“To know the invisible,
look carefully at the visible "
Talmud
Like any three-dimensional object, each package has its own original shape and through it affects us and the surrounding space. Previously, we described the form of packaging only in connection with its convenience in the process of using the product, logistics, consumer perception. And never - her own influence on a person and space. This topic is the area of responsibility of sacred geometry, an interesting and immense science. Today we will only try to touch it and consider some classical geometric bodies. Perhaps tomorrow many packaging manufacturers, possessing this information, will be more likely to design packaging that can only harmonize the world with its shape and thus make it a little better. Sacred geometry is a teaching about the forms of space and the laws of the development of the Universe in accordance with these forms. The term "sacred geometry" is used by archaeologists, anthropologists, philosophers and cultural scientists. It is used to capture the system of religious, philosophical and spiritual archetypes that are observed in various cultures throughout human history and are in one way or another associated with geometric views regarding the structure of the universe and man. This term covers all Pythagorean and Neoplatonic geometry, referring also to the geometry of concave spaces and fractals.
In ancient Greece, the study of the essence of beauty, the mystery of beauty, based on certain geometric patterns, formed a separate branch of science - aesthetics, which, among ancient philosophers, was inextricably linked with cosmology. The ancient Greeks possessed a geometric vision of a universal order. They perceived the Universe as a vast space of various interconnected elements. Sacred geometry unites the wisdom of many schools, both those that existed long before our era, and modern ones, linking esotericism with the latest achievements of quantum physics. This amazing science recognizes all typical forms of manifestation of higher knowledge, considering them as cups containing information about the manifested world and about a person's place in it. Everything is energy, vibration, harmony and dissonance of frequency; everything is geometry.
Sacred geometric shapes are an important tool for spiritual growth. A person who does not imagine the power contained in geometric forms, does not realize that with their help he comes into contact with a fantastically rich information and energy world, is deprived of a lot. He loses the ability to feed on earthly and cosmic energy, which will inevitably affect his physical and spiritual development. Understanding the simple truths of sacred geometry leads to the development of consciousness and the opening of the heart, which is the next step in human development. Sacred geometry has played and continues to play a major role in the art, architecture and philosophy of many cultures for thousands of years.
Polyhedra in nature
Some of the regular and semi-regular bodies occur naturally in the form of crystals. Crystals are bodies with a multifaceted shape. Here is one example of such bodies: a pyrite crystal (pyrite FeS) - a natural model of a dodecahedron. Fullerene, one of the forms of carbon, is also a polyhedron. Fullerenes are molecular compounds belonging to the class of allotropic forms of carbon (others are diamond, carbyne and graphite) and are convex closed polyhedrons composed of an even number of three-coordinated carbon atoms. These connections owe their name to the engineer and designer R. Buck Minster Fuller, whose geodetic structures are built on this principle. Initially, this class of joints was limited only to structures that include only pentacular and hexagonal faces. It was discovered while trying to simulate space processes. Later, scientists in terrestrial laboratories managed to synthesize and study numerous derivatives of these spherical molecules.
The chemistry of fullerenes arose. But fullerenes, as it turned out, are also in terrestrial rocks, in particular, in shungites, the healing properties of which have been known since the time of Peter the Great. At the microscopic level, the dodecahedron and icosahedron are the relative parameters of DNA that all life is built upon. The DNA molecule is a rotating cube. When the cube is rotated sequentially by 72 ° according to a certain model, an icosahedron is obtained, which, in turn, is a pair of a dodecahedron. Thus, the double strand of the DNA helix is built according to the principle of two-way correspondence: the dodecahedron follows the icosahedron, then the icosahedron again, and so on. How fullerenes affect a person was found out by Peter the Great on marcial waters. And now a whole state corporation is studying this influence from all points of view and introducing nanotechnology into life.
Big sister of packaging - architecture
Sacred geometry manifested itself most vividly in the architecture of different cultures. When the Indians were going to erect any religious building, they first performed a simple geometric drawing on the ground, properly defining the directions to the east and west and building a square on their base. After that, the entire building was erected. The geometric calculations were accompanied by chants and prayers. The Christian religion uses the cross as its main symbol (in ancient times it appeared in the form of an unfolded cube). Many Gothic cathedrals were built using cube calculations. The ancient Egyptians discovered that regular polygons could be enlarged by the addition of a strictly designated area (which would later be called "gnomon" by the Greeks). The spirals on the pillars of ancient Greek temples were placed according to the principle of a rotating rectangle, a method of creating a logarithmic spiral. One of the types of early constructions of sacred architecture that have come down to us is the observatory. They were not only structures for observing the starry sky, but were also centers of spiritual knowledge. The modern architecture of large cities, focused on the construction of box houses and monotonous structures, has a very dangerous effect on humans. A person moves into an artificial habitat, completely technocratic, where reinforced concrete houses dominate. Violation of the laws of sacred architecture leads to the fact that the standardized environment with its ridiculous forms has a destructive effect on the psyche, causing negative emotions and provoking unmotivated actions.
Isn't this the case with packaging today? Feng Shui is also used to adjust buildings. The provisions combined under this term represent a set of sacred architecture and geometry requirements as applied to the energetic modeling of a living space. The applicability of feng shui ideas to construction helps people to resonate with natural human and earthly rhythms. The interaction of feng shui and sacred geometry is manifested in the generality of methods for determining the direction of the flow of vital energy, work with the subtle world. This is an ancient geomancy that studies the connection of the vital energy of qi with the landscape, its layout, location, internal design, i.e. with the environment of a person. The form of packaging, just like architecture, affects a person, with the only difference that we cannot feel its influence from the inside, but this influence must be studied from both sides in order to understand the influence of form on a packaged product in the future. ... After all, it is known that in a properly constructed pyramid, the meat does not deteriorate, but the blades are sharpened. Can you imagine what possibilities packaging has? Let's take a closer look at some of the classic geometric shapes in this regard.
Platonic solids and others
Platonic solids are a collection of all regular polyhedra, volumetric (three-dimensional) bodies, bounded by equal regular polygons, first described by Plato. The final, XIII book "Beginnings" by Platonov's pupil of Euclid is also dedicated to them. With all the infinite variety of regular polygons (two-dimensional geometric figures bounded by equal sides, adjacent pairs of which form equal angles in pairs), there are only five volumetric polygons, in accordance with which, since Plato's time, the five elements of the universe have been put. An interesting connection exists between the hexahedron and the octahedron, as well as between the dodecahedron and the icosahedron: the geometric centers of the faces of each first are the vertices of every second.
Scientists have long been interested in "ideal" or regular polygons, that is, polygons with equal sides and equal angles. An equilateral triangle can be considered the simplest regular polygon, since it has the smallest number of sides that can limit part of the plane. The general picture of regular polygons of interest to us, along with an equilateral triangle, is: a square (four sides), a pentagon (five sides), a hexagon (six sides), an octagon (eight sides), a decagon (ten sides), etc. Obviously, there is theoretically no limitation on the number of sides of a regular polygon, that is, the number of regular polygons is infinite.
What is a regular polyhedron? A regular polyhedron is a polyhedron of which all faces are equal (or congruent) to each other and, at the same time, are regular polygons. How many regular polyhedra are there? At first glance, the answer to this question is very simple: as many as there are regular polygons. However, it is not. In the "Principles of Euclid" we find a rigorous proof that there are only five regular polyhedra, and their faces can only be three types of regular polygons: triangles, squares and pentagons. Each form radiates its own energy and affects the person and space in different ways. So, the cross protects, the triangle charges, the circle aligns the Yin-Yang energies. Let's try to consider Platonic solids from this point of view. Plato, as well as the Pythagoreans, thoroughly studied the philosophical, mathematical and magical aspects of regular convex polyhedra. There are five such regular convex polyhedra. Each of these polyhedrons corresponds to a certain element and concentrates its energy. The vertices of the polyhedrons emit energy, and the centers of the faces absorb.
Further, the energy characteristics of polygons are considered from the point of view of the Chinese teaching "Wu-shing". Knowing the Yin or Yang nature of the radiation of polyhedrons, as well as the energies of their elements, doctors of Chinese medicine will be able to operate with them as means that harmonize human energy. So, a hexahedron (cube) has 8 energy-emitting points-vertices and 6 faces, in which energy is absorbed. Since there are more emitting points than absorbing ones, in accordance with the Chinese teaching "Wu-Xing" the cube refers to the masculine principle of "Yang". The octahedron has 6 points-vertices of radiation and 8 points-centers of absorption edges. Consequently, the octahedron absorbs more energy than it emits, so it belongs to the female Yin principle. The tetrahedron has 4 vertices and 4 faces, which results in Yin-Yang equality. The icosahedron has 12 vertices and 20 faces that look like regular triangles, so it expresses the Yin principle. The dodecahedron has 20 vertices and 12 faces and therefore expresses the Yang principle. Its 12 faces are in the form of regular pentagons. The dodecahedron is shaped like a soccer ball. The dodecahedron has a center of symmetry and 15 axes of symmetry. Each of the axes passes through the midpoints of opposite parallel ribs. The dodecahedron has 15 planes of symmetry. Any of the planes of symmetry passes in each face through the vertex and midpoint of the opposite edge.
In terms of sacred forces, the dodecahedron is the most powerful polyhedron. No wonder Salvador Dali chose this figure for his "Last Supper". In it, from 12 pentagons, also a strong figure, forces are concentrated at one point - on Jesus Christ. In the Pythagorean school, the word "dodecahedron" was killed for mentioning the word "dodecahedron" outside the school walls. This figure was considered so sacred. Two hundred years later, during the life of Plato, they talked about her, but only very carefully. Why? It is believed that the dodecahedron is located at the outer edge of the human energy field and is the highest form of consciousness. Regular polyhedra attract with the perfection of their forms, complete symmetry.
Generators from the developers of epam-technologies
According to the scientists Skvortsov A.V. and Khmelinskaya E.V., who have developed unique preparations "Epam", some geometric objects have the properties of harmonizing man and space:
- the truncated octahedron neutralizes the energy impact from the outside, increases the energy level of the brain, helps to work on an intuitive level and cleans the energy structure of a place within a radius of 500 m;
- an icosahedron with a side of 5 cm eliminates psychological dependencies, restores biostructure, harmonizes personality, cleans the structure of a place within a radius of 100 m;
- an icosahedron with a side of 3 cm improves communication with the subconscious, harmonizes relationships with other people, increases the energy level within a radius of 200 m, restores human connection with the earth and space, restores the thyroid gland; contributes to the implementation of their own mission in accordance with the implementation program;
- an icosahedron with a side of 1 cm enhances the energy power and intelligence of a person, improves fate, restores the energy of the place, aligns the psyche;
- a ten-sided pyramid protects against man-made radiation, activates self-regulation of the body, restores human energy exchange, enhances human energy, increases the energy level of a place (70 m), restores the human endocrine system, neutralizes geomagnetic radiation, harmonizes relationships between people;
- The twelve-sided pyramid harmonizes relations between people, restores human energy channels, turns on adaptation systems, improves self-regulation, attunes with the terrain, promotes creative processes, neutralizes geomagnetic radiation, restores human connection with space and natural biostructures.
The convex shape of the body without edges allows you to accumulate energy and transfer it to the owner. This form can contribute to a change in any structure or leisurely work. This form "softens" those who, for whatever reason, are harsh and unbalanced or are mired in internal contradictions. The lack of directional angles prevents energy from being unconsciously directed. This form stabilizes, soothes, concentrates strength. The oval shape allows the object to exchange energy with the person. It has a positive effect mainly on the psyche and behavior.
The round shape condenses energy in the best way. Serves mainly to enhance health. A geometric object in the form of a lentil or a drop communicates energetically with a person on an equal footing. They exchange energy, but do not merge. This form is capable of responding to thoughts. If a person decided to do something from the area of influence of this form, then it will help him. At other times, it just has a good effect on well-being. Objects with a flat bottom and rounded top reveal the magical power of the material they are made of. The shapes of the Chinese pagoda and the Tibetan stupa have ideal harmonizing effects. They are often located in the garden near the house, and small models are located inside the dwelling.
Mantra wheels?
Mantra wheels are known in Tibet and neighboring countries since ancient times, they are considered as generators of blissful energy that helps all living beings. Mantra wheels are a hollow cylinder rotating on an axle. The dimensions of such a cylinder can vary from a few centimeters to several meters. Tibetans wear small mantra wheels in their hand, rotating with a slight swing of the wrist. Larger wheels are located in huge numbers near temples and other sacred structures. In addition, they can be located in different parts of the area, sometimes very remote from a person's dwelling, rotating by the energy of wind or water in a mountain stream. These wheels are connected to a small turbine and rotate day and night.
It should be noted that all mantra wheels rotate clockwise when viewed from above. Studies of the so-called torsion fields arising from the rotation of massive cylinders, cones and other objects have shown that they have a pronounced biological and physicochemical effect. Moreover, it has now been shown that this is a completely new type of physical fields associated with the spin polarization of the physical vacuum. The mantra wheel is a kind of ecological device, a kind of "entropy pump" that reduces chaos and disorganization of the environment. However, these devices, discovered in ancient times, still have a number of know-how that are absent in modern spin-torsion generators. First of all, these are mantras that serve as a kind of modulator of the spin-torsion field. Actually, the type of such a mantra determines the nature of the action of such a generator. In other words, the main effect here is not associated with the energy of radiation, but with its information component - the semantic structure of the mantra. In this regard, the study of ancient archetypal signs, symbols and mantric formulas deserves a separate description, which we will certainly do. We will also return to the topic of the harmonizing influence of the form more than once, although it is possible that, when developing the next packaging design, you will do it before us, but for now ... look what your next box of Lemon Slices marmalade looks like and open it clockwise arrow several times a day!
Olga Gulinkina,
based on materials from open
summaries of other presentations"Circle 9 grade" - 2. Equation of a circle. Tasks. O (xo, yo) is the center of the circle, A (x; y) is the point of the circle. Let d be the distance from the center of the circle to a given point of the plane, R be the radius of the circle. № 1 Fill in the table according to the following data: Grade 9. № 2 Derive the equation of a circle centered at the point M (-3; 4) passing through the origin.
"The middle line of the trapezoid" - MN =? AB. D. Determination of the midline of the trapezoid. Continue the sentence: A. In the triangle, you can build ... middle lines. The middle line of the trapezoid. The theorem on the middle line of a trapezoid. MN - middle line of trapezoid ABCD. The middle line of a triangle has the property… MN || AB.
"Symmetry about a straight line" - Line a - axis of symmetry. http://www.potolok-spb.ru/art/images/butterfly/butterfly14.jpg. http://www.idance.ru/articles/20/767p_sy4.jpg. Which letters have an axis of symmetry? In fact, a person's face is not perfectly symmetrical. http://www.indostan.ru/indiya/foto-video/2774/3844_9_o.jpg. Injection. Isosceles triangle. Ray. Construct segment A1B1 symmetrical to segment AB relative to a straight line. How many axes of symmetry does each figure have?
"Amazing squares" - 1. Crossword puzzle. Basic forms. 3. A bit of history about origami. Boat. Flowers: A square is a rectangle in which all sides are equal. Show how amazing such a simple figure as a square is. Tasks with matches. Squared cutting. The size of the figurine depends on the size of the square, and then it is a matter of technique and taste. Boat station. Seal. Amazing square. 4. Envelope.
"Mapping a plane to itself" - Mapping a plane to itself. C1. Motion. Axial symmetry. IN 1. ... A1. Central symmetry. S. A. V.
"Regular polygons" - The purpose of the lesson: 1. 2. 5. Geometry - grade 9. Lesson progress: Work on cards. Competition "Fill in the table". Tasks for the finished drawing. 3. Lesson summary. Regular Polygons. Mathematical dictation. 6. Generalizing lesson
Russian Egor, Tarasov Dmitry
The world around us is a world of forms, it is very diverse and amazing. We are surrounded by household items of various types. After studying this topic, we really saw that polygons surround us everywhere and are found in various spheres of life.
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Slide captions:
Regular polygons
Amazing polygon
Star polygons
Polygons in nature
Polygons in nature
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Regular polygons in science and some other spheres of life Authors of the project: pupils of the 8th grade of the Russians Yegor Tarasov Dmitry. Scientific adviser: teacher of mathematics Rakhmankulova E.R.
Problematic question. What place do polygons occupy in our life? Research object: polygons. Research subject: the practical application of polygons in the world around us.
Purpose: systematization of knowledge on this topic and obtaining new information about polygons and their practical application. Objectives: 1. Study literature on the topic. 2. Show the practical application of regular polygons in the world around us.
Research methods: 1. Scientific (study of literature); 2. Research. Hypothesis: Polygons create beauty in a person's environment.
Regular polygons
Magic square 4 9 2 3 5 7 8 1 6
Amazing polygon
Star polygons
Polygons in nature P3: P4: P6 = 1: 0.877: 0.816
Polygons in nature
Polygons in nature
Polygons around us Parquet
Conclusion Without geometry, there would be nothing, everything that surrounds us is geometric shapes. But we forget to pay attention to this.
Conclusion The world around us is a world of forms, it is very diverse and amazing. We are surrounded by household items of various types. After studying this topic, we really saw that polygons surround us everywhere and are found in various spheres of life.
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What if there was only one type of shape in the world, for example, a shape such as a rectangle? Some things wouldn't change at all: doors, cargo trailers, soccer fields - they all look the same. But what about doorknobs? They would be a little weird. And what about the wheels of cars? It would be ineffective. And what about football? It's hard to even imagine. Fortunately, the world is full of many different forms. Do they exist in nature? Yes, and there are a lot of them.
What is a polygon?
For a shape to be a polygon, certain conditions are necessary. First, there must be many sides and corners. Also, it must be a closed form. is a shape with all sides and angles equal. Accordingly, in the wrong one, they may be slightly deformed.
Types of regular polygons
What is the minimum number of sides a regular polygon can have? One line cannot have many sides. The two sides also cannot meet and form a closed shape. And three sides can - this is how you get a triangle. And since we are talking about regular polygons where all sides and angles are equal, we mean
If you add one more side, you get a square. Can a rectangle where the sides are not equal be a regular polygon? No, this shape will be called a rectangle. If you add the fifth side, you get a pentagon. Accordingly, there are hexagons, heptagons, octagons and so on ad infinitum.
Elementary geometry
Polygons come in different types: open, closed, and self-intersecting. In elementary geometry, a polygon is a flat figure that is bounded by a finite chain of straight line segments in the form of a closed polyline or contour. These line segments are its edges or sides, and the points where two edges meet are its vertices and corners. The interior of a polygon is sometimes referred to as its body.
Polyhedrons in nature and human life
While many living forms abound in pentagonal patterns, the mineral world prefers double, triple, fourfold, and sixfold symmetry. The hexagon is a dense shape that maximizes structural efficiency. It is very common in the field of molecules and crystals, in which pentagonal shapes are almost never found. Steroids, cholesterol, benzene, vitamins C and D, aspirin, sugar, graphite are all manifestations of sixfold symmetry. Where do regular polyhedra occur in nature? The most famous hexagonal architecture is created by bees, wasps and hornets.
Six water molecules form the core of each snow crystal. This makes a snowflake. The facets of the fly's eye form a tightly packed hexagonal arrangement. What other regular polyhedra are there in nature? These are crystals of water and diamond, basalt columns, epithelial cells in the eye, some plant cells and much more. Thus, polyhedrons created by nature, both living and inanimate, are present in human life in a huge number and variety.
What is the reason for the popularity of hexagons?
Snowflakes, organic molecules, quartz crystals and columnar basalts are hexagons. The reason for this is their inherent symmetry. The most striking example is the honeycomb, the hexagonal structure of which minimizes the spatial disadvantage, since the entire surface is used very rationally. Why divide into identical cells? Bees create regular polyhedrons in nature in order to use them for their needs, including for storing honey and laying eggs. Why does nature prefer hexagons? The answer to this question can be given by elementary mathematics.
- Triangles. Take 428 equilateral triangles with sides about 7.35 mm. Their total length is 3 * 7.35mm * 428/2 = 47.2cm.
- Rectangles. Take 428 squares with a side of about 4.84 mm, their total length is 4 * 4.84 m * 428/2 = 41.4 cm.
- Hexagons. And finally, take 428 hexagons with a side of 3 mm, their total length is 6 * 3 mm * 428/2 = 38.5 cm.
The victory of the hexagons is obvious. It is this shape that helps to minimize the space as much as possible and allows you to place as many figures as possible in a smaller area. The honeycomb in which the bees store their amber nectar is a marvel of precision engineering, an array of prismatic cells with a perfectly hexagonal cross-section. The wax walls are made to a very precise thickness, the cells are carefully tilted to prevent viscous honey from falling out, and the entire structure is aligned with the Earth's magnetic field. In a surprising way, bees work simultaneously, coordinating their efforts.
Why hexagons? This is simple geometry
If you want to bring together cells of the same shape and size so that they fill the entire plane, then only three regular shapes (with all sides and with the same angles) will work: equilateral triangles, squares and hexagons. Of these, hexagonal cells require the smallest overall wall length compared to triangles or squares in the same area.
So the bees' choice of hexagons makes sense. Back in the 18th century, the scientist Charles Darwin declared that hexagonal honeycombs were "absolutely ideal in saving labor and wax." He believed that natural selection endowed bees with instincts to create these wax chambers, which had the advantage of being less energy and time consuming than other forms.
Examples of polyhedra in nature
The compound eyes of some insects are packed in a hexagonal, where each facet is a lens connected to a long, thin retinal cell. The structures that are formed by clusters of biological cells often have shapes governed by the same rules as bubbles in soapy water. The microscopic structure of the facet of the eye is one of the best examples. Each facet contains a cluster of four light-sensitive cells, which have the same shape as a cluster of four regular bubbles.
What determines these rules for soap films and bubble shapes? Nature is even more concerned about economy than bees. Bubbles and soap films are made of water (with the addition of soap), and surface tension pulls the surface of the liquid in such a way as to give it as little area as possible. This is why droplets are (more or less) spherical when they fall: a sphere has a smaller surface area than any other shape with the same volume. On a sheet of wax, water droplets are drawn into small beads for the same reason.
This surface tension explains the bubble raft and foam patterns. The foam will look for the structure that has the lowest total surface tension, which will provide the smallest wall area. Although the geometry of soap films is dictated by the interaction of mechanical forces, it does not tell us what the shape of the foam will be. Typical foam contains polyhedral cells of various shapes and sizes. If you take a closer look, the regular polyhedra in nature are not so correct. Their edges are rarely perfectly straight.
Correct bubbles
Let's say you can make a “perfect” foam in which all the bubbles are the same size. What is the perfect cell shape that makes the total area of the bubble wall as small as possible? This has been discussed for many years, and for a long time it was believed that the ideal cell shape is a 14-sided polyhedron with square and hexagonal sides.
In 1993, a more economical, albeit less ordered structure was discovered, consisting of a repeating group of eight different cell shapes. This more sophisticated model was used as inspiration for the foamy design of the swimming stadium during the 2008 Beijing Olympics.
The rules for cell formation in foam also control some of the patterns observed in living cells. Not only does the compound eye of flies show the same hexagonal packing of facets as the flat bubble. The light-sensitive cells inside each of the individual lenses also combine into groups that look like soap bubbles.
The world of polyhedra in nature
The cells of many different types of organisms, from plants to rats, contain membranes with such microscopic structures. No one knows what they are for, but they are so widespread that it is fair to assume that they have a useful role to play. Perhaps they isolate one biochemical process from another, avoiding cross-intervention.
Or maybe this is just an effective way to create a large working plane, since many biochemical processes take place on the surface of membranes, where enzymes and other active molecules can be incorporated. Whatever the function of polyhedra in nature, you shouldn't bother creating complex genetic instructions, because the laws of physics will do it for you.
Some butterflies have winged scales that contain an ordered maze of durable material called chitin. Exposure to light waves bouncing off normal ridges and other structures on the wing's surface causes some wavelengths (that is, some colors) to disappear while others reinforce each other. Thus, the polygonal structure offers an excellent means of producing animal color.
To make ordered networks of hard mineral, some organisms appear to form a shape from soft flexible membranes and then crystallize solid material inside one of the interpenetrating networks. The honeycomb structure of hollow microscopic channels within the chitinous spines of the unusual known as the sea mouse, transforms these hair-like structures into natural optical fibers that can direct light, changing it from red to bluish-green depending on the direction of illumination. This color change can serve to deter predators.
Nature knows better
The flora and fauna are replete with examples of polyhedra in living nature, as well as the inanimate world of stones and minerals. From a purely evolutionary point of view, the hexagonal structure is the leader in energy optimization. In addition to the obvious advantages (space saving), polyhedral meshes provide a large number of edges, therefore, the number of neighbors increases, which has a beneficial effect on the entire structure. The end result of this is that information travels much faster. Why are regular hexagonal and irregular star polyhedra so common in nature? Probably so it is necessary. Nature knows better, she knows better.
Main goal: Expansion and systematization of information about polygons.
Learning objectives:
Educational: Review the formulas for calculating the areas of polygons with students. Polygon properties.
Educational: Show students the practical application of polygons in human life.
Developing: Practical application and development of logical thinking.
Guys, the purpose of our lesson is to repeat the definitions, properties of polygons and answer the question: Why do we need this knowledge? During the lesson, you will perform various tasks, and enter the results on the control sheet. One correct answer to a question is one point. At the end of the lesson, according to the number of points scored, each of you will receive a corresponding mark.
I wish you all success!
II Repetition of what has been learned:
1. Guys, you are presented with various polygons. (Slide 2)
Write down the numbers:
- Triangles
- Parallelograms
- Trapezoid
- Rhombov
Swap notebooks with your deskmate and check. Count the number of correct answers and write it down on the checklist. (Slide 3)
2). The second task is to test your knowledge of the definitions of polygons.
Complete the sentences or insert the missing word. (Slide 4)
Swap notebooks with your deskmate and check. Count the number of correct answers and write it down on the checklist.
3. Guys, imagine that all the polygons gathered in a forest glade and began to discuss the issue of choosing their king. They argued for a long time and could not come to a consensus in any way. And then one old parallelogram said: “Let's all go to the kingdom of polygons. Whoever comes first will be the king ”(Slide 5) Everyone agreed. Early in the morning everyone set off on a long journey. (Slide 6) On the way of the travelers they met a river that said: “Only those whose diagonals intersect and the intersection point is divided in half will swim across me.” Some of the figures remained on the bank, the rest swam safely and went on. On the way, they met a high mountain, which said that it would only allow those with equal diagonals to pass. Several travelers stayed at the mountain, the rest continued on their way. We reached a large cliff, where there was a narrow bridge. The bridge said that it will allow those whose diagonals intersect at right angles. Only one polygon crossed the bridge, which was the first to reach the kingdom and was proclaimed king.
Question: Who became king?
Additional question: Why did the square become king?
(Since the square has more properties)
4. We have repeated the definitions, properties of polygons, but you must still be able to calculate the areas of these shapes. (Slide 7) Your attention is offered a set of figures and formulas for calculating areas. Establish a correspondence between them.
Check it out. Count the number of correct matches and enter the result on the control sheet.
III. Practical application of the knowledge gained.
1. Often in life we are faced with tasks in which we must be able to find the area of a particular figure.
I have a piece of cloth measuring 38 square meters. units (Slide 8)
Will this fabric be enough for me to make an appliqué made from these shapes?
The solution of the problem. Examination. Results in the control sheet.
2. The applique is made up of figures that can be folded into a square called "Tangram". (Slide 9)
Tangram is a world famous game based on ancient Chinese puzzles. According to legend, 4 thousand years ago, one man lost a ceramic tile from his hands and broke into 7 parts. Excited, he tried to collect it with his staff. But from the newly composed parts, each time he received new interesting images. This lesson soon turned out to be so exciting, puzzling, that the compiled square of seven geometric shapes was called the Board of Wisdom. If you cut the square, as shown in the picture above, you get the popular Chinese puzzle TANGRAM, which in China is called "chi tao tu", i.e. mental puzzle of seven parts. The name "tangram" originated in Europe most likely from the word "tan", which means "Chinese" and the root "gram". We now have it distributed under the name "Pythagoras"
Drawings composed of various polygons are also used in such a modern construction industry as parquetry. (Slide10)
The parquet floor has always been considered a symbol of prestige and good taste. The use of precious woods for the production of elite parquet flooring and the use of various geometric patterns add sophistication and respectability to the room.
The very history of art parquet is very ancient - it dates back to about the 12th century. It was then that new trends at that time began to appear in noble and noble mansions, palaces, castles and family estates - monograms and heraldic distinctions on the floor of halls, halls and vestibules, as a sign of a special belonging to the powerful of this world. The first artistic parquet was laid out quite primitively, from the point of view of modern times, from ordinary wooden pieces that matched the color. Today, the formation of complex ornaments and mosaic combinations is available. This is achieved through high precision laser and mechanical cutting.
I want to offer you the task of creating a parquet floor (Slide 11)
Students are divided into three teams. Each team is given a package with a set of triangles, parallelograms, trapezoids and a sheet of 280x120 mm. It is necessary to cover the “floor” with parquet, after making calculations. (See slide 12)
Students who are part of the winning team write down 5 points on the control sheet, 2nd place - 4 points, 3rd place - 3 points.
IV. Summarizing
You coped with all the tasks with dignity, let's remember, but what is the purpose of our lesson? Can you now answer the question "Why do we need polygons?" (Slide 13)
I would like to give a few more examples of the application of knowledge about polygons in our life.
When conducting trainings: Polygons are drawn by people who are quite demanding of themselves and others, who achieve success in life not only thanks to patronage, but also to their own strengths. When polygons have five, six or more corners, and are connected with decorations, then we can say that they were drawn by an emotional person who sometimes makes intuitive decisions.
VALUES of fortune telling on coffee - The correct quadrangle is the best sign. Your life will be happy and you will be financially secure, there are profits.
Summarize your work on the checklist and give yourself a final grade. (Slide 14)
V Reflection
Lesson is assessed by children through Smilies with different moods (Slide 15)