Axial and central symmetry drawings are easy. Types of symmetry

Consider axial and central symmetries as properties of some geometric figures; Consider axial and central symmetries as properties of some geometric figures; Be able to construct symmetrical points and be able to recognize figures that are symmetrical with respect to a point or line; Be able to construct symmetrical points and be able to recognize figures that are symmetrical with respect to a point or line; Improving problem solving skills; Improving problem solving skills; Continue to work on accurately recording and completing geometric drawings; Continue to work on accurately recording and completing geometric drawings;

Oral work “Gentle questioning” Oral work “Gentle questioning” What point is called the middle of the segment? Which triangle is called isosceles? What properties do the diagonals of a rhombus have? State the bisector property of an isosceles triangle. Which lines are called perpendicular? Which triangle is called equilateral? What properties do the diagonals of a square have? What figures are called equal?

What new concepts did you learn about in class? What new concepts did you learn about in class? What new things have you learned about geometric shapes? What new things have you learned about geometric shapes? Give examples of geometric shapes that have axial symmetry. Give examples of geometric shapes that have axial symmetry. Give an example of figures that have central symmetry. Give an example of figures that have central symmetry. Give examples of objects from the surrounding life that have one or two types of symmetry. Give examples of objects from the surrounding life that have one or two types of symmetry.

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.

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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 an interesting type of 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 a person's symmetry is the most important factor that 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 a nearly perfect circular web with radial supporting threads at nearly equal distances and spiral webs 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 like 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 become solid by forming weak hydrogen bonds, they align in an orderly arrangement that balances the forces of attraction and repulsion, forming the hexagonal shape of a 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. A new section at the edge of the Milky Way Galaxy has recently been discovered, and astronomers believe that 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 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 to two years, a precise alignment occurs and we witness a spectacular event known as a total solar eclipse. Astronomers don't know how common this symmetry is among other planets, but they think it's quite rare occurrence. 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.

Central symmetry. Central symmetry is movement.

Picture 9 from the presentation “Types of symmetry” for geometry lessons on the topic “Symmetry”

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Symmetry

"Symmetry in Nature"- In the 19th century, in Europe, isolated works appeared on the symmetry of plants. . Axial Central. One of the main properties of geometric shapes is symmetry. The work was carried out by: Zhavoronkova Tanya Nikolaeva Lera Supervisor: Artemenko Svetlana Yuryevna. By symmetry in a broad sense we understand any regularity in internal structure bodies or figures.

"Symmetry in Art"- II.1. Proportion in architecture. Each end of the pentagonal star represents a golden triangle. II. Central axis symmetry present in almost every architectural object. Place des Vosges in Paris. Periodicity in art. Content. Sistine Madonna. Beauty is multifaceted and many-sided.

"Point of Symmetry"- Crystals of rock salt, quartz, aragonite. Symmetry in the animal world. Examples of the above types of symmetry. B A O Any point on a line is a center of symmetry. This figure has central symmetry. A circular cone has axial symmetry; the axis of symmetry is the axis of the cone. An equilateral trapezoid has only axial symmetry.

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

"Mathematical Symmetry"- Symmetry. Symmetry in mathematics. Types of symmetry. In x and m and i. Rotational. Mathematical symmetry. Central symmetry. Rotational symmetry. Physical symmetry. The mystery of the mirror world. However, complex molecules generally lack symmetry. HAS A LOT IN COMMON WITH PROGRESSAL SYMMETRY IN MATHEMATICS.

"Symmetry around us"- Central. One kind of symmetry. Axial. In geometry there are figures that have... Rotations. Rotation (rotary). Symmetry on a plane. Horizontal. Axial symmetry is relatively straight. The Greek word symmetry means “proportion”, “harmony”. Two types of symmetry. Central relative to a point.

There are a total of 32 presentations in the topic

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 given command and extend sides to the required value. 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 encounter 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 must first 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 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 justified mathematically. 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 correct one five pointed star, into a regular pentagon. This is exactly the way I built it