see movement in nature?

“Symmetry in art” - Levitan. RAPHAEL. II.1. Proportion in architecture. Rhythm is one of the main elements of melody expressiveness. R. Descartes. Ship Grove. A.V. Voloshinov. Velazquez "Surrender of Breda" Externally, harmony can manifest itself in melody, rhythm, symmetry, proportionality. II.4.Proportion in literature.

There are a total of 32 presentations in the topic

For centuries, symmetry has remained a subject that has fascinated philosophers, astronomers, mathematicians, artists, architects and physicists. The ancient Greeks were completely obsessed with it - and even today we tend to encounter symmetry in everything from furniture arrangement to haircuts.

Just keep in mind that once you realize this, you'll probably feel an overwhelming urge to look for symmetry in everything you see.

(Total 10 photos)

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1. Broccoli Romanesco

Perhaps you saw Romanesco broccoli in the store and thought it was another example of a genetically modified product. But in fact, this is another example of the fractal symmetry of nature. Each broccoli floret has a logarithmic spiral pattern. Romanesco is similar in appearance to broccoli, but in taste and consistency - cauliflower. It is rich in carotenoids, as well as vitamins C and K, which makes it not only beautiful, but also healthy food.

For thousands of years, people have marveled at the perfect hexagonal shape of honeycombs and asked themselves how bees could instinctively create a shape that humans could only reproduce with a compass and ruler. How and why do bees have a passion for creating hexagons? Mathematicians believe that this is perfect shape, which allows them to store the maximum possible amount of honey using minimum quantity wax. Either way, it's all a product of nature, and it's damn impressive.

3. Sunflowers

Sunflowers boast radial symmetry and interesting guy symmetry known as the Fibonacci sequence. Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. (each number is determined by the sum of the two previous numbers). If we took our time and counted the number of seeds in a sunflower, we would find that the number of spirals grows according to the principles of the Fibonacci sequence. There are many plants in nature (including Romanesco broccoli) whose petals, seeds and leaves correspond to this sequence, which is why it is so difficult to find a clover with four leaves.

But why do sunflowers and other plants follow mathematical rules? Like the hexagons in a hive, it's all a matter of efficiency.

4. Nautilus Shell

In addition to plants, some animals, such as the Nautilus, follow the Fibonacci sequence. The shell of the Nautilus twists into a Fibonacci spiral. The shell tries to maintain the same proportional shape, which allows it to maintain it throughout its life (unlike humans, who change proportions throughout life). Not all Nautiluses have a Fibonacci shell, but they all follow a logarithmic spiral.

Before you envy the math clams, remember that they don’t do this on purpose, it’s just that this form is the most rational for them.

5. Animals

Most animals have bilateral symmetry, which means they can be split into two identical halves. Even humans have bilateral symmetry, and some scientists believe that human symmetry is the most important factor, which influences the perception of our beauty. In other words, if you have a one-sided face, you can only hope that it is compensated by other good qualities.

Some go to complete symmetry in an effort to attract a mate, such as the peacock. Darwin was positively annoyed by the bird, and wrote in a letter that "The sight of the tail feathers of a peacock, whenever I look at it, makes me sick!" To Darwin, the tail seemed cumbersome and made no evolutionary sense, as it did not fit with his theory of “survival of the fittest.” He was furious until he came up with the theory of sexual selection, which states that animals develop certain functions to increase your chances of mating. Therefore, peacocks have various adaptations to attract a partner.

There are about 5,000 types of spiders, and they all create an almost perfect circular web with radial supporting threads almost equal distance and spiral fabric for catching prey. Scientists aren't sure why spiders love geometry so much, as tests have shown that a round cloth won't lure food any better than a canvas irregular shape. Scientists theorize that radial symmetry evenly distributes the impact force when prey is caught in the net, resulting in fewer breaks.

Give a couple of tricksters a board, mowers, and the safety of darkness, and you'll see that people create symmetrical shapes, too. Due to the complexity of the design and incredible symmetry of crop circles, even after the creators of the circles confessed and demonstrated their skills, many people still believe that they were made by space aliens.

As the circles become more complex, their artificial origin becomes increasingly clear. It's illogical to assume that aliens will make their messages increasingly difficult when we couldn't even decipher the first ones.

Regardless of how they came to be, crop circles are a joy to look at, mainly because their geometry is impressive.

Even tiny formations such as snowflakes are governed by the laws of symmetry, since most snowflakes have hexagonal symmetry. This occurs in part because of the way water molecules line up when they solidify (crystallize). Water molecules acquire solid state, forming weak hydrogen bonds, they align in an orderly arrangement that balances the forces of attraction and repulsion, forming the hexagonal shape of the snowflake. But at the same time, each snowflake is symmetrical, but not one snowflake is similar to the other. This happens because as each snowflake falls from the sky, it experiences unique atmospheric conditions that cause its crystals to arrange themselves in a certain way.

9. Milky Way Galaxy

As we have already seen, symmetry and mathematical models exist almost everywhere, but are these laws of nature limited to our planet? Obviously not. Recently opened a new section at Galaxy's Edge Milky Way, and astronomers believe the galaxy is an almost perfect mirror image of itself.

10. Sun-Moon Symmetry

Considering that the Sun has a diameter of 1.4 million km, and the Moon - 3474 km, it seems almost impossible that the Moon could block sunlight and provide us with about five solar eclipses every two years. How does this work? Coincidentally, while the Sun is about 400 times wider than the Moon, the Sun is also 400 times farther away. Symmetry ensures that the Sun and Moon are the same size when viewed from Earth, so that the Moon can obscure the Sun. Of course, the distance from the Earth to the Sun can increase, which is why we sometimes see annular and partial eclipses. But every one or two years a fine alignment occurs, and we witness a spectacular event known as complete solar eclipse. Astronomers don't know how common this symmetry is among other planets, but they think it's quite rare. However, we should not assume that we are special, as it is all a matter of chance. For example, every year the Moon moves about 4 cm away from the Earth, meaning that billions of years ago every solar eclipse would have been a total eclipse. If things continue like this, total eclipses will eventually disappear, and this will be accompanied by the disappearance of annular eclipses. It turns out that we are simply in the right place at the right time to see this phenomenon.