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By famous places peace the world Grade 3 Completed Oksana Nikolaevna Konovalova, primary school teacher of the first qualification category

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Taj Mahal The mausoleum of the Taj Mahal, located in Agra, is one of the most recognizable landmarks not only in India, but throughout the world. The construction was built by Emperor Shah Jahan in memory of his third wife, Mumtaz Mahal, who died during childbirth. The Taj Mahal is considered one of the most beautiful buildings in the world, as well as a symbol eternal love. In 1983, the Taj Mahal was included in the list of objects world heritage UNESCO.

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Taj Mahal Inside the mausoleum are two tombs - the shah and his beloved wife. Muslim tradition forbids the decoration of graves. Consequently, the bodies of Mumtaz and Shah Jahan were placed in a simple crypt.

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The pedestal and tomb of the Taj Mahal were built in 12 years, while the rest of the complex took another 10 years to complete. For the construction of the complex, materials from all over Asia were used. More than a thousand elephants were used for transportation. In total, twenty-eight types of precious and semi-precious stones were set in white marble. Shortly after the completion of the Taj Mahal, Shah Jahan was overthrown by his own son Aurangzeb and arrested at the Delhi Fort. After his death, he was buried in a mausoleum next to his beloved wife. By the end of the 19th century, parts of the building fell into disrepair. The Taj Mahal was looted by British soldiers and officials who carved precious materials from the walls of the building. In 1942, the government erected scaffolding in an attempt to disguise the Taj Mahal from attack by Luftwaffe and Japanese air forces. The same action was taken during the India-Pakistan Wars of 1965 and 1971. This had an effect, and the structure remained unharmed. The history of the construction of the Taj Mahal

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The Taj Mahal is the main attraction of India. The complex is currently threatened by pollution. environment. Due to the pollution of the Jumna River, there is a threat of its shallowing and soil erosion. Cracks began to appear in the walls of the tomb, and the mausoleum began to subside. Due to air pollution, the building began to lose its whiteness, a yellow coating appeared, which has to be cleaned every year. The Indian government is taking urgent action by shutting down hazardous industries in Agra and expanding the conservation area, but this has not yet had an effect. Taj Mahal

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Egyptian pyramids Egyptian pyramids are the greatest architectural monuments ancient egypt, among which one of the "seven wonders of the world" - the pyramid of Cheops and an honorary candidate of the "new seven wonders of the world" - the Pyramids of Giza. Pyramids are huge pyramid-shaped stone structures used as tombs for the pharaohs of ancient Egypt.

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Egyptian pyramids The Great Pyramid of Cheops, which is known anywhere in the world. It is she who is the face of the Egyptian pyramids and the largest building of antiquity, giving rise to many secrets and legends around her. The construction of the pyramid took two whole decades and was completed in 2560 BC. Inside the pyramid of Cheops there are three chambers - tombs. The lowest one is carved into the base of the rock on which the pyramid is built. For unknown reasons, its construction was not completed. Above it are the chamber of the Queen and the chamber of the Pharaoh.

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Egyptian pyramids The second most important is the pyramid of Khafre, the son of Cheops. It was built on a 10-meter plateau, because it seems to be higher than the pyramid of Cheops, but it is not. Its height is 136.4 meters, while that of Cheops is 146.5 meters. Not far from the pyramid of Khafre is the Great Sphinx - a monument carved into the rock. The facial features of the Sphinx repeat the face of Pharaoh Khafre.

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Egyptian Pyramids The third great pyramid is the Pyramid of Menkaure. It is the smallest of them, and built the latest. In height, it is only 66 meters, and the length of the base is 108.4 meters. Despite its small size, it is considered to be the most beautiful of the three pyramids. In addition, the pyramid of Menkaure marked the end of an era great pyramids. All subsequent buildings were small in size.

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Egyptian pyramids The Egyptian pyramids do not end there, we move from Giza to other parts of Egypt. The Step Pyramid of Djoser is one of the largest in Egypt. It is located in the village of Sakkara, and was built for Pharaoh Djoser by Imhotep himself. It occupies an area of ​​125 by 115 meters, and its height is 62 meters. This is the first pyramid in Egypt, and also very well preserved.

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Egyptian Pyramids The pyramid in Meidum can be safely called the most unusual in form. It is located 100 kilometers south of the capital of Egypt, it was built for Pharaoh Huni, but his son Sneferu completed it. It originally had 8 steps, but in our time only the last 3 are visible. After construction, its height was 118 meters, and the area was 146 by 146 meters.

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Egyptian pyramids pink pyramid unusual in that it has a pink tint due to the special stone used for construction. This is the third highest pyramid after Cheops and Khafre, 104.4 meters high. Researchers believe that this pyramid was also built by Pharaoh Sneferu, already known to us.

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Egyptian pyramids Not far from the Pink is broken pyramid built in the 26th century. BC e. She got her name because of the irregular shape. It was built in 3 stages, at each of which it was given different angles of inclination.

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The Statue of Liberty The Statue of Liberty was built on October 28, 1886. The French gave the statue to the American people as a token of friendship between France and America. Over the past years, the monument has been recognized not only as the personification of the friendship of the two peoples, but also as a symbol of the freedom of the American people, a symbol of the United States and New York as a whole.

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Statue of Liberty The history of the Statue of Liberty, as well as the place where it was built, is closely connected with the history of the formation of the United States. The monument was placed on a pedestal inside Fort Wood, which was built specifically for the war of 1812 in the shape of a star. For the first few decades, the complex was serviced by the US Lighthouse Service, and only in 1924 it became a National Monument. And in 1937, the area of ​​the monument was expanded to the borders of the entire island of Bedloe, in 1956 it was renamed Liberty Island.

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The Statue of Liberty The Statue of Liberty is located on Liberty Island in New York State. The goddess of freedom holds a torch in her right hand and a tablet in her left. With one foot, "Freedom" stands on broken chains. 354 steps lead to the top of the statue, and 192 steps to the top of the pedestal. There are 25 windows in the crown, which symbolizes the precious stones of the Earth and the rays of the sun that illuminate the world. The seven rays on the crown of the statue symbolize the 7 seas and 7 continents. The plaque the statue holds in its left hand reads: "July 4, 1776". This date is the day of the adoption of the Declaration of Independence of the United States. 31 tons of copper were used to cast the statue, and the total weight of the steel structure was 125 tons.

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Sydney Opera House Sydney Opera House is a musical theater in Sydney, one of the most famous and easily recognizable buildings in the world, which is a symbol of largest city Australia and one of the main attractions of the continent - the sail-like shells that form the roof make this building unlike any other in the world. The Opera House is recognized as one of the outstanding buildings modern architecture in the world and since 1973 has been calling card Sydney. Since June 28, 2007, the theater has been under the protection of UNESCO as a World Heritage Site.

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Sydney Opera At the heart of the Opera House project is the desire to bring people from the world of daily routine into the world of fantasy, where musicians and actors live. Every year, 1600 performances are held here - symphony concerts, ballets, operas, theatrical, dance and musical productions, comedy and children's shows. This has made the Sydney Opera House one of the most creative and busiest centers for the performing arts in the world. The Australian Ballet, the Sydney Dance Company and the Australian Opera Company and other companies perform here regularly.

Our project is called "Egyptian Pyramids"

We chose this project because in the history lessons we already read about the famous Egyptian pyramids and the nature of Egypt, and in the lessons of the “Visual” geometry course we studied the topic “Geometry around us”

We were attracted by the mysteriousness and significance of these structures; it is not for nothing that they are referred to as the Wonders of the World. And we decided to touch the history of Ancient Egypt and create a layout using pyramid models.

Thus, objective of the project: creating a layout of the Egyptian pyramids using handmade geometric models

Tasks: a) collect material about the pyramids of Egypt

b) learn all about the pyramids and make models of the pyramids

c) make a layout

We worked according to the plan:

Ø Determined the purpose and objectives of the work.

Ø Divide responsibilities.

Ø Collected information on this topic in encyclopedias and the Internet

Ø Calculated and made a sweep and glued the models.

Ø Designed the layout.

Ø We collected everything we found in the project folder

Ø Prepare for defense.

While working on the project, we learned a lot of interesting things about the pyramids.

Now we will tell you about the Egyptian pyramids

1 slidePyramids are considered to be the hallmark of Egypt. The pyramids and the Sphinx belong to the most characteristic monumental monuments of Ancient Egypt.

Pyramids were a classic type of royal tomb in the era ancient kingdom, and were also erected for the pharaohs of the Middle Kingdom.

Archaeologists have counted 80 pyramids. Not all of them have survived to this day.

3 SLIDE All early pyramids were built of stone, but most of the kings used special brick for their tombs, which did not collapse with time. Outside, it was covered with limestone slabs. The pyramids were built using mounds, although there are various theories about how they were built.

The oldest - pyramid of pharaoh Djoser - erected about five thousand years ago. Its height is 60 meters. The builder of the first pyramid, Imhotep, was an architect, physician, astronomer, writer, adviser to the pharaoh, and for many centuries was considered the greatest sage of antiquity.

The pyramids served the pharaohs, according to their religion, as a ladder by which they ascended to heaven. Therefore, the most ancient pyramids were stepped, had the form of stairs, and only later ones had smooth walls.

4 SLIDE Most the famous three great pyramids near Giza: the pyramids of the pharaoh Cheops, his son Khafre and his grandson Mekerina. The largest of them was built first - this the Pyramid of Cheops. Initially, it rose to 147 m, but due to the advance of the sands, its height decreased to 137 m. Each side of the square base of the pyramid is 233 m. The area of ​​\u200b\u200bthe pyramid is more than 50,000 m2. Pyramid of Cheops almost solid masonry. Her internal premises occupy a very small volume - no more than 3 - 4%.

The pyramid consists of two million three hundred thousand cubic limestone blocks with smoothly polished sides. It was calculated that each block weighs mainly 2.5 tons, and the heaviest - 15 tons. The total weight of the pyramid is about 5.7 million tons. Its stones are held by their own weight - there is no binding material. Despite this, the blocks are so carefully fitted to each other that the gap between them is no more than five millimeters.

5 SLIDE The pyramids are located as shown on the plan.

Three Great Pyramids were built at Giza, exactly aligned with the positions of the three stars in Orion's Belt. The Egyptians believed that this constellation was a harbinger of a great event - the summer solstice. In other words, the appearance of Orion in the sky was a sign of the beginning of the revival of nature, the revival of Egypt.

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Until the end of the 19th century. The Pyramid of Cheops was the tallest building in the world.

If the pyramid were hollow, it would include the entire site with a launch tower for a space rocket or St. Peter's Cathedral in Rome.

Until the 50s of the 20th century, the weight of the pyramid exceeded the tonnage of the entire US Navy, including aircraft carriers, and its stone blocks would be enough to build a city with a hundred thousand inhabitants or enclose the whole of France with a three-meter-high wall.

Scientists believe that even the atomic bomb dropped on Hiroshima could not destroy the creation of Cheops.

7 SLIDE The Pyramid of Cheops was built in the form of a regular pyramid with a square at the base.

1. The sides of the base are 230.35 meters each (b = 230.35 m).

2. The height of the pyramid of Cheops: 146.71 meters (h = 146.71 m).

3. Total triangular faces 4

4. A pyramid is built from cubic blocks of limestone, the largest of which has a rib length of 1.5 meters.

5. Probably, initially, 210 steps led to the top of the pyramid.

8-11 SLIDE Inside the pyramid are is a complex structure. The slides show the passages, shafts, ventilation structures where the pyramid walker passed.

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The most popular hypothesis asserts that a huge structure, similar to the petrified ray of the life-giving Egyptian sun, was created in twenty years.

The ancient Greek scientist Herodotus claims that the construction of the pyramid was worked continuously during each three months 100 thousand people.

14-15 SLIDE Speaking about the pyramids of Egypt, one cannot but touch on the riddle of the great Sphinx

And the ruler appointed one guard to each pyramid, ready to pounce on anyone who approaches, wrap around his neck, strangle, they could knock down and kill any person, and then return to their refuge ...

The famous soothsayer Edgar Cayce claimed that the Sphinx was created around the same time as the pyramid of Cheops. “The Sphinx faces exactly that point in the sky,” he said, where, around 10450 BC, three stars from Orion's belt shone in a strictly defined place above the horizon.

So what is the purpose of the Sphinx? The same Edgar Cayce wrote: “The most important information for modern mankind should be found at the base of the left front paw of the sphinx, but not in underground tunnels under him. The information is embedded in the cornerstone of the foundation of this paw. The tunnels under the sphinx, which are not yet known to you, also carry an informational load in their configurations. However, the capsule with the message to descendants is located under the left front paw ... "

There is a lot of talk these days about secrets. famous pyramids.Exist, scientific hypotheses that if you build a pyramid in a certain ratio of its elements, it will have unique properties.

Now let's move on to our layout.

On our layout, you see 3 pyramids and the Sphinx, as well as a caravan of camels going through the sands of Egypt to the city of Giza. We made all the pyramids out of paper, having previously unfolded them, and then gluing them together. The pyramids are regular quadrangular, the base is a square, and the side faces are isosceles triangles. We did not succeed right away, we had to redo the pyramids, they did not stick together. But gradually everything began to work out, and we came up with the idea of ​​adding the Sphinx first, and then the camel caravan, in order to better represent the nature and originality of Egypt.

In custody we would like to present our project portfolio, where we have enclosed the project passport, all the collected information, developments, sketches and materials for the presentation.

And further.

We decided to donate our project to the mathematics classroom in order to use it as a visual aid.

She is not torn, not a mantle, but an outfit of eternal stone.
She stands here not tired, even though many centuries have passed,
She reached her head up to the grayest clouds.
What did she save people?
Great stones oblivion?
The pupils of the yellow Nile look lazily at her.
Who sleeps in this ancient tomb?
Will the stones tell
How all of them were salted with tears and then crumbled with blood.
There is a pyramid on the ground, and the Gods talk about it.
She is not torn, not mantle, but an outfit of eternal stone

Municipal Autonomous Educational Institution "Lyceum No. 21", Pervouralsk

Project on

"The Geometric Secrets of the Egyptian Pyramids"

Fulfilled

Panov Dmitry

8th grade student

Supervisor

Krotova Irina Leonidovna

2016

CONTENT

Introduction……………………………………………………………………...……3

Theoretical part

What are the Egyptian pyramids……………………………………………………6

Practical part

The presence of mathematics in the Egyptian pyramids………………………………9

Geometry in pyramids…………………………………………………..……12

Conclusion………………………………………………………………..……15

Materials used……………………………………………………..17

INTRODUCTION

Ancient Egypt is known primarily for its stone giants - pyramids that served as the burial places of Egyptian kings and pharaohs. However, not all Egyptian rulers found rest inside the pyramids, and this is not the only mystery of the Egyptian pyramids. And although scientists have been studying the pyramids for more than a century, only recently they managed to lift the veil of secrecy over how the Egyptians built them and why they abandoned construction.

Modern history is replete with white spots and this is not surprising, because if modern people knew about the technologies and capabilities of people who lived in the distant past, they would have a different look at the "development" of modern society. There is an understanding that in ancient times people knew much more than modern science is trying to show us, and this knowledge was not limited only to the external material world, but led to an understanding of the internal, spiritual processes in a person. Ancient people possessed a perfect science of man. If we talk about the monumental buildings of the past, then many amazing architectural monuments have come down to our time. Among them are stone structures of unimaginable scale.

The oldest Egyptian mathematical texts that have come down to us date back to the beginning of the 2nd millennium BC. Mathematics was then used in astronomy, navigation, land surveying, in the construction of houses, dams, canals and military fortifications. The Egyptians wrote on papyrus, which is poorly preserved, and therefore, at present, there is much less knowledge about the mathematics of Egypt than about the mathematics of Babylon or Greece. It was probably better developed than can be imagined from the documents that have come down to us. This is indirectly confirmed by the fact that Greek mathematicians studied with the Egyptians.

The Egyptian pyramids arouse curiosity and admiration, and at the same time they are an intellectual challenge to humanity, demonstrating its impotence in unraveling the mysteries of these ancient structures. The very existence of the Egyptian pyramids, the mystery of their origin and purpose, worried the best minds of mankind for many millennia. Many scientists devoted their lives to revealing these secrets, but despite the long history of excavations and research, despite the development of science and technology modern world, humanity is not yet able to answer the huge number of mysteries that are hidden in the Egyptian pyramids. The main questions that have always worried the researchers of the pyramids can be reduced to four main points: who, when, how and, most importantly, why built these majestic structures. Since the time of Herodotus, who visited Egypt in the 5th century BC, and until now, scientists have practically not found answers to these questions.

There is no doubt that the main problem on the way to solving the riddle of the Great Pyramids is the complete inconsistency of the actions of various researchers and, as a result, the lack of systematic research on the issue. Specialists in the history and culture of ancient Egypt confine themselves to the official version, which is not free from obvious contradictions, and refuse to acknowledge the existence of a problem, much less discuss it with "non-specialists". Non-specialists in the history and culture of Egypt, but experts in any other field of knowledge sin with one-sided approaches that do not take into account the complex nature of the problem. The provocative nature of the problem itself attracts people with an artistic mindset who are ready to put forward fantastic hypotheses and wishful thinking too often and uncritically.

Therefore, the first step is to formulate a set of key circumstances that require a satisfactory explanation in the development and consideration of any hypothesis - this will create certain restrictions on the proposed hypotheses, which will allow them to be more tolerant. It is advisable to adhere to Occam's principle and not so much to look for an initial premise from which the existence and purpose of the pyramids can follow, but to take care that what is observed cannot be explained otherwise than with the help of the proposed idea. Secondly, any hypothesis that can be not only taken into account, but discussed, must contain specific proposals for its verification with a prediction of possible results.

It is both of these circumstances that should form the methodological essence of the approach to the study of the problem of the Great Pyramids of Egypt.

The relevance of this topic is that, on the other hand, people remain people: they, as always, try to find answers, each at their own mundane level, based only on the knowledge and experience that they used to rely on. But what if something more is hidden in the mystery of the Egyptian pyramids, which scientists, due to the limited knowledge and experience and the dogmatic nature of their thinking, are simply not able to know? For example, in the drawings of the Nazca desert, UFOs and crop circles: the elementary narrowness of thinking and the conservatism of pundits does not allow them to penetrate the secrets of Being, and therefore they continue to insist on their theories, in which there is no and cannot be a place for another Existence, different from physical. It resembles walking in a circle, when it seems that the next circle will be the final one. Unfortunately, marking time does not move progress.

The purpose of this work: to study the secrets of the Egyptian pyramids from the point of view of mathematics.

To achieve this goal, it is necessary to solve the following tasks:

1) Find general information about the Egyptian pyramids

2) Analyze the found information and choose the one that can be consistent with the geometry

3) study this information and draw appropriate conclusions

WHAT ARE THE EGYPTIAN PYRAMIDS

The Egyptian pyramids are the greatest architectural monuments of Ancient Egypt, including one of the "seven wonders of the world" - the Cheops pyramid and an honorary candidate for the "new seven wonders of the world" - the Pyramids of Giza. Pyramids are huge pyramid-shaped stone structures used as tombs for the pharaohs of ancient Egypt. The word "pyramid" - Greek, means a polyhedron. According to some researchers, a large pile of wheat became the prototype of the pyramid. According to other scientists, this word comes from the name of the funeral cake of a pyramidal shape. A total of 118 pyramids have been discovered in Egypt.

When referring to the Egyptian pyramids, as a rule, they mean the Great Pyramids located in Giza, not far from Cairo. But they are not the only pyramids in Egypt. During the period of the first dynasties, special “houses after life” appeared - mastabas - funerary buildings, consisting of an underground burial chamber and a stone structure above the ground. The term itself already refers to the Arab time and is due to the fact that the shape of these tombs, similar in section to a trapezoid, reminded the Arabs of large benches called “mastaba”.

The first pharaohs also built mastabas for themselves. The oldest royal mastabas, dating back to the times of the 1st dynasty, were built from adobes - unbaked bricks made of clay or river silt. They were built in Nagadei Abydos in Upper Egypt, as well as in Saqqara, where the main necropolis of Memphis, the capital of the rulers of the first dynasties, was located. In the ground part of these buildings there were chapels and rooms with grave goods, and in the underground part there were actually burial chambers.

The purpose of the pyramids was multifunctional. They served not only as the tombs of the pharaohs, but were also attributes of the greatness, power and wealth of the country, cultural monuments, repositories of the country's history and information about the life of the pharaoh and the people, a collection of household items.

It is clear that the pyramids had a deep "scientific content", embodied in their shape, size and orientation to the terrain. Every detail of the pyramid, every element of the form was chosen carefully and had to demonstrate the high level of knowledge of the pyramid makers. After all, they were built for millennia, "forever." And it is not for nothing that the Arabic proverb says: "Everything in the world is afraid of time. Time is afraid of the pyramids."

Among the grandiose pyramids of Egypt, the Great Pyramid of Pharaoh Cheops (Khufu) occupies a special place. Before proceeding to the analysis of the shape and size of the pyramid of Cheops, we should remember what system of measures the Egyptians used. The Egyptians had three units of length: "cubit" (466 mm), equal to seven "palms" (66.5 mm), which, in turn, was equal to four "fingers" (16.6 mm).

The pyramids were built about 4.5 thousand years ago. Before erecting the pyramid, the builders leveled the terrain on which the pyramid was to be erected. The central axis of the pyramid was determined, around which the rock was removed under the blocks, which were the basis for its installation.

Over 800 kilometers from the installation site, 130 granite blocks were mined, each of which weighs from 12 to 70 tons. These blocks have been delivered. Subsequently, for the construction of internal chambers, they had to be raised by 65 meters and installed.

The lower, middle and upper chambers were built, which are negligible in comparison with the entire huge mass of the monument. A passage was built 90 meters long and less than a meter wide, this passage leading to the lower chamber was cut right into the rock. Precise tools were required to keep the angle of the passage accurate and constant along its entire length. Also, in order to gouge a tunnel of this length, lighting was required, but no traces of soot were found. So what did the ancient Egyptians use to illuminate and control the angle of immersion?

The pyramid consists of more than 2 million blocks of different shapes and sizes, although, as we know from the experience of modern builders, it is easier to work with blocks of the same size. The average mass of blocks is 2.5 tons. The accuracy of the overall design of the pyramid is striking, it meets modern standards. For example, the top camera is perfectly aligned horizontally and vertical planes. The accuracy of the installation of blocks exceeds the standards of modern construction. The error is only half a millimeter. This is amazing accuracy for such an ancient object. Modern builders using the most modern equipment cannot achieve this. The pyramid has survived at least three large-scale earthquakes in its history, one of which razed the city of Cairo to the ground in the 13th century. What was the level of knowledge of the builders if everything inside the pyramid remained in place?

The geographic orientation of the pyramid along the magnetic poles of the planet was also performed in a jeweler's way. It has a deviation of the western side 2'30" and the eastern side 5'30" from the true direction to the north, and the northern and southern sides by 2'28" and 1'57" respectively from the Equator line.

THE PRESENCE OF MATH IN THE EGYPTIAN PYRAMIDS

One of interesting features this pyramid is that it has not four, but eight faces, because. each of the four sides is slightly concave in the middle, which further complicates the construction process.

Let's analyze the dimensions of the pyramid of Cheops.

Most researchers agree that the length of the side of the base of the pyramid, for example, GF = 233.16 m. This value corresponds almost exactly to 500 "cubits". There will be full compliance with 500 "cubits" if the length of the "cub" is considered equal to 0.4663 m. The height of the pyramid (AC) is estimated by researchers differently from 146.6 to 148.2 m. elements. What is the reason for the differences in the estimate of the height of the pyramid? The fact is that, strictly speaking, the pyramid of Cheops is truncated. Its upper platform today has a size of approximately 10 * 10 m, and a century ago it was 6 * 6 m. It is obvious that the top of the pyramid was dismantled, and it does not correspond to the original one. Estimating the height of the pyramid, it is necessary to take into account such a physical factor as the "draft" of the structure. For a long time under the influence of colossal pressure (up to 500 tons per 1 m 2 lower surface) the height of the pyramid has decreased compared to the original height. What was the original height of the pyramid? This height can be recreated if you find the basic "geometric idea" of the pyramid.

In 1837, the English Colonel G. Wise measured the angle of inclination of the faces of the pyramid: it turned out to be equal to a = 51 ° 51 ". This value is still recognized by most researchers today. The indicated value of the angle corresponds to the tangent (tg a), equal to 1.27306. This value corresponds to the ratio of the height of the pyramid AC to half of the side of its base CB, i.e.AC/CB=H/(L/2)=2H/L. And here the researchers were in for a big surprise! The point is that iftake the square root of the golden ratio, then we get the following result = 1.272. Comparing this value with the value tg a = 1.27306, we see that these values ​​are very close to each other. If we take the angle a \u003d 51 ° 50 ", that is, reduce it by only one arc minute, then the value of a will become equal to 1.272, that is, it will coincide with the valueWHAT???.It should be noted that in 1840 G. Wise repeated his measurements and clarified that the value of the angle a = 51°50". These measurements led the researchers to the following very interesting hypothesis: the relation AC / CB = 1,272!

Consider now a right triangle ABC, in which the ratio of the legsAC / CB = rootZ. If now the lengths of the sidesrectangleABC denote by x, y, z, and also take into account that the ratio y / x = rootz, then in accordance with the Pythagorean theorem, the length z can be calculated by the formula:z= sum rootxsquared andyin a square. Let us now derive some other relations for the pyramid of Cheops, which follow from the "golden" hypothesis. In particular, we find the relationouter areapyramid to the area of ​​its base. To do this, we take the length of the leg CB as a unit, that is: CB = 1. But then the length of the side of the base of the pyramid GF = 2,and the base area of ​​EFGH will be equal to SEFGH = 4.

Let us now calculate the area of ​​the side face of the Cheops pyramid SD.Insofar asheight AB of triangle AEF is equal to t, then the area of ​​the side face will be equal to SD = t. Then the total area of ​​all four side faces of the pyramid will be equal to 4t, and the ratio of the total external area of ​​the pyramid to the base area will beequal to the golden ratio! This is the main geometric secret of the Cheops pyramid!

An analysis of other Egyptian pyramids shows that the Egyptians always sought to embody some important mathematical knowledge in their pyramids. In this regard, the pyramid of Khafre is very interesting. Measurements of the pyramid showed that the angle of inclination of the side faces in it is 53 ° 12 ", which corresponds to the ratio of the legs of a right triangle 4: 3. This ratio of the legs corresponds to the well-known right triangle with sides 3: 4: 5, which is called "perfect", " sacred" or "Egyptian" triangle. According to historians, the "Egyptian" triangle was given a magical meaning. Plutarch wrote that the Egyptians compared the nature of the Universe with the "sacred" triangle; they symbolically likened the vertical leg to the husband, the base to the wife, and the hypotenuse to the one that is born from both.

For a 3:4:5 triangle, the equality is true: 3 squared + 4 squared = 5 squared, which expresses the Pythagorean theorem. Is it not this theorem that the Egyptian priests wanted to perpetuate by erecting a pyramid on the basis of the triangle 3:4:5? It is difficult to find a better example to illustrate the Pythagorean theorem, which was known to the Egyptians long before its discovery by Pythagoras.

Thus, the ingenious creators of the Egyptian pyramids sought to impress distant descendants with the depth of their knowledge, and they achieved this by choosing as the "main geometric idea" for the pyramid of Cheops - the "golden" right-angled triangle, and for the pyramid of Khafre - the "sacred" or "Egyptian" triangle.

GEOMETRY DEPTH

Let's look at the pyramid of Cheops again. But we will only act not according to algebraic formulas, but from the point of view of geometry.

Many Egyptologists and researchers of the Cheops pyramid assume that the builders of ancient Egypt knew the numerical value of the golden section and the number "pi". And also Egyptologists suggest that these golden numbers were key in the architectural design of the Egyptian pyramids, but in reality the knowledge of these numbers is not necessary, since for the construction of the pyramids it is enough to know the proportions of the heptagon and use the ratio of the lines that exist in the geometric figure of the heptagon.

The AEK triangle in the figure is an approximate silhouette of the side faces of the Cheops pyramid.

The shown silhouette of the side faces is approximate, since the angle of the EKA heptagon with the apex at point K has a value of 360/7 = 51.429 degrees (51 degrees 25.71 minutes), and the actual angle of inclination of the Cheops pyramid faces has a value of 51 degrees 50 minutes.

The builders of the pyramid of Cheops compensated for the existing difference by adding to the height of the triangle AEK the value of human growth AH. That is, the builders conventionally placed a human figure on the top of the pyramid, and as a result they received 51 degrees 50 minutes as the value of the angle ECX with the apex at point K. Namely, if the height of the ECX triangle has a value of 318 cubits, then the height of the EKA triangle is approximately 314 cubits, and the value human height approximately 4 cubits. Egyptian builders increased the correct angle of the heptagon, as if there was a person at the top of the pyramid of Cheops, and as a result, the golden ratio was encrypted in the ratios of the EK / KX lines and the proportions of the human body were laid in the general geometry of the future pyramidal structure, which was the idea of ​​​​the architectural design of the future pyramid . Essentially, the builders and creators of the architectural plan great pyramid Cheops inscribed a double heptagonal or diheptagonal network of lines into a living circle, in which the vertical diameter differed from the horizontal diameter by the relative size of human growth.

The diheptagonal network of lines is inscribed in the space of a living circle, which has a ratio of vertical and horizontal diameters of approximately 15/14, which corresponds to the proportions of the male and female bodies. The blue outline indicates the approximate silhouette of the side faces of the pyramid, while the white outline indicates the approximate silhouette of the diagonal edges.

The angle DLA with the apex at point L has a value of 40 degrees 59 minutes and corresponds to the inclination of the diagonal edges of the Cheops pyramid.

The values ​​of the angles DLA (slope of the diagonal edges) and EKA (slope of the side faces) with vertices at points L and K differ from the angular values ​​of a regular bigeptagon by the relative size of human height, since the bigeptagon in the design of the Egyptian pyramid of Cheops is inscribed in the space of a living circle, which is transformed in accordance with with human growth. Namely, it is transformed in accordance with the height of the human body, which has a size of 4 Egyptian cubits and is conventionally located at the top of the pyramid.

INTERESTING EGYPTIAN PYRAMIDS OF EGYPT

The Pyramid of Cheops

One of the largest pyramids in the world. Built by pharaoh Khufu, the second pharaoh of the fourth dynasty, whose Greek name was Cheops. Its height reached 146.6 m, and the size of the base - 230 m. The pyramid consists of blocks of limestone, granite and basalt. Inside it there are 3 burial chambers, which are designed one above the other. The entrance to the pyramid is located at a height of 15 meters on the north side, but tourists enter it through a gap made by Caliph Abu Jafar al-Ma'mun in order to search for countless treasures.

Pyramid of Khafre (or Khafre)

The second largest pyramid of Ancient Egypt, next to which is the sculpture of the world-famous Great Sphinx - a formidable lion with a human head. Its height reached 143.5 m, however, unlike the pyramid of Khufu, its inner space is more modest. There are two chambers and two crossing passages. There is also a burial chamber located under its base.

pink pyramid

One of the largest pyramids in Dahshur and the third largest in Egypt after the two pyramids of Giza. It got its name due to the color of the stone blocks, which acquire a pink tint in the rays of the setting sun. In addition, this structure is considered the first successful attempt to build an isosceles regular pyramid.

The final pyramid in the ensemble of grandiose structures of Giza. It is located 200 m from the pyramid of Khafre, however, it is much smaller in size than its predecessors - only 66.5 m in height. Inside it, two small chambers were also found, but there are no traces of burial. The pyramid has 3 companions, surrounded by a common fence.

CONCLUSION

In the work done, I learned and got acquainted with some of the secrets of the Egyptian pyramids. Coordinated them with mathematics and carried out several tasks. I also found and got acquainted with information about the greatness and features of each Egyptian pyramids.

The art of Ancient Egypt is a reflection of concrete, real reality at different stages of its historical development. The ancient Egyptians considered earthly life to be just a brief episode of eternal existence. That is why for the construction of houses, residences of nobles and royal palaces, they used not very durable materials. Things looked different with the construction of dwellings for the immortal soul of the pharaoh and the nobility, which, like the temples of the gods, were built of stone. The monuments that have survived to this day testify to the strength of these buildings. Egyptian pyramids at Giza, tombs of Egyptian pharaohs. The largest of them - the pyramids of Cheops, Khafre and Mykerin in ancient times were considered one of the Seven Wonders of the World. The construction of the pyramid, in which the Greeks and Romans already saw a monument to the unprecedented pride of the kings and cruelty, which doomed the entire people of Egypt to senseless construction, according to scientists, was the most important cult act and should have expressed, apparently, the mystical identity of the country and its ruler. The population of the country worked on the construction of the tomb in the part of the year free from agricultural work. A number of texts testify to the attention and care that the kings themselves (albeit of a later time) paid to the construction of their tomb and its builders. From ancient times to the present day, many have explored the pyramids, because the builders of the pyramids have not left us any explanation on this matter. The first among such seekers was the indefatigable Greek traveler and historian Herodotus, who, back in the middle of the 5th century. BC e. visited Egypt and suggested that the pyramids were built using wooden machines that lifted blocks from ledge to ledge. What these cars looked like was unknown, apparently, to himself. The very process of their construction remains a mystery to this day - even with modern technology, creating something similar is a difficult task. But the Egyptians did not have any cranes or powerful dump trucks. They did not have not only steel cables, but also the simplest iron tools. But they literally lifted mountains of stone and laid these monoliths with amazing precision. And yet, of greatest interest is the reason for the appearance of such structures, which no one had created before the Egyptians. Why was it necessary to build them so large and at the same time calculate all the proportions with such amazing accuracy? This remains unexplained. https://ru.wikipedia.org/wiki/Ancient_Egypt

Encyclopedia "Wonders of the World" V.V. Kiselev, S.V. Andreeva. UDC 087.5 BBK 9 Ch-81

Encyclopedia "I know everything" Z.V. Nikolaevich MFR 98-B

Martin Gardner. Mathematical wonders and secrets - M.: Nauka, 1978. 128 p.

Dictionary

"About the pyramids" - Energy of the pyramids. Mayan pyramids in El Salvador. Currently known different types pyramids. A pyramid is a polyhedron whose base is a polygon. By the number of corners of the base, pyramids are triangular, quadrangular, etc. Enormous spiritual energy flows from the top of the pyramid into the room.

"Pyramid of Cheops" - Here came the turn of the most dangerous and hard work. Pyramid building. Interesting to know. Egypt Pyramid of Cheops. Not a single pyramid today has been preserved at the top. The sailboat transported the workers and the block to the other side of the Nile. Description. Inside the pyramid. Pyramid of Pharaoh Cheops, built over 4000 years ago.

"Pyramid Grade 10" - Hexagonal pyramid. What is the total surface area of ​​a pyramid? C. A polyhedron composed of an n-gon А1А2…An and n triangles is called a pyramid. What is the lateral surface area of ​​a regular pyramid? Correct pyramid? What is the lateral surface area of ​​a pyramid called?

"Pyramid lesson" - Establish connections between the locations of the pyramids. The study of the properties of the pyramids. Assignment to the group "Investigators of the properties of the pyramids." Independently choose the content of the electronic accompaniment of the lesson. The glass pyramid is the new entrance to the Louvre, Paris. Study the pyramid as a geometric body. The use of ICT allows the teacher.

"Ancient pyramids" - Architecture of ancient Egypt. Mastaba consists of two parts - ground and underground. Sphinx. Some of the theories consider the pyramid astronomical observatory. Description of the pyramid. The top of the pyramid was crowned with a gilded stone - a pyramidion. Grotto, Grand Gallery and the Chambers of the Pharaoh. Legends about the sphinx are found in ancient Egypt (the period of the Old Kingdom).

"Pyramids in Egypt" - Pyramids of Ancient Egypt. How were the pyramids built? Pyramid of Khafre. Who built the pyramids? Man is afraid of Time, and Time is afraid of the Pyramids (Arabic proverb). Two types of pyramids. The most famous pyramids Pyramid of Djoser. The structure of the pyramids. Step pyramids. Pyramids at Giza. There are approximately 2 million such blocks in the pyramid.