Physics 11th grade
Geometric shapes and forms. Amazing figures in geometry. Fig.8. Parquet made from a Greek cross

Geometric shapes and forms. Amazing figures in geometry. Fig.8. Parquet made from a Greek cross

Geometry – precise mathematical science, which studies spatial and other similar relationships and forms. But it is often called "dry" because it is not able to describe the shape of many natural objects, because clouds are not spheres, mountains are not cones, and lightning does not travel in straight lines. Many objects in nature have complex shapes compared to standard geometry.

However, there are a number of amazing figures that are not usually studied in school geometry lessons, but they surround a person in the real world: in nature and architecture, puzzles, computer games etc.

The main property of this complex geometric figure is self-similarity, that is, it consists of several parts, each of which is similar to the whole object. It is this property that distinguishes fractals from objects of classical (or, as they say, Euclidean) geometry.

Moreover, the term “fractal” itself is not mathematical and does not have an unambiguous definition, therefore it can be applied to objects that are self-similar or approximately self-similar. It was invented in 1975 by Benoit Mandelbrot, borrowing the Latin word “fractus” (broken, crushed).

Fractal shapes are the best way to describe real world and are often found among natural objects: snowflakes, plant leaves, the blood vessel system of humans and animals.

This is one of the most extraordinary three-dimensional shapes in geometry, which is easy to make at home. To do this, it is enough to take a paper strip, the width of which is 5-6 times less than its length, and, twisting one of the ends 180°, glue them together.

If everything is done correctly, you can check its amazing properties yourself:

The presence of only one side (without division into internal and external). This can be easily checked if you try to paint over one of its sides with a pencil. Regardless of where and in what direction you start painting, the end result will be that the entire tape will be painted with the same color.
Continuity: If you draw a line along the entire surface with a pen, its end will connect to the starting point without crossing the boundaries of the surface.
Two-dimensionality (connectedness): when cutting a Möbius strip lengthwise, it remains intact, new shapes are simply obtained (for example, when cut in half, one larger ring is obtained).
Lack of orientation. A journey along such a Mobius strip will always be endless, it will lead to the starting point of the path, only in a mirror image.

Mobius strips are widely used in industry and science (in conveyor belts, matrix printers, sharpening mechanisms, etc.). In addition, there is a scientific hypothesis according to which the Universe itself is also a Mobius strip of incredible size.

Polyomino

It's flat geometric figures, which are formed by connecting several squares of equal sizes along their sides.

The names of polyominoes depend on the number of squares from which they are formed:

monomino – 1;
domino – 2;
trimino – 3;
tetromino – 4, etc.

Moreover, for each variety there is a different number of types of figures: dominoes have 1 type, triminos have 3 types, hexaminos (of 6 squares) have 35 types. The number of different variations depends on the number of squares used, but no scientist has yet been able to find an amazing formula that will express this dependence. From polyomino parts you can lay out both geometric shapes and images of people, animals, and objects. Despite the fact that these will be sketchy silhouettes, the main features and shapes of the objects make them quite recognizable.

Polyamond

Along with polyominoes, there is another amazing geometric figure used to compose other shapes - polyamong. It is a polygon formed from several equilateral triangles of equal size.

The name was invented by the mathematician T. O'Beirne based on one of the names of the rhombus in English language– a diamond that can be made from 2 equilateral triangles. By analogy, O’Beirne called a figure of 3 equilateral triangles a triamond, a figure of 4 - a tetriamond, etc.

The main question of their existence remains the question of the possible number of polyamides that can be made from a certain number of triangles. The use of polyamunds in real life also similar to using polyominoes. These can be various kinds of puzzles and logical tasks.

Reuleaux triangle

As surprising as it sounds, you can drill a square hole with a drill, and the Reuleaux triangle helps with this. It represents the area formed by the intersection of 3 equal circles, the centers of which are the vertices of a regular triangle, and the radii are equal to its sides.

The Reuleaux triangle itself is named after the German scientist-engineer, who was the first to study its features in more detail and use it for his mechanisms at the turn of the 19th-20th centuries. century, although its amazing properties were already known to Leonardo da Vinci. Whoever was its discoverer, modern world This figure is widely used in the form:

Watts drill, which allows you to drill holes of an almost perfect square shape, only with slightly rounded edges;
a mediator necessary for playing plucked musical instruments;
cam mechanisms used to create zigzag seams in sewing machines, as well as German watches;
pointed arches, characteristic of the Gothic style in architecture.

Impossible figures

The so-called impossible figures deserve special attention - amazing optical illusions, which at first glance seem to be a projection of a three-dimensional object, but upon closer inspection, unusual combinations of elements become noticeable. The most popular of them are:

Tribar, created by father and son Lionel and Roger Penrose, which is an image of an equilateral triangle, but has strange patterns. The sides that form the top of the triangle appear perpendicular, but the right and left sides at the bottom also appear perpendicular. If we consider each part of this triangle separately, we can still recognize their existence, but in reality such a figure cannot exist, since the correct elements were incorrectly connected when it was created.

The Endless Staircase, the authorship of which also belongs to the father and son Penroses, is therefore often called by their name - the “Penrose Staircase”, as well as the “Eternal Staircase”. At first glance, it looks like an ordinary staircase leading up or down, but a person walking along it will continuously ascend (counterclockwise) or descend (clockwise). If you visually travel along such a staircase, then at the end of the “journey” your gaze stops at the starting point of the path. If such a staircase existed in reality, it would have to be climbed and descended an infinite number of times, which can be compared to an endless Sisyphean task.

Impossible trident - amazing object, looking at which it is impossible to determine where the middle tooth begins. It is also based on the principle of irregular connections, which can only exist in two dimensions, but not three-dimensional space. Looking at the parts of the trident separately, 3 round teeth are visible on one side, and 2 rectangular ones on the other side.

Thus, the parts of the figure enter into a kind of conflict: firstly, the foreground and background change, and secondly, the round teeth in the lower part are transformed into flat ones in the upper part.

Here you and your child can learn geometric shapes and their names with fun picture activities. But learning will be most effective if you also add various samples of geometric shapes to the printed assignment. Suitable items for this purpose include balls, pyramids, cubes, inflated balloons (round and oval), tea mugs (standard, cylinder-shaped), oranges, books, balls of thread, square cookies and much more - everything whatever your imagination tells you.

All of the items listed will help the child understand what a three-dimensional geometric figure means. Flat figures You can prepare by cutting out the desired geometric shapes from paper, after painting them in different colors.

The more different materials you prepare for the lesson, the more interesting it will be for your child to learn new concepts.

You may also like our online math simulator for grade 1 “Geometric Shapes”:

The online mathematics trainer "Geometric Shapes 1st Grade" will help first-graders practice their ability to distinguish basic geometric shapes: square, circle, oval, rectangle and triangle.

Geometric shapes and their names - We conduct a lesson with the child:

So that your child can easily and naturally remember geometric shapes and their names, first download the picture with the task in the attachments at the bottom of the page, print it on a color printer and place it on the table along with colored pencils. Also, by this time, you should already have prepared the various items that we listed earlier.

Stage 1. First, let the child complete the tasks on the printed sheet - say the names of the shapes out loud and color all the pictures.
Stage 2. It is necessary to clearly show the child the differences between three-dimensional figures and flat ones. To do this, lay out all the sample objects (both three-dimensional and cut out of paper) and move away with the child from the table to such a distance from which all three-dimensional figures are clearly visible, but all flat samples are lost from sight. Draw your child's attention to this fact. Let him experiment, coming closer to the table, then further, telling you about his observations.
Stage 3. Then the activity needs to be turned into a kind of game. Ask your child to look carefully around him and find objects that have the shape of some geometric shapes. For example, a TV is a rectangle, a clock is a circle, etc. On each piece you find, clap your hands loudly to add enthusiasm to the game.
Stage 4. Carry out research and observational work with the sample materials that you have prepared for the lesson. For example, place a book and a flat rectangle of paper on the table. Invite your child to touch them, look at them from different angles and tell you their observations. In the same way, you can examine an orange and a paper circle, a children's pyramid and a paper triangle, a cube and a paper square, balloon oval shaped and oval cut out of paper. You can add to the list of items yourself.
Stage 5. Place various three-dimensional samples in an opaque bag and ask the child to touch a square object, then a round one, then a rectangular one, and so on.
Stage 6. Place several on the table in front of your child. various items of those participating in the lesson. Then have the child turn away for a few seconds while you hide one of the objects. Turning to the table, the child must name the hidden object and its geometric shape.

You can download geometric shapes and their names - Task form - in the attachments at the bottom of the page.

Names of geometric shapes - Printable cards

When studying geometric shapes with your child, you can use printable cards from Little Fox Bibushi during classes. . Download the attachments, print out a form with cards on a color printer, cut out each card along the outline - and start learning. Cards can be laminated or glued to thicker paper to preserve appearance pictures, because they will be used repeatedly.

The first six cards will give you the opportunity to study the following shapes with your child: oval, circle, square, rhombus, rectangle and triangle; under each shape in the cards you can read its name.

After the child remembers the name a certain figure, ask him to do the following: circle all the samples of the figure being studied on the card, and then color them in the color of the main figure located in the upper left corner.

You can download the names of geometric shapes - Printable cards - in the attachments at the bottom of the page

With the help of the following six cards, your child will be able to become familiar with the following geometric shapes: parallelogram, trapezoid, pentagon, hexagon, star and heart. As in the previous material, under each figure you can find its name.

To diversify activities with your child, combine learning with drawing - this method will prevent the child from getting overtired, and the child will be happy to continue studying. Make sure that when tracing the figures, the child does not rush and completes the task carefully, because such exercises not only develop fine motor skills, they can affect the baby’s handwriting in the future.

You can download printable cards with images of flat geometric shapes in the attachments

In the process of how you will study three-dimensional geometric shapes and their names with your child, using the new six cards from Bibushi with images of a cube, cylinder, cone, pyramid, ball and hemisphere, purchase the figures you are studying in the store, or use objects in the house that have a similar shape.

Show your child with examples what three-dimensional figures look like in real life; the child should touch and play with them. First of all, this is necessary in order to use the child’s visual and effective thinking, with the help of which it is easier for the child to understand the world around him.

Download - Volumetric geometric shapes and their names - you can find them in the attachments at the bottom of the page

You will also find other materials on studying geometric shapes useful:

Fun and colorful tasks for children "Drawings from geometric shapes" are a very convenient educational material for preschool and younger children school age for learning and memorizing basic geometric shapes:

The tasks will familiarize the child with the basic shapes of geometry - circle, oval, square, rectangle and triangle. Only here there is no boring memorization of the names of figures, but a kind of coloring game.

As a rule, geometry begins to be studied by drawing flat geometric figures. The perception of the correct geometric shape is impossible without drawing it with your own hands on a sheet of paper.

This activity will greatly amuse your young mathematicians. After all, now they will have to find familiar shapes of geometric figures among many pictures.

Layering shapes on top of each other is a geometry activity for preschoolers and junior schoolchildren. The point of the exercise is to solve addition examples. Just this unusual examples. Instead of numbers, you need to add geometric shapes.

This task is designed in the form of a game in which the child will have to change the properties of geometric shapes: shape, color or size.

Here you can download tasks in pictures that show how to count geometric shapes for math classes.

In this task, the child will become familiar with the concept of drawings of geometric bodies. Essentially, this lesson is a mini-lesson on descriptive geometry.

Here we have prepared for you three-dimensional geometric paper shapes that need to be cut and glued. Cube, pyramids, rhombus, cone, cylinder, hexagon, print them on cardboard (or colored paper and then paste them on cardboard), and then give them to the child to memorize.

Here we have posted for you counting to 5 - pictures with mathematical tasks for kids, thanks to which your children will practice not only their counting skills, but also their ability to read, write, distinguish geometric shapes, draw and color.

And you can also play online math games from little fox Bibushi:

In this developing online game The child will have to determine which is odd among the 4 pictures. In this case, it is necessary to be guided by the characteristics of geometric shapes.

Lesson topic

Geometric figures

What is a geometric figure

Geometric figures are a collection of many points, lines, surfaces or bodies that are located on a surface, plane or space and form a finite number of lines.

The term “figure” is to some extent formally applied to a set of points, but as a rule, a figure is usually called a set that is located on a plane and is limited by a finite number of lines.

A point and a straight line are the basic geometric figures located on a plane.

The simplest geometric figures on a plane include a segment, a ray and a broken line.

What is geometry

Geometry is a mathematical science that deals with the study of the properties of geometric figures. If we literally translate the term “geometry” into Russian, it means “land surveying,” since in ancient times the main task of geometry as a science was the measurement of distances and areas on the surface of the earth.

The practical application of geometry is invaluable at all times and regardless of profession. Neither a worker, nor an engineer, nor an architect, nor even an artist can do without knowledge of geometry.

In geometry there is a section that deals with the study of various figures on a plane and is called planimetry.

You already know that a figure is an arbitrary set of points located on a plane.

Geometric figures include: point, straight line, segment, ray, triangle, square, circle and other figures that planimetry studies.

Dot

From the material studied above, you already know that the point refers to the main geometric figures. And although this is the smallest geometric figure, it is necessary for constructing other figures on a plane, drawing or image and is the basis for all other constructions. After all, the construction of more complex geometric figures consists of many points characteristic of a given figure.

In geometry, points represent in capital letters Latin alphabet, for example, such as: A, B, C, D....

Now let's summarize, and so, with mathematical point From a point of view, a point is an abstract object in space that does not have volume, area, length or other characteristics, but remains one of the fundamental concepts in mathematics. A point is a zero-dimensional object that has no definition. According to Euclid's definition, a point is something that cannot be defined.

Straight

Like a point, a straight line refers to figures on a plane, which has no definition, since it consists of an infinite number of points located on one line, which has neither beginning nor end. It can be argued that a straight line is infinite and has no limit.

If a straight line begins and ends with a point, then it is no longer a straight line and is called a segment.

But sometimes a straight line has a point on one side and not on the other. In this case, the straight line turns into a beam.

If you take a straight line and put a point in its middle, then it will split the straight line into two oppositely directed rays. These rays are additional.

If in front of you there are several segments connected to each other so that the end of the first segment becomes the beginning of the second, and the end of the second segment becomes the beginning of the third, etc., and these segments are not on the same straight line and when connected they have common point, then such a chain is a broken line.

Exercise

Which broken line is called unclosed?
How is a straight line designated?
What is the name of a broken line that has four closed links?
What is the name of a broken line with three closed links?

When the end of the last segment of a broken line coincides with the beginning of the 1st segment, then such a broken line is called closed. An example of a closed polyline is any polygon.

Plane

Like a point and a straight line, so a plane is a primary concept, has no definition and it cannot be seen either beginning or end. Therefore, when considering a plane, we consider only that part of it that is limited by a closed broken line. Thus, any smooth surface can be considered a plane. This surface can be a sheet of paper or a table.

Corner

A figure that has two rays and a vertex is called an angle. The junction of the rays is the vertex of this angle, and its sides are the rays that form this angle.

Exercise:

1. How is an angle indicated in the text?
2. What units can you use to measure an angle?
3. What are the angles?

Parallelogram

A parallelogram is a quadrilateral whose opposite sides are parallel in pairs.

Rectangle, square and rhombus are special cases of parallelogram.

A parallelogram with right angles equal to 90 degrees is a rectangle.

A square is the same parallelogram; its angles and sides are equal.

As for the definition of a rhombus, it is a geometric figure whose all sides are equal.

In addition, you should know that every square is a rhombus, but not every rhombus can be a square.

Trapezoid

When considering a geometric figure such as a trapezoid, we can say that, in particular, like a quadrilateral, it has one pair of parallel opposite sides and is curvilinear.

Circle and circle

Circle - the geometric locus of points in the plane equidistant from given point, called the center, to a given non-zero distance, called its radius.

Triangle

The triangle you have already studied also belongs to simple geometric figures. This is one of the types of polygons in which part of the plane is limited by three points and three segments that connect these points in pairs. Any triangle has three vertices and three sides.

Exercise: Which triangle is called degenerate?

Polygon

Polygons include geometric figures of different shapes that have a closed broken line.

In a polygon, all points that connect the segments are its vertices. And the segments that make up a polygon are its sides.

Did you know that the emergence of geometry goes back centuries and is associated with the development of various crafts, culture, art and observation of the surrounding world. And the name of geometric figures is confirmation of this, since their terms did not arise just like that, but due to their similarity and similarity.

After all, the term “trapezoid” translated from ancient Greek language from the word “trapezion” means table, meal and other derivative words.

“Cone” comes from the Greek word “konos,” which means pine cone.

“Line” has Latin roots and comes from the word “linum”, translated it sounds like linen thread.

Did you know that if you take geometric figures with the same perimeter, then among them the owner of the most large area turned out to be a circle.

At the same time as learning colors, you can start showing your child cards of geometric shapes. On our website you can download them for free.

How to study figures with your child using Doman cards.

1) You need to start with simple shapes: circle, square, triangle, star, rectangle. As you master the material, begin to study more complex shapes: oval, trapezoid, parallelogram, etc.

2) You need to work with your child using Doman cards several times a day. When demonstrating a geometric figure, clearly pronounce the name of the figure. And if during classes you also use visual objects, for example, collecting inserts with figures or a toy sorter, then your child will master the material very quickly.

3) When the child remembers the name of the figures, you can move on to more difficult tasks: Now showing the card, say - this is a blue square, it has 4 equal sides. Ask your child questions, ask him to describe what he sees on the card, etc.

Such activities are very useful for the development of a child’s memory and speech.

Here you can download Doman's cards from the series “Flat geometric shapes” There are 16 pieces in total, including cards: flat geometric shapes, octagon, star, square, ring, circle, oval, parallelogram, semicircle, rectangle, right triangle, pentagon, rhombus, trapezoid, triangle, hexagon.

Classes according to Doman cards They perfectly develop the child’s visual memory, attentiveness, and speech. This is a great exercise for the mind.

You can download and print everything for free Doman cards flat geometric shapes

Right-click on the card and click “Save Image As...” so you can save the image to your computer.

How to make Doman cards yourself:

Print cards on thick paper or cardboard, 2, 4 or 6 pieces per sheet. To conduct classes using the Doman method, the cards are ready, you can show them to your child and say the name of the picture.

Good luck and new discoveries to your baby!

Educational video for children (toddlers and preschoolers) made according to the Doman method “Prodigy from the cradle” - educational cards, educational pictures on various topics from part 1, part 2 of the Doman method, which can be watched for free here or on our Channel Early childhood development on youtube