“Polygons” - Material for self-study on the topic “Polygons” Tasks for the game. Triangle (equilateral). Broken. Non-convex. Compiled by Soloninkina T.V. The finite portion of a plane bounded by a polygon. Draw a convex pentagon. Pentagon. Regular polygons. Expert 2.
“Measuring the area of a polygon” - Learning something new. 1. How to measure the area of a figure? -Everyone knows the concept of area from life experience. Abu r-Rayhan al-Buruni. 3. Lesson objectives: From today we will learn to calculate the areas of various geometric shapes. We often hear: “the area of our apartment is 63 m2.” Cherevina Oksana Nikolaevna.
“Areas of figures geometry” - Figures having equal areas are called equal in area. H. S=(a?b):2. Rectangle, triangle, parallelogram. C. S=a?b. D. Teacher: Ivniaminova L.A. Areas of figures. A. B. b. Authors: Zyryanova N. Jafarova A, grade 8b.
“Regular polygon” - Corollary 1. Regular polygons. Basic formulas. R. Regular triangle. Corollary 2. A circle circumscribed about a regular polygon. r. Consequences. A circle inscribed in a regular polygon. Regular hexagon. O. Application of formulas. In any regular polygon you can inscribe a circle, and only one.
"Parallelogram" - Parallelogram. If a quadrilateral has opposite sides equal in pairs, then the quadrilateral is a parallelogram. If two sides of a quadrilateral are equal and parallel. What is a parallelogram? Signs of a parallelogram. In a parallelogram, opposite sides and opposite angles are equal. The diagonals of a parallelogram are divided in half by the point of intersection.
“Rectangle rhombus square” - Solving problems on the topic “Rectangle. A. Answers to the screening test. Find: MD + DN. Rhombus. Purpose of the lesson: To consolidate theoretical material on the topic “Rectangle. Theoretical independent work Fill out the table, marking the signs + (yes), - (no). Correct answers to theoretical independent work.
There are 19 presentations in total
Russkikh Egor, Tarasov Dmitry
The world around us is a world of forms, it is very diverse and amazing. We are surrounded by household objects of various types. Having studied this topic, we really saw that polygons surround us everywhere and are found in various spheres of life.
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Regular polygons
Amazing polygon
Star polygons
Polygons in nature
Polygons in nature
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Regular polygons in science and some other areas of life Project authors: 8th grade Russian students Egor Tarasov Dmitry. Scientific supervisor: mathematics teacher Rakhmankulova E.R.
Problematic question. What place do polygons occupy in our lives? Object of study: polygons. Subject of research: practical application of polygons in the world around us.
Goal: systematize knowledge on this topic and obtain 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 (literature study); 2. Research. Hypothesis: Polygons create beauty in human surroundings.
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 objects of various types. Having studied this topic, we really saw that polygons surround us everywhere and are found in various spheres of life.
Thank you for your attention!
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Regional scientific and practical conference
Section Mathematics
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Stages of research work:
· selection of a research topic of interest,
· discussion of the research plan and intermediate results,
· work with different information sources;
· intermediate consultations with the teacher,
· public speaking with demonstration of presentation material.
Equipment used: Digital camera, multimedia equipment.
Hypothesis:
Polygons create beauty in human surroundings.
Research topic
Properties of polygons in everyday life, life, nature.
Note: All completed work contains not only informational, but also scientific material. Each section has a computer presentation that illustrates each area of research.
Experimental base. The successful completion of the research work was facilitated by a lesson in the “Geometry Around Us” circle and lessons in geometry, geography, and physics.
Brief literature review: We became acquainted with polygons in geometry lessons. Additionally, we learned from the book “Entertaining Geometry”, the magazine “Mathematics at School”, the newspaper “Mathematics”, and the encyclopedic dictionary of a young mathematician edited. Some data was taken from the magazine “Read, Learn, Play”. Much information is obtained from the Internet.
Personal contribution: In order to connect the properties of polygons with life, we began to talk with students and teachers whose grandparents or other relatives were engaged in carving, embroidery, knitting, patchwork, etc. From them we received valuable information.
Polygons
We decided to explore the geometric shapes that are found around us. Having become interested in the problem, we drew up a work plan. We decided to study: the use of polygons in practical human activities. To answer the questions posed, we had to: think on our own, ask another person, consult books, conduct observations. We looked for answers to questions in books. - What polygons have we studied? We conducted an observation to answer the question. - Where can I see this? During the lesson, an extracurricular event in mathematics “Parade of Quadrilaterals” was held, where they learned about the properties of quadrilaterals.
Geometry in architecture. Modern architecture boldly uses a variety of geometric shapes. Many residential buildings are decorated with columns. Geometric figures of various shapes can be seen in the construction of cathedrals and bridge designs.
Geometry in nature. There are many wonderful geometric shapes in nature itself. The polygons created by nature are incredibly beautiful and varied.
I.Regular polygons
Geometry is an ancient science and the first calculations were made over a thousand years ago. Ancient people made ornaments of triangles, rhombuses, and circles on the walls of caves. Since ancient times, regular polygons have been considered a symbol of beauty and perfection. Over time, man learned to use the properties of figures in practical life. Geometry in everyday life. The walls, floor and ceiling are rectangles. Many things resemble a square, a rhombus, a trapezoid.
Of all the polygons with a given number of sides, the most pleasing to the eye is the regular polygon, in which all sides are equal and all angles are equal. One of these polygons is a square, or in other words, a square is a regular quadrilateral.
A square can be defined in several ways: a square is a rectangle in which all sides are equal, and a square is a rhombus in which all angles are right.
From a school geometry course we know: a square has all sides equal, all angles are right angles,
The diagonals are equal, mutually perpendicular, the point of intersection bisects and bisects the corners of the square.
The square has a number of interesting properties. So, for example, if you need to enclose a quadrangular area of the largest area with a fence of a given length, then you should choose this area in the form of a square.
The square has symmetry, which gives it simplicity and a certain perfection of form: the square serves as a standard for measuring the areas of all figures.
The book “The Amazing Square” sets out in detail the proofs of some properties of the square, gives an example of a “perfect square” and a solution to one problem of cutting a square by the 10th-century Arab mathematician Abul Vefa.
I. Lehman’s book “Fascinating Mathematics” contains several dozen problems, including some that are thousands of years old. For a complete understanding of the construction by folding a square piece of paper, the book “Apply Math” was used. Here you can list a number of square puzzles: magic squares, tangrams, pentominoes, tetrominoes, polyominoes, stomachions, origami. I want to talk about some of them.
1. Magic squares
Sacred, magical, mysterious, mysterious, perfect... As soon as they were called. “I don’t know anything more beautiful in arithmetic than these numbers, called by some planetary and by others magic,” wrote the famous French mathematician, one of the creators of number theory, Pierre de Fermat, about them. Attractive with natural beauty, filled with inner harmony, accessible, but still incomprehensible, hiding many secrets behind the apparent simplicity...
Meet magic squares - amazing representatives of the imaginary world of numbers.
Magic squares originated in ancient times in China. Probably the “oldest” of the magic squares that have come down to us is the Lo Shu table (c. 2200 BC). It is 3x3 in size and filled with natural numbers from 1 to 9.
2. Tangram
Tangram is a world-famous game based on ancient Chinese puzzles. According to legend, 4 thousand years ago, a ceramic tile fell out of one man’s hands and broke into 7 pieces. Excited, he tried to collect it with his staff. But from the newly composed parts I received new interesting images each time. This activity soon turned out to be so exciting and puzzling that the square made up of seven geometric shapes was called the Board of Wisdom. If you cut a square, you get the popular Chinese puzzle TANGRAM, which in China is called "chi tao tu", i.e. a mental puzzle with seven parts. The name "tangram" originated in Europe most likely from the word "tan", which means "Chinese" and the root "gram". In our country it is now common under the name "Pythagoras"
3. Star polygons
In addition to the usual regular polygons, there are also stellate ones.
The term "stellate" has a common root with the word "star", and this does not indicate its origin.
The star pentagon is called a pentagram. The Pythagoreans chose a five-pointed star as a talisman; it was considered a symbol of health and served as an identification mark.
There is a legend that one of the Pythagoreans was sick in the house of strangers. They tried to get him out, but the disease did not subside. Without the means to pay for treatment and care, the patient, before his death, asked the owner of the house to draw a five-pointed star at the entrance, explaining that by this sign there would be people who would reward him. And in fact, after some time, one of the traveling Pythagoreans noticed a star and began asking the owner of the house how it appeared at the entrance. After the owner's story, the guest generously rewarded him.
The pentagram was well known in Ancient Egypt. But it was adopted directly as an emblem of health only in Ancient Greece. It was the five-pointed star of the sea that “suggested” to us the golden ratio. This ratio was later called the “golden ratio”. Where it is present, beauty and harmony are felt. A well-built man, a statue, the magnificent Parthenon created in Athens are also subject to the laws of the golden ratio. Yes, all human life needs rhythm and harmony.
4. Star polyhedra
Many forms of stellate polyhedra are suggested by nature itself. Snowflakes are star-shaped polyhedra. Several thousand different types of snowflakes are known. But Louis Poinsot managed to discover two other stellate polyhedra 200 years later. Therefore, stellated polyhedra are now called Kepler–Poinsot bodies. With the help of star-shaped polyhedra, unprecedented cosmic forms burst into the boring architecture of our cities. The unusual polyhedron “Star” of the Doctor of Art History inspired the architect to create the project for the National Library in Damascus.
The great Johannes Kepler’s book “Harmony of the World” is known, and in his work “On Hexagonal Snowflakes” he wrote: “The construction of a pentagon is impossible without the proportion that modern mathematicians call “divine.” He discovered the first two regular stellated polyhedra.
Star-shaped polyhedra are very decorative, which allows them to be widely used in the jewelry industry in the manufacture of all kinds of jewelry. They are also used in architecture.
Conclusion: There are a shockingly small number of regular polyhedra, but this very modest squad managed to get into the very depths of various sciences.
The star polyhedron is a delightfully beautiful geometric body, the contemplation of which gives aesthetic pleasure.
Ancient people saw beauty on the walls of caves in patterns of triangles, rhombuses, and circles. Since ancient times, regular polygons have been considered a symbol of beauty and perfection.
The star-shaped pentagon - the pentagram was considered a symbol of health and served as an identification mark of the Pythagoreans.
II.Polygons in nature
1. Honeycomb
Regular polygons are found in nature. One example is the honeycomb, which is a polygon covered with regular hexagons. Of course, they did not study geometry, but nature endowed them with the talent to build houses in the form of geometric shapes. On these hexagons, bees grow cells from wax. The bees deposit honey in them, and then cover them again with a solid rectangle of wax.
Why did the bees choose the hexagon?
To answer this question, you need to compare the perimeters of different polygons that have the same area. Let a regular triangle, a square and a regular hexagon be given. Which of these polygons has the smallest perimeter?
Let S be the area of each of the named figures, side and n be the corresponding regular n-gon.
To compare the perimeters, we write down their ratio: P3: P4: P6 = 1: 0.877: 0.816
We see that of the three regular polygons with the same area, the regular hexagon has the smallest perimeter. Therefore, wise bees save wax and time for building honeycombs.
The mathematical secrets of bees don't end there. It is interesting to further explore the structure of bee honeycombs. Smart bees fill the space so that there are no gaps left, saving 2% of wax. How to disagree with the opinion of the Bee from the fairy tale “A Thousand and One Nights”: “My house was built according to the laws of the strictest architecture. Euclid himself could learn from the geometry of my honeycomb.” Thus, with the help of geometry, we touched upon the secret of mathematical masterpieces made of wax, once again making sure of the comprehensive effectiveness of mathematics.
So, the bees, not knowing mathematics, correctly “determined” that a regular hexagon has the smallest perimeter among figures of equal area.
Beekeeper Nikolai Mikhailovich Kuznetsov lives in our village. He has been involved with bees since early childhood. He explained that when building honeycombs, bees instinctively try to make them as large as possible, while using as little wax as possible. The hexagonal shape is the most economical and efficient shape for honeycomb construction.
Cell volume is about 0.28 cm3. When building honeycombs, bees use the earth's magnetic field as a guide. Cells of honeycombs are drone, honey and brood. They differ in size and depth. Honey ones are deeper, drone ones are wider.
2. Snowflake.
A snowflake is one of nature's most beautiful creatures.
Natural hexagonal symmetry stems from the properties of the water molecule, which has a hexagonal crystal lattice held together by hydrogen bonds, allowing it to have a structural form with minimal potential energy in the cold atmosphere.
The beauty and variety of geometric shapes of snowflakes is still considered a unique natural phenomenon.
The mathematicians were especially struck by the “tiny white dot” found in the middle of the snowflake, as if it were the trace of the leg of a compass used to outline its circumference.” The great astronomer Johannes Kepler in his treatise “New Year's Gift. On Hexagonal Snowflakes” explained the shape of crystals by the will of God. Japanese scientist Nakaya Ukichiro called snow “a letter from heaven, written in secret hieroglyphs.” He was the first to create a classification of snowflakes. The world's only snowflake museum, located on the island of Hokkaido, is named after Nakai.
So why are snowflakes hexagonal?
Chemistry: In the crystal structure of ice, each water molecule participates in 4 hydrogen bonds directed to the vertices of the tetrahedron at strictly defined angles equal to 109°28" (in ice structures I, Ic, VII and VIII this tetrahedron is regular). In the center of this tetrahedron is an oxygen atom, in two vertices - a hydrogen atom, the electrons of which are involved in the formation of a covalent bond with oxygen. The two remaining vertices are occupied by pairs of valence electrons of oxygen, which do not participate in the formation of intramolecular bonds. Now it becomes clear why the ice crystal is hexagonal.
The main feature that determines the shape of a crystal is the connection between water molecules, similar to the connection of links in a chain. In addition, due to the different ratios of heat and moisture, the crystals, which in principle should be the same, take on different shapes. Colliding with supercooled small droplets on its way, the snowflake simplifies its shape while maintaining symmetry.
Geometry: The formative principle chose a regular hexagon not out of necessity determined by the properties of matter and space, but only because of its inherent property to completely, without a single gap, cover the plane and be closest to a circle of all the figures that have the same property.
Physics teacher – N
At temperatures below 0°C, water vapor immediately turns into a solid state and ice crystals form instead of droplets. The main water crystal has the shape of a regular hexagon in the plane. New crystals are then deposited on the vertices of such a hexagon, new crystals are deposited on them, and this is how those various shapes of stars - snowflakes, which are familiar to us, are obtained.
Mathematic teacher -
Of all the regular geometric figures, only triangles, squares and hexagons can fill a plane without leaving voids, with the regular hexagon covering the largest area. In winter we have a lot of snow. That's why nature chose hexagonal snowflakes to take up less space.
Chemistry teacher –
The hexagonal shape of snowflakes is explained by the molecular structure of water, but the question of why snowflakes are flat has not yet been answered.
E. Yevtushenko expresses the beauty of snowflakes in his poem.
From snowflake to ice
He lay down on the ground and on the roofs,
Amazing everyone with whiteness.
And he was really magnificent
And he was really beautiful...
.
III. Polygons around us
"The art of ornament contains in an implicit form the most ancient part of the higher mathematics known to us"
Hermann Weil.
1. Parquet
The lizards depicted by the Dutch artist M. Escher form, as mathematicians say, a “parquet”. Each lizard fits snugly against its neighbors without the slightest gap, like parquet flooring.
A regular division of a plane, called a “mosaic,” is a set of closed figures that can be used to tile the plane without intersections of the figures and gaps between them. Typically, mathematicians use simple polygons, such as squares, triangles, hexagons, octagons, or combinations of these figures, as shapes to make mosaics.
Beautiful parquet floors are made from regular polygons: triangles, squares, pentagons, hexagons, octagons. For example, circles cannot form parquet.
Parquet flooring has always been considered a symbol of prestige and good taste. The use of valuable wood species for the production of luxury parquet and the use of various geometric patterns give the room sophistication and respectability.
The history of artistic parquet itself is very ancient - it dates back to approximately 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 insignia on the floor of halls, halls and vestibules, as a sign of special affiliation with the powers that be. The first artistic parquet was laid out quite primitively, from a modern point of view - from ordinary wooden pieces that matched the color. Today, the formation of complex ornaments and mosaic combinations is available. This is achieved thanks to high precision laser and mechanical cutting.
At the beginning of the 19th century, instead of the refined lines of the parquet design, simple lines, clean contours and regular geometric shapes appeared, and strict symmetry in the compositional structure.
All aspirations in decorative art are aimed at displaying the heroism and uniquely meaningful classical antiquity. The parquet acquired a harsh geometry: now solid checkers, now circles, now squares or polygons with their division into narrow stripes in different directions. In newspapers of that time one could find advertisements in which it was proposed to choose parquet of exactly this pattern.
A characteristic parquet flooring of the Russian classics of the 19th century is the parquet designed by the architect Voronikhin in the Stroganov house on Nevsky Prospekt. The entire parquet consists of large shields with precisely repeated obliquely placed squares, at the crosshairs of which four-petal rosettes, lightly traced with graphemes, are modestly given.
The most typical parquet floors from the early 19th century are those designed by the architect C. Rossi. Almost all the drawings in them are distinguished by great laconicism, repetition, geometricism and clear division with straight or oblique slats that united the entire parquet floor of the apartment.
Architect Stasov chose parquet floors that consisted of simple shapes of squares and polygons. In all of Stasov’s projects one can feel the same rigor as Rossi’s, but the need to carry out restoration work, which fell to his lot after the fire of the palace, makes it more versatile and broader.
Just like Rossi's, Stasov's parquet flooring in the Blue Drawing Room of the Catherine Palace was built from simple squares united by horizontal, vertical or diagonal slats, forming large cells dividing each square into two triangles.
Geometricism is also observed in the parquet floors of Maria Feodorovna's library, where only the variety of color of the parquet - rosewood, amaranth, mahogany, rosewood, etc. - brings some animation.
The predominant color of the parquet is mahogany, on which the sides of the rectangles and squares are given by pear wood, framed by a thin layer of ebony, which gives even greater clarity and linearity to the entire pattern. Maple throughout the parquet is abundantly given in the form of ribbons, oak leaves, rosettes and ionites.
All of these parquet floors do not have a main central pattern; they all consist of repeating geometric motifs. A similar parquet was preserved in Yusupov’s former house in St. Petersburg.
Architects Stasov and Bryullov restored the apartments of the Winter Palace after the fire of 1837. Stasov created the parquets of the Winter Palace in the solemn, monumental and official style of Russian classics of the 30s of the 19th century. The colors of the parquet were also chosen exclusively classic.
In choosing parquet, when it was not necessary to combine the parquet with the pattern of the ceiling, Stasov remained true to his compositional principles. For example, the parquet flooring of the gallery of 1812 is distinguished by its dry and solemn majesty, which was achieved by the repetition of simple geometric shapes framed by a frieze.
2. Tessellation
Tessellations, also known as tiling, are collections of shapes that cover the entire mathematical plane, fitting together without overlap or gaps. Regular tessellations consist of figures in the form of regular polygons, when combined, all corners have the same shape. There are only three polygons suitable for use in regular tessellations. These are a regular triangle, a square and a regular hexagon. Semi-regular tessellations are those in which regular polygons of two or three types are used and all vertices are the same. There are only 8 semi-regular tessellations. Together, the three regular tessellations and eight semi-regular ones are called Archimedean. Tessellation, in which individual tiles are recognizable figures, is one of the main themes of Escher's work. His notebooks contain more than 130 variations of tessellations. He used them in a huge number of his paintings, including “Day and Night” (1938), the series of paintings “The Limit of the Circle” I-IV, and the famous “Metamorphoses” I-III (). The examples below are paintings by contemporary authors Hollister David and Robert Fathauer.
3. Patchwork from polygons
If stripes, squares and triangles can be done without special preparation and without skills using a sewing machine, then polygons will require a lot of patience and skill from us. Many quilters prefer to assemble polygons by hand. The life of every person is a kind of patchwork canvas, where bright and magical moments alternate with gray and dark days.
There is a parable about patchwork. “One woman came to the sage and said: “Teacher, I have everything: a husband, children, and a house - a full cup, but I began to think: why all this? And my life fell apart, everything is not a joy!” The sage listened to her, thought about it and advised her to try to sew her life together. The woman left the sage in doubt, but she tried. She took a needle and thread and sewed a piece of her doubts onto a piece of blue sky that she saw in the window of her room. Her little grandson laughed, and she sewed a piece of laughter onto her canvas. And so it went. The bird sings - and another piece is added; they will offend you to tears - another one.
The patchwork fabric was used to make blankets, pillows, napkins, and handbags. And everyone they came to felt how pieces of warmth settled in their souls, and they were never lonely again, and life never seemed empty and useless to them.”
Each craftswoman, as it were, creates the canvas of her life. You can verify this at work.
She passionately works creating patchwork quilts, bedspreads, rugs, drawing inspiration from each of her works.
4. Ornament, embroidery and knitting.
1). Ornament
Ornament is one of the oldest types of human visual activity, which in the distant past carried a symbolic magical meaning, a certain symbolism. The design was almost exclusively geometric, consisting of strict forms of circle, semicircle, spiral, square, rhombus, triangle and their various combinations. Ancient man endowed his ideas about the structure of the world with certain signs. With all this, the ornamentist has a wide scope when choosing motives for his composition. They are supplied to him in abundance by two sources - geometry and nature.
For example, a circle is the sun, a square is the earth.
2). Embroidery
Embroidery is one of the main types of Chuvash folk ornamental art. Modern Chuvash embroidery, its ornamentation, technique, and color scheme are genetically related to the artistic culture of the Chuvash people in the past.
The art of embroidery has a long history. From generation to generation, patterns and color schemes were refined and improved, and embroidery samples with characteristic national features were created. The embroidery of the peoples of our country is distinguished by great originality, a wealth of technical techniques, and color schemes.
Each nation, depending on local conditions, peculiarities of life, customs and nature, created its own embroidery techniques, pattern motifs, and their compositional structure. In Russian embroidery, for example, a large role is played by geometric patterns and geometrized forms of plants and animals: rhombuses, motifs of a female figure, birds, and also a leopard with a raised paw.
The sun was depicted in the shape of a diamond, a bird symbolized the arrival of spring, etc.
Of great interest are the embroideries of the peoples of the Volga region: Mari, Mordovians and Chuvash. The embroideries of these peoples have many common features. The differences lie in the motifs of the patterns and their technical execution.
Embroidery patterns composed of geometric shapes and highly geometric motifs.
Old Chuvash embroidery is extremely diverse. Various types of it were used in the manufacture of clothing, in particular canvas shirts. The shirt was richly decorated with embroidery on the chest, hem, sleeves, and back. And therefore, I believe that Chuvash national embroidery should begin with a description of the women’s shirt as the most colorful and richly decorated with ornaments. On the shoulders and sleeves of this type of shirt there is embroidery of geometric, stylized plant, and sometimes animal patterns. Shoulder embroidery is different in nature from sleeve embroidery, and it is like a continuation of the shoulder embroidery. On one of the old shirts, embroidery along with braid stripes, going down from the shoulders, goes down and ends at the chest with an acute angle. The stripes are arranged in the form of rhombuses, triangles, and squares. Inside these geometric figures there is small, mesh embroidery, and large hook-shaped and star-shaped figures are embroidered along the outer edge. Such embroideries were preserved in the Nikolaevs’ house. My relative embroidered them.
Another type of women's needlework is crocheting. Since ancient times, women have knitted a lot and tirelessly. This type of needlework is no less exciting than embroidery. Here is one of Tamara Fedorovna's works. She shared with us her memories of how every girl in the village was taught to cross-stitch on canvas and satin stitch, and knit stitches. By the number of knitted stitches, by things decorated with embroidery and lace, a girl was judged as a bride and future housewife. The stitching patterns were different, they were passed down from generation to generation, they were invented by the craftswomen themselves. The floral motif, geometric shapes, dense columns, covered and uncovered gratings are repeated in the stitching ornament. At 89 years old, Tamara Fedorovna is engaged in crocheting. Here are her handicrafts. She knits for children, relatives, and neighbors. He even takes orders.
Conclusion: Knowing about polygons and their types, you can create very beautiful decorations. And all this beauty surrounds us.
People have had the need to decorate household items for a long time.
5. Geometric carving
It so happened that Rus' is a country of forests. And such a fertile material as wood was always at hand. With the help of an ax, a knife and some other auxiliary tools, a person provided himself with everything necessary for: life: he erected housing and outbuildings, bridges and windmills, fortress walls and towers, churches, made machines and tools, ships and boats, sleighs and carts , furniture, dishes, children's toys and much more.
On holidays and leisure hours, he amused his soul with his rollicking tunes on wooden musical instruments: balalaikas, pipes, violins, and whistles.
Even ingenious and reliable door locks were made from wood. One of these castles is kept in the State Historical Museum in Moscow. It was made by a master woodworker back in the 18th century, lovingly decorated with triangular-notched carvings! (This is one of the names of geometric carvings,)
Geometric carving is one of the most ancient types of wood carving, in which the depicted figures have a geometric shape in various combinations. Geometric carving consists of a number of elements that form various ornamental compositions. Squares, triangles, trapezoids, rhombuses and rectangles are an arsenal of geometric elements that make it possible to create original compositions with a rich play of light and shadow.
I could see this beauty since childhood. My grandfather, Mikhail Yakovlevich Yakovlev, worked as a technology teacher at the Kovalinskaya school. According to my mother, he taught carving classes. I did this myself. The daughters of Mikhail Yakovlevich have preserved his works. The box is a gift for the eldest granddaughter on her 16th birthday. A backgammon box for the eldest grandson. There are tables, mirrors, photo frames.
The master tried to add a piece of beauty to each product. First of all, great attention was paid to shape and proportions. For each product, wood was selected taking into account its physical and mechanical properties. If the beautiful texture of wood in itself could decorate the products, then they tried to identify and emphasize it.
IV. Examples from life
I would like to give a few more examples of applying knowledge about polygons in our lives.
1/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 strength. When polygons have five, six or more angles, and are connected with decorations, then we can say that they were drawn by an emotional person who sometimes makes intuitive decisions.
2/Coffee fortune telling meanings:
If there is no quadrangle, this is a bad omen, warning of future troubles.
A regular quadrilateral is the best sign. Your life will pass happily, and you will be financially secure and have profits.
Summarize your work on the control sheet and give yourself a final mark.
The quadrangle is the space on the palm between the head line and the heart line. It is also called the hand table. If the middle of the quadrilateral is wide on the side of the thumb and even wider on the side of the palm, this indicates very good organization and composition, truthfulness, fidelity and a generally happy life.
3/ Palmistry - fortune telling by hand
The figure of the quadrangle (it also has another name - “hand table”) is placed between the lines of the heart, mind, fate and Mercury (liver). In case of weak expression or complete absence of the latter, its function is performed by the Apollo line.
A quadrilateral that is large in size, regular in shape, has clear boundaries and extends towards the Mount of Jupiter indicates good health and good character. Such people are ready to sacrifice themselves for the sake of others, they are open, unhypocritical, for which they are respected by others.
If the quadrangle is wide, a person’s life will be filled with various joyful events, he will have many friends. The overly modest size of the quadrangle or the curvature of the sides clearly states that the person who has it is infantile, indecisive, selfish, and his sensuality is undeveloped.
The abundance of small lines within the quadrangle is evidence of the limitations of the mind. If a cross in the shape of an “x” is visible inside the figure, this indicates the eccentric nature of the person being examined and is a bad sign. A cross that has the correct shape indicates that he is inclined to be interested in mysticism.
1. Amazing polygon
In addition to the theory of qi, the principles of yin and yang and Tao, there is another fundamental concept in the teachings of feng shui: the “sacred octagon”, called ba gua. Translated from Chinese, this word means “dragon body.” Guided by the principles of Ba Gua, you can plan the furnishings of the room so that it creates an atmosphere that promotes maximum spiritual comfort and material well-being. In Ancient China, it was believed that the octagon was a symbol of prosperity and happiness.
Characteristics of the ba-gua sectors.
Career - North
Sector color is black. The element that promotes harmonization is Water. The sector is directly related to our type of activity, place of work, realization of work potential, professionalism and earnings. Success or failure in this regard directly depends on the prosperity in the area of this sector.
Knowledge - Northeast
Sector color is blue. The element is Earth, but it has a rather weak effect. The sector is associated with the mind, the ability to think, spirituality, the desire for self-improvement, the ability to assimilate received information, memory and life experience.
Family - East
The color of the sector is green. The element that promotes harmonization is Wood. The direction is associated with family in the broadest sense of the word. This means not only your household, but also all relatives, including distant ones.
Wealth - southeast
The color of the sector is purple. The element – Wood – has a weak effect. The direction is associated with our financial condition, it symbolizes well-being and prosperity, material wealth and abundance in absolutely all areas.
Slava - south
Color – red. The element that makes this sphere active is Fire. This sector symbolizes your fame and reputation, the opinion of your loved ones and acquaintances.
Marriage - southwest
The color of the sector is pink. Element – Earth. The sector is associated with your loved one and symbolizes your relationship with him. If there is no such person in your life at the moment, this sector represents a void waiting to be filled. The state of the direction will tell you what your chances are of quickly realizing your potential in the field of personal relationships.
Children - West
The color of the sector is white. Element – Metal, but has a weak effect. Symbolizes your ability to reproduce in any area, both physical and spiritual. We can talk about children, creative self-expression, the implementation of various plans, the result of which will please you and those around you and will serve as your calling card in the future. Among other things, the sector is associated with your ability to communicate and reflects your ability to attract people to you.
Helpful People – North-West
Sector color is gray. Element – Metal. The direction symbolizes people you can rely on in difficult situations; it shows the presence in your life of those who are able to come to the rescue, provide support, and become useful to you in one area or another. In addition, the sector is associated with travel and the male half of your family.
Health is the center
The color of the sector is yellow. It does not have a specific element, it is connected with all elements as a whole, and from each it takes the necessary share of energy. The area symbolizes your mental and spiritual health, connection and harmony in all aspects of life.
2. Pi and regular polygons.
On March 14 this year, Pi Day will be celebrated for the twentieth time - an informal holiday of mathematicians dedicated to this strange and mysterious number. The “father” of the holiday was Larry Shaw, who drew attention to the fact that this day (3.14 in the American date system) falls, among other things, on Einstein’s birthday. And, perhaps, this is the most appropriate moment to remind those who are far from mathematics about the wonderful and strange properties of this mathematical constant.
Interest in the value of the number π, which expresses the ratio of the circumference to the diameter, arose in ancient times. The well-known formula for the circumference L = 2 π R is also the definition of the number π. In ancient times it was believed that π = 3. For example, this is mentioned in the Bible. In the Hellenistic era it was believed that, and this meaning was used by both Leonardo da Vinci and Galileo Galilei. However, both approximations are very rough. A geometric drawing depicting a circle circumscribed about a regular hexagon and inscribed in a square immediately gives the simplest estimates for π: 3< π < 4. Использование буквы π для обозначения этого числа было впервые предложено Уильямом Джонсом (William Jones, 1675–1749) в 1706 году. Это первая буква греческого слова περιφέρεια
Conclusion: We answered the question: “Why study mathematics?” Because in the depths of our souls, each of us lives a secret hope to know ourselves, our inner world, to improve ourselves. Mathematics provides such an opportunity - through creativity, through a holistic view of the world. The octagon is a symbol of prosperity and happiness.
V. Regular polygons in architecture
Sculptors, architects, and artists also showed great interest in the forms of regular polyhedra.
In geometry lessons we learned the definitions, characteristics, properties of various polygons.
After reading the literature on the history of architecture, we came to the conclusion that the world around us is a world of forms, it is very diverse and amazing. We saw that buildings have a wide variety of shapes.
We are surrounded by household objects of various types. After studying this topic, we really saw that polygons are all around us. In Russia, buildings have very beautiful architecture, both historical and modern, in each of which you can find different types of polygons.
1. Architecture of Moscow and other cities of the world.
How beautiful the Moscow Kremlin is. Its towers are beautiful! How many interesting geometric shapes are used as their basis! For example, the Alarm Tower. On a high parallelepiped there is a smaller parallelepiped, with openings for windows, and a quadrangular truncated pyramid is erected even higher. There are four arches on it, topped by an octagonal pyramid. Geometric figures of various shapes can be recognized in other remarkable structures erected by Russian architects. St. Basil's Cathedral)
The expressive contrast of a triangle and a rectangle on the facade attracts the attention of visitors to the Groningen Museum (Holland) (Fig. 9). Round, rectangular, square - all these shapes coexist perfectly in the building of the Museum of Modern Art in San Francisco (USA). The building of the Georges Pompidou Center for Contemporary Art in Paris is a combination of a giant transparent parallelepiped with openwork metal fittings.
2. Architecture of the city of Cheboksary
The capital of the Chuvash Republic is the city of Cheboksary (Chuv. Shupashkar), located on the right bank of the Volga, has a centuries-old history. In written sources, Cheboksary has been mentioned as a settlement since 1469 - then Russian soldiers stopped here on their way to the Kazan Khanate. This year is considered to be the time of the founding of the city, but historians are already insisting on revising this date - materials found during the latest archaeological excavations indicate that Cheboksary was founded in the 13th century by settlers from the Bulgarian city of Suvar.
The city was universally famous for its bell production - Cheboksary bells were known both in Russia and in Europe.
The development of trade, the spread of Orthodoxy and the mass baptism of the Chuvash people also led to the architectural flourishing of the city - the city was replete with churches and temples, in each of which various polygons are visible
Cheboksary is a very beautiful city. In the capital of Chuvashia, the novelty of a modern metropolis and antiquity, where geometricism is expressed, are surprisingly intertwined. This is expressed primarily in the architecture of the city. Moreover, a very harmonious interweaving is perceived as a single ensemble and only complements each other.
3. Architecture of the village of Kovali
You can see beauty and geometricism in our village. Here is a school that was built in 1924, a monument to soldiers - soldiers.
Conclusion:
Without geometry there would be nothing, because all the buildings that surround us are geometric shapes.
Conclusion
After conducting research, we came to the conclusion that, indeed, knowing about polygons and their types, you can create very beautiful decorations and build diverse and unique buildings. And all this is the beauty that surrounds us.
Human ideas about beauty are formed under the influence of what a person sees in living nature. In her various creations, very far from each other, she can use the same principles. And we can say that polygons create beauty in art, architecture, nature, and in human surroundings.
Beauty is everywhere. It exists in science, and especially in its pearl - mathematics. Remember that science, led by mathematics, will reveal fabulous treasures of beauty to us.
List of used literature.
1. Models of polyhedra. Per. from English . M., "Mir", 1974
2. Mathematical novels. Per. from English . M., "Mir", 1974.
3. M. Introduction to geometry. M., Nauka, 1966.
4. Mathematical kaleidoscope. Per. from Polish. M., Nauka, 1981.
5., Erganzhiev geometry: Textbook for grades 5-6. –
Smolensk: Rusich, 1995.
6. , Orlova on wood. M.: Art
At the beginning of the last century, the great French architect Corbusier once exclaimed: “Everything around is geometry!” Today we can repeat this exclamation with even greater amazement. In fact, look around - geometry is everywhere! Geometric knowledge and skills are today professionally significant for many modern specialties, for designers and constructors, for workers and scientists. A person cannot truly develop culturally and spiritually if he has not studied geometry at school; geometry arose not only from the practical, but also from the spiritual needs of man.
Geometry is a whole world that surrounds us from birth. After all, everything we see around us relates to geometry in one way or another, nothing escapes its attentive gaze. Geometry helps a person walk through the world with his eyes wide open, teaches him to look carefully around and see the beauty of ordinary things, to look, think and draw conclusions.
“A mathematician, just like an artist or poet, creates patterns. And if his patterns are more stable, it is only because they are composed of ideas... The patterns of a mathematician, just like the patterns of an artist or a poet, must be beautiful; an idea, just like colors or words, must be harmonious with each other. Beauty is the first requirement: there is no place in the world for ugly mathematics.”
Relevance of the selected topic
In geometry lessons we learned the definitions, characteristics, properties of various polygons. Many objects around us have a shape similar to the geometric shapes already familiar to us. The surfaces of a brick or a piece of soap consist of six sides. Rooms, cabinets, drawers, tables, reinforced concrete blocks resemble in their shape a rectangular parallelepiped, the edges of which are familiar quadrangles.
Polygons undoubtedly have beauty and are used very widely in our lives. Polygons are important to us, without them we would not be able to build such beautiful buildings, sculptures, frescoes, graphics and much more. I became interested in the topic “Polygons” after a lesson - a game, where the teacher presented us with a task - a fairy tale about choosing a king.
All the polygons gathered in a forest clearing and began to discuss the issue of choosing their king. They argued for a long time and could not come to a common opinion. And then one old parallelogram said: “Let's all go to the kingdom of polygons. Whoever comes first will be the king.” Everyone agreed. Early in the morning everyone set off on a long journey. On the way, the travelers met a river that said: “Only those whose diagonals intersect and are divided in half by the point of intersection will swim across me.” Some of the figures remained on the shore, the rest swam safely and moved on. On the way they met a high mountain, which said that it would only allow those with equal diagonals to pass. Several travelers remained near the mountain, the rest continued on their way. We reached a large cliff where there was a narrow bridge. The bridge said it would allow those whose diagonals intersect at right angles to pass. Only one polygon crossed the bridge, who was the first to reach the kingdom and was proclaimed king. So they chose the king. I also chose a topic for my research work.
Purpose of the research work: Practical application of polygons in the world around us.
Tasks:
1. Conduct a literature review on the topic.
2. Show the practical application of polygons in the world around us.
Problematic question: How
Correct parquet floors. The project was prepared by a student of Municipal Educational Institution-Secondary School No. 6, Marx Zhilnikova Nastya Supervisor: Martyshova Lyudmila Iosifovna Goals and objectives Find out which regular convex polygons can be used to make a regular parquet. Consider all types of correct parquets and answer the question about their quantity. Consider examples of the use of regular polygons in nature. . We often encounter parquet in everyday life: they cover floors in houses, cover the walls of rooms with various tiles, and often decorate buildings with ornaments. . . . . . . . . . . The first question that interests us and which can be easily solved is the following: from what regular convex polygons can a parquet be made? Sum of angles of a polygon. Let the parquet slab be a regular n-gon. The sum of all the angles of an n-gon is 180(n-2), and since all the angles are equal, each of them is 180(n-2)/n. Since an integer number of angles meet at each vertex of the parquet, the number 360 must be an integer multiple of 180(n-2)/n. Transforming the ratio of these numbers, we get 360n/ 180(n-2)= 2n/ n-2. 180(n-2), n is the number of sides of the polygon. It is quite simple to make sure that no other regular polygon forms the parquet. And here we need the formula for the sum of the angles of a polygon. If the parquet is made up of n-gons, then k 360: a n polygons will converge at each vertex of the parquet, where a n is the angle of a regular n-gon. It is easy to find that a 3 = 60°, a 4 = 90°, a 5 = 108°, a 6 = 120°. 360° is divisible by a n only when n = 3; 4; 6. It is clear from this that n-2 can only take the values 1, 2 or 4; therefore, the only possible values for n are 3, 4, 6. Thus, we get parquets made up of regular triangles, squares or regular hexagons. Other parquets made from regular polygons are impossible. PARQUETS - TERMINATION OF A PLANE WITH POLYGONS Already the Pythagoreans knew that there are only three types of regular polygons with which a plane can be completely paved without gaps or overlaps - triangle, square and hexagon. PARQUETS - PLANE TILES WITH POLYGONS You can require that the parquet be regular only “at the vertices”, but allow the use of different types of regular polygons. Then eight more parquet floors will be added to the original three. . Parquets from different regular polygons. First, let's find out how many different regular polygons (with the same side lengths) can be around each point. The angle of a regular polygon must be in the range from 60° to 180° (not including); therefore, the number of polygons located in the vicinity of a point must be greater than 2 (360°/180°) and cannot exceed 6 (360°/60°). Parquets from different regular polygons. It can be shown that there are the following ways to lay parquet using combinations of regular polygons: (3,12,12); (4,6,12); (6,6,6); (3,3,6,6) - two parquet options; (3,4,4,6) - four options; (3,3,3,4,4) - four options; (3,3,3,3,6); (3,3,3,3,3,3) (the numbers in brackets are the designations of polygons converging at each vertex: 3 - regular triangle, 4 square, 6 - regular hexagon, 12 regular dodecagon). Coverings of a plane with regular polygons meet the following requirements: 1 The plane is covered entirely with regular polygons, without gaps or double coverings, i.e. two covering polygons either have a common side, or have a common vertex, or have no common points at all. This covering is called parquet. 2 Around all vertices, regular polygons are arranged in the same way, i.e. Around all vertices, polygons of the same names follow in the same order. For example, if around one vertex the polygons are arranged in the sequence: triangle - square - hexagon - square, then around any other vertex of the same covering the polygons are arranged in exactly the same sequence. Regular parquet Thus, a parquet can be superimposed on itself in such a way that any given vertex of it is superimposed on any other previously given vertex. This kind of parquet is called correct. How many regular parquets are there and how are they arranged? Let us divide all regular parquets into groups according to the number of different regular polygons included in the parquet 1.a). Hexagons b). Squares c). Triangles 2.a). Squares and triangles b). Squares and octagons c). Triangles and hexagons d). Triangles and dodecagons 3.a). Squares, hexagons and dodecagons b). Squares, hexagons and triangles Regular parquets made from one regular polygon Group1 a). Hexagons b). Squares c). Triangles 1a. A coating consisting of regular hexagons. 1b. Parquet consisting only of squares. 1st century Parquet consisting of only triangles. Regular parquets composed of two regular polygons Group 2 a). Squares and triangles b). Squares and octagons c). Triangles and hexagons d). Triangles and dodecagons 2a. Parquets consisting of squares and triangles. View I. Arrangement of polygons around the vertex: triangle - triangle - triangle - square - square 2a. Type II. Parquets consisting of squares and triangles Arrangement of polygons around the top: triangle – triangle – square – triangle – square 2 b. Parquet consisting of squares and octagons 2c. Parquet consisting of triangles and hexagons. Type I and type II. Regular parquets composed of three regular polygons Group 3 a). Squares, hexagons and dodecagons b). Squares, hexagons and triangles 2d. Parquet consisting of dodecagons and triangles 3a.Parquet consisting of squares, hexagons and dodecagons. 3b. Parquet consisting of squares, hexagons and triangles Covering in the form of a sequence: triangle - square - hexagon - square This is impossible: Parquet consisting of regular pentagons does not exist. Coverings in the form of a sequence are not possible: 1) triangle – square – hexagon – square; 2) triangle – triangle – square – dodecagon; 3) triangle – square – triangle – dodecagon. Conclusions Pay attention to parquets that are made up only of regular polygons of the same name - equilateral triangles, squares and regular hexagons. Among these shapes (if all sides are equal), the regular hexagon covers the largest area. Therefore, if we want, for example, to divide an endless field into sections of 1 hectare in size so that as little material as possible is spent on fencing, then the sections need to be shaped into regular hexagons. . Another interesting fact: it turns out that the cut of a honeycomb also looks like a plane covered with regular hexagons. Bees instinctively strive to build the largest possible honeycombs in order to store more honey. . Conclusion So, all possible combinations have been considered. This is how 11 correct parquet floors turned out. They are very beautiful, aren't they? Which parquet floor did you like best? . . Sources A.N. Kolmogorov “Parquets made of regular polygons”. "Quantum" 1970 No. 3. Internet resources: htt://www. arbuz. uz/v parket. html. virlib.eunnet.net/mif/text/n0399/1.html nordww.narod.ru/…/laureat08/1549parket.htm Group of Companies "Amber Strand - Parquet". Product Catalog.