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Why is the Egyptian triangle called that? This amazing Egyptian triangle

It is possible that the term “Egyptian triangle” gave Pythagoras, having visited at the insistence Thales in Egypt…

“... in this essay we are interested in precisely the non-practical, non-applied aspect of mathematics; we assume that it will be very, very instructive to include in the “gentleman’s set” of mathematical concepts the knowledge of why a triangle with sides 3, 4, 5 is called Egyptian.

The whole point is that the ancient Egyptian pyramid builders needed a way to construct a right angle. Here is the required method. The rope is divided into 12 equal parts, the boundaries between adjacent parts are marked, and the ends of the rope are connected. The rope is then tensioned by three people so that it forms a triangle, and the distances between adjacent tensioners are 3 parts, 4 parts and 5 parts, respectively. In this case, the triangle will be right-angled, in which sides 3 and 4 will be the legs, and side 5 will be the hypotenuse, so the angle between sides 3 and 4 will be right.

I am afraid that most readers will answer the question “Why will the triangle be right-angled?” will refer to the Pythagorean theorem: after all, three squared plus four squared equals five squared. However, the Pythagorean theorem states that if a triangle is right-angled, then in this case the sum of the squares of its two sides is equal to the square of the third.

Here we use the theorem inverse to the Pythagorean theorem: if the sum of the squares of two sides of a triangle is equal to the square of the third, then in this case the triangle is right-angled. (I’m not sure this converse theorem has its proper place in the school curriculum.).”

Uspensky V.A. , Apology of mathematics, or about mathematics as part of spiritual culture, magazine “New World”, 2007, N 11, p. 131.

Let's say we have a line to which we need to set a perpendicular, i.e. another line at an angle of 90 degrees relative to the first. Or we have an angle (for example, the corner of a room) and we need to check whether it is equal to 90 degrees.

All this can be done with just a tape measure and a pencil.

There are two great things, such as the Egyptian Triangle and the Pythagorean Theorem, that will help us with this.

Once causes and goals are found, the search for innovative knowledge will be a natural consequence. You have to be optimistic, but that's not enough. Beliefs must be turned into actions. If possible, not in isolated actions. If the classroom is the only space you need to have, you need to occupy it wisely and make real what you once dreamed of.

The origin of geometry is somewhat obscure, as one of the many knowledges of mathematics, in which it is impossible to credit one person with its discovery. However, its beginnings in Egypt and the earliest evidence of modern geometry are thought to date back to around 600 BC.

So, Egyptian triangle is a right triangle with the ratio of all sides equal to 3:4:5 (side 3: side 4: hypotenuse 5).

The Egyptian triangle is directly related to the Pythagorean theorem - the sum of the squares of the legs is equal to the square of the hypotenuse (3*3 + 4*4 = 5*5).

How can this help us? Everything is very simple.

Task No. 1. You need to construct a perpendicular to a straight line (for example, a line at 90 degrees to the wall).

Despite its importance in the historical and cultural context, geometry has not been sufficiently studied. At the same time, the skills that will be developed in students are outdated. According to the teaching proposal of Santa Catarina regarding the teaching of geometry and the competencies that must be developed in the student, certain factors must be taken into account.

The study or exploration of physical space and forms. Orientation and visualization and representation of physical space. Visualizing and understanding geometric shapes. Naming and recognizing forms according to their characteristics. Classification of objects according to their shapes.

Step 1. To do this, from point No. 1 (where our angle will be), we need to measure on this line any distance that is a multiple of three or four - this will be our first leg (equal to three or four parts, respectively), we get point No. 2.

To simplify calculations, you can take a distance, for example 2m (this is 4 parts of 50cm each).

Studying the properties of figures and the relationships between them. Construction of geometric figures and models. Constructing and justifying relationships and prepositions based on hypothetical deductive reasoning. To achieve this, competences related to geometry must be transferred from the second year of primary school, taking into account the student's content absorption level.

It is accepted and accepted in society that the principle of “doing math is solving problems.” In this regard, solving the problem is a subject for researchers and mathematicians. Understanding the difficulties faced by most students in this vital activity is a major challenge. The first, of course, is an accurate understanding of the problem. For Lakatos and Marconi, "a problem is a difficulty, theoretical or practical, in knowing something of real significance for which a solution must be found", and this understanding is fundamental for students to work through the resolution of the problem.

Step 2. Then from the same point No. 1 we measure 1.5 m (3 parts of 50 cm each) upward (we set an approximate perpendicular), draw a line (green).

Step 3. Now from point No. 2 you need to put a mark on the green line at a distance of 2.5 m (5 parts of 50 cm each). The intersection of these marks will be our point No. 3.

By connecting points No. 1 and No. 3 we get a line perpendicular to our first line.

First, it can be said that problem solving, as a strategy for the development of mathematics education, must get rid of this feeling of “necessary evil” created by the endless list of “problems” that, as a rule, at the end of each unit of the program, the teacher presents to the students.

The traditional use of problems, reduced to the application and systematization of knowledge, attracts hostility and disinterest in the student, preventing their full intellectual development. Excessive preparation of definitions, methods and demonstrations becomes a routine and mechanical activity in which only the final product is evaluated. Failure to follow the stages of research and communication of logical-mathematical ideas does not allow the construction of concepts. Thus, “mathematical knowledge does not represent the student as a system of concepts that allows him to solve many problems, but as endless symbolic, abstract, incomprehensible speech.”

Task No. 2. The second situation is that there is an angle and you need to check whether it is straight.

This is our corner. It is much easier to check with a large square. What if he is not there?

>>Geometry: Egyptian triangle. Complete lessons

Mathematical knowledge has evolved only from many answers to many questions asked throughout history. Creativity, critical census, curiosity and pleasure were the fuel that fueled this process of discovery. According to Paul, a problem solving scheme.

The systematic use of this scheme helps the student organize his thinking. Confronting his original solution idea with a colleague's or group's solution promotes learning, thereby re-emphasizing the teacher's role. The earliest evidence of the rudiments of trigonometry arose in both Egypt and Babylon, from the calculation of the relationships between numbers and between the sides of similar triangles.

Lesson topic

Lesson Objectives

Get acquainted with new definitions and remember some already studied.
Deepen your knowledge of geometry, study the history of origin.
To consolidate students' theoretical knowledge about triangles in practical activities.
Introduce students to the Egyptian triangle and its use in construction.
Learn to apply the properties of shapes when solving problems.
Developmental – to develop students’ attention, perseverance, perseverance, logical thinking, mathematical speech.
Educational - through the lesson, cultivate an attentive attitude towards each other, instill the ability to listen to comrades, mutual assistance, and independence.

Lesson Objectives

Test students' problem-solving skills.

Lesson Plan

Introduction.
It's useful to remember.
Toegon.

introduction

Did they know mathematics and geometry in ancient Egypt? They not only knew it, but also constantly used it when creating architectural masterpieces and even... during the annual marking of fields where flood water destroyed all the boundaries. There was even a special service of surveyors who quickly, using geometric techniques, restored the boundaries of fields when the water subsided.

The Achemic Papyrus is the most extensive Egyptian document on mathematics that has survived to this day. Who was in the power of the scribe Ahmes. The Babylonians had a great interest in astronomy, both for religious reasons and for connections to the calendar and planting seasons. It is impossible to study the phases of the Moon, cardinal points and seasons of the year without the use of triangles, a system of units of measurement and scale.

This study is further divided into two parts: plane trigonometry and spherical trigonometry. The use of trigonometry in various fields of exact sciences is an indisputable fact. Knowing this truth is fundamental to high school students, and it is the responsibility of the mathematics teacher to teach this subject to the best of his ability, creating the necessary connection to future career choices. Currently, trigonometry is not limited to the study of triangles. Its application extends to other areas of mathematics such as "Analysis" and other areas of human endeavor such as electricity, mechanics, acoustics, music, topography, civil engineering, etc.

It is not yet known what we will call our younger generation, which grows up on computers that allow us not to memorize the multiplication table and not perform other elementary mathematical calculations or geometric constructions in our heads. Maybe human robots or cyborgs. The Greeks called those who could not prove a simple theorem without outside help ignoramuses. Therefore, it is not surprising that the theorem itself, which was widely used in applied sciences, including for marking fields or building pyramids, was called by the ancient Greeks “the bridge of donkeys.” And they knew Egyptian mathematics very well.

It is noted, however, that one of the biggest difficulties faced by secondary school students as discussed in Trigonometry relates to the fact of memorizing formulas. However, not remembering would require time to infer during tests, which would make the situation unfeasible.

Here we present some of the basic relationships and theorems associated with geometry and, more specifically, trigonometry. Recall that the causes and, respectively, representing sine, cosine and tangent are valid for the previously discovered triangle and do not need to be decorated or taken, as a rule, thus the concept is assessed rather than memorization of the formula.

Useful to remember

Triangle

Triangle rectilinear, a part of the plane limited by three straight segments (sides of the Triangle (in geometry)), each having one common end in pairs (vertices of the Triangle (in geometry)). A triangle whose lengths of all sides are equal is called equilateral, or correct, Triangle with two equal sides - isosceles. The triangle is called acute-angled, if all its angles are sharp; rectangular- if one of its angles is right; obtuse-angled- if one of its angles is obtuse. A triangle (in geometry) cannot have more than one right or obtuse angle, since the sum of all three angles is equal to two right angles (180° or, in radians, p). The area of the Triangle (in geometry) is equal to ah/2, where a is any of the sides of the Triangle, taken as its base, and h is the corresponding height. The sides of the Triangle are subject to the following condition: the length of each of them is less than the sum and greater than the difference in the lengths of the other two sides.

The major evolution of trigonometric concepts occurred after the use of the trigonometric cycle, formerly called the trigonometric circle. These are “coordinate axes that have as a unit of measurement the radius of an oriented circle coinciding with the coordinate center of the coordinate axes.”

Euler, born in Basel, was one of the best and most productive mathematicians in history, and with his above-mentioned contributions he agreed to use one beam for the trigonometric cycle. Thus, "as the cycle is oriented, each measure of degrees will correspond to one point in the cycle."

Triangle- the simplest polygon having 3 vertices (angles) and 3 sides; part of the plane bounded by three points and three segments connecting these points in pairs.

With this definition, one can establish the same concepts for sine, cosine and tangent as follows. Let's look at the figure to the side where the trigonometric circle is depicted. That is: the cosine of a right triangle is equal to the adjacent leg divided by its hypotenuse, the hypotenuse being the opposite of the right angle.

Recall that the radius of a trigonometric circle is 1, it is concluded that the sine and cosine of the arc are real numbers that vary in the real interval from -1 to. The scale adopted on the tangent axis is the same as for the abscissa and ordinate axes.

Three points in space that do not lie on the same straight line correspond to one and only one plane.
Any polygon can be divided into triangles - this process is called triangulation.
There is a section of mathematics entirely devoted to the study of the laws of triangles - Trigonometry.

Types of Triangles

By type of angles

Considering the following representation for the law of breasts. The proportions related to the law of the mammary gland indicated above are determined by the following definition. Given the following representation for the cosine law. According to the law of cosines, as indicated above, a triangle is any square measure of one side equal to the sum of the squares of the measures of the other two sides minus twice the product of the measures of these sides by the cosine of the angle they form.

The purpose of this chapter is to develop a curriculum for trigonometry content based on problematization, contextualization, and historical inquiry to enable learning on the part of students. It is emphasized that it is understood that a teaching plan is a prerequisite for guiding the educational process through the teaching of any content, it emphasizes, as we will see below, the content, the objectives, the development of the plan, the materials that should be And how to assess the content that needs to be administered.

Since the sum of the angles of a triangle is 180°, at least two angles in the triangle must be acute (less than 90°). The following types of triangles are distinguished:

If all the angles of a triangle are acute, then the triangle is called acute;
If one of the angles of a triangle is obtuse (more than 90°), then the triangle is called obtuse;
If one of the angles of a triangle is right (equal to 90°), then the triangle is called right-angled. The two sides that form a right angle are called legs, and the side opposite the right angle is called the hypotenuse.

According to the number of equal sides

Based on the thematic project, trigonometry emerged: problematization and contextualization. Contextualize subject matter trigonometry using a historical approach and exploring physical space and shapes present in the environment. Provide opportunities for students to learn the basics of trigonometry.

Recognize where it is spreading and the impact it is causing. Provide students with techniques to facilitate understanding, interpretation and problem solving. The trigonometry content will be applied according to the material designed to track the content, which will follow the following steps.

A scalene triangle is one in which the lengths of the three sides are pairwise different.
An isosceles triangle is one in which two sides are equal. These sides are called lateral, the third side is called the base. In an isosceles triangle, the base angles are equal. The altitude, median and bisector of an isosceles triangle lowered to the base are the same.
An equilateral triangle is one in which all three sides are equal. In an equilateral triangle, all angles are equal to 60°, and the centers of the inscribed and circumscribed circles coincide.

In terms of research, this can be done in groups and divided by topic. Socialization can be accomplished through presentation worthy of each group's creativity and interest. After the presentation, the teacher can make their placements, prioritizing the importance of the content.

Trigonometry is a branch of mathematics that studies triangles, especially triangles in the plane, where one of the angles of the triangle measures 90 degrees. It also specifically studies the relationships between the sides and angles of triangles; Trigonometric functions and calculations based on them. The trigonometric approach makes its way into other areas of geometry, such as the study of spheres using spherical trigonometry.

– a right triangle with an aspect ratio of 3:4:5. The sum of these numbers (3+4+5=12) has been used since ancient times as a unit of multiplicity when constructing right angles using a rope marked with knots at 3/12 and 7/12 of its length. The Egyptian triangle was used in the architecture of the Middle Ages to construct proportional schemes.

The origins of trigonometry are unknown. A triangle is a geometric figure with three sides and three angles. To form a triangle, simply connect all three points with segments if they are not aligned. Below are the triangles. The aperture obtained by two lines connected by the same point is called an angle, which has radians as the international measurement system, and the degree is also very useful. In triangles, the sum of their interior angles is 180°.

A right angle is indicated by a symbol. In a right triangle, the opposite side of the right angle is called the hypotenuse. Some authors believe that Pythagoras was a student of Tales, Eve, when he said that "he was fifty years younger than this and lived near Miletus, where Thales lived." Boyer says that "although some of the statements claim that Pythagoras was a student of the Tales, this hardly gives a difference of half a century between his ages."

So where to start? Is it because of this: 3 + 5 = 8. and the number 4 is half the number 8. Stop! The numbers 3, 5, 8... Don't they resemble something very familiar? Well, of course, they are directly related to the golden ratio and are included in the so-called “golden series”: 1, 1, 2, 3, 5, 8, 13, 21 ... In this series, each subsequent term is equal to the sum of the previous two: 1 + 1= 2. 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8 and so on. It turns out that the Egyptian triangle is related to the golden ratio? And did the ancient Egyptians know what they were dealing with? But let's not rush to conclusions. It is necessary to find out more details.

The expression “golden ratio”, according to some, was first introduced in the 15th century Leonardo da Vinci . But the “golden series” itself became known in 1202, when the Italian mathematician first published it in his “Book of Counting” Leonardo of Pisa . Nicknamed Fibonacci. However, almost two thousand years before them, the golden ratio was known Pythagoras and his students. True, it was called differently, as “division in the average and extreme ratio.” But the Egyptian triangle with its The “golden ratio” was known back in those distant times when the pyramids were built in Egypt when Atlantis flourished.

To prove the Egyptian triangle theorem, it is necessary to use a line segment of known length A-A1 (Fig.). It will serve as a scale, a unit of measurement, and will allow you to determine the length of all sides of the triangle. Three segments A-A1 are equal in length to the smallest side of triangle BC, whose ratio is 3. And four segments A-A1 are equal in length to the second side, whose ratio is expressed by the number 4. And, finally, the length of the third side is equal to five segments A -A1. And then, as they say, it’s a matter of technique. On paper we will draw a segment BC, which is the smallest side of the triangle. Then, from point B with a radius equal to the segment with ratio 5, we draw a circular arc with a compass, and from point C, an arc of a circle with a radius equal to the length of the segment with ratio 4. If we now connect the intersection point of the arcs with lines to points B and C, we get a right triangle aspect ratio 3:4:5.

Q.E.D.

The Egyptian triangle was used in the architecture of the Middle Ages to construct proportional schemes and to construct right angles by surveyors and architects. The Egyptian triangle is the simplest (and first known) of the Heronian triangles - triangles with integer sides and areas.

The Egyptian Triangle - a mystery of antiquity

Each of you knows that Pythagoras was a great mathematician who made invaluable contributions to the development of algebra and geometry, but he gained even more fame thanks to his theorem.

And Pythagoras discovered the Egyptian triangle theorem at the time when he happened to visit Egypt. While in this country, the scientist was fascinated by the splendor and beauty of the pyramids. Perhaps this was precisely the impetus that exposed him to the idea that some specific pattern was clearly visible in the shapes of the pyramids.

History of discovery

The Egyptian triangle received its name thanks to the Hellenes and Pythagoras, who were frequent guests in Egypt. And this happened approximately in the 7th-5th centuries BC. e.

The famous pyramid of Cheops is actually a rectangular polygon, but the pyramid of Khafre is considered to be the sacred Egyptian triangle.

The inhabitants of Egypt compared the nature of the Egyptian triangle, as Plutarch wrote, with the family hearth. In their interpretations one could hear that in this geometric figure its vertical leg symbolized a man, the base of the figure related to the feminine principle, and the hypotenuse of the pyramid was assigned the role of a child.

And already from the topic you have studied, you are well aware that the aspect ratio of this figure is 3: 4: 5 and, therefore, that this leads us to the Pythagorean theorem, since 32 + 42 = 52.

And if we take into account that the Egyptian triangle lies at the base of the Khafre pyramid, we can conclude that the people of the ancient world knew the famous theorem long before it was formulated by Pythagoras.

The main feature of the Egyptian triangle was most likely its peculiar aspect ratio, which was the first and simplest of the Heronian triangles, since both the sides and its area were integers.

Features of the Egyptian Triangle

Now let's take a closer look at the distinctive features of the Egyptian triangle:

First, as we have already said, all its sides and area consist of integers;

Secondly, by the Pythagorean theorem we know that the sum of the squares of the legs is equal to the square of the hypotenuse;

Thirdly, with the help of such a triangle you can measure right angles in space, which is very convenient and necessary when constructing structures. And the convenience is that we know that this triangle is right-angled.

Fourthly, as we also already know, even if there are no appropriate measuring instruments, this triangle can be easily constructed using a simple rope.

Application of the Egyptian triangle

In ancient centuries, the Egyptian triangle was very popular in architecture and construction. It was especially necessary if a rope or cord was used to build a right angle.

After all, it is known that laying a right angle in space is quite a difficult task, and therefore enterprising Egyptians invented an interesting way of constructing a right angle. For these purposes, they took a rope, on which they marked twelve even parts with knots, and then from this rope they folded a triangle, with sides that were equal to 3, 4 and 5 parts, and in the end, without any problems, they got a right triangle. Thanks to such an intricate tool, the Egyptians measured the land with great precision for agricultural work, built houses and pyramids.

This is how a visit to Egypt and studying the features of the Egyptian pyramid prompted Pythagoras to discover his theorem, which, by the way, was included in the Guinness Book of Records as the theorem that has the largest amount of evidence.

Triangular Reuleaux wheels

Wheel- a round (as a rule), freely rotating or fixed on an axis disk, allowing a body placed on it to roll rather than slide. The wheel is widely used in various mechanisms and tools. Widely used for transporting goods.

The wheel significantly reduces the energy required to move a load on a relatively flat surface. When using a wheel, work is performed against the rolling friction force, which in artificial road conditions is significantly less than the sliding friction force. Wheels can be solid (for example, a wheel pair of a railway carriage) and consisting of a fairly large number of parts, for example, a car wheel includes a disk, rim, tire, sometimes a tube, fastening bolts, etc. Car tire wear is almost a solved problem (if the wheel angles are set correctly). Modern tires travel over 100,000 km. An unsolved problem is the wear of tires on airplane wheels. When a stationary wheel comes into contact with the concrete surface of the runway at a speed of several hundred kilometers per hour, the tire wear is enormous.

In July 2001, an innovative patent was received for the wheel with the following wording: “a round device used for transporting goods.” This patent was issued to John Kao, a lawyer from Melbourne, who wanted to show the imperfections of Australian patent law.
In 2009, the French company Michelin developed a mass-produced car wheel, the Active Wheel, with built-in electric motors that drive the wheel, spring, shock absorber and brake. Thus, these wheels make the following vehicle systems unnecessary: engine, clutch, gearbox, differential, drive and drive shafts.
In 1959, the American A. Sfredd received a patent for a square wheel. It easily walked through snow, sand, mud, and overcame holes. Contrary to fears, the car on such wheels did not “limp” and reached speeds of up to 60 km/h.

Franz Relo(Franz Reuleaux, September 30, 1829 - August 20, 1905) - German mechanical engineer, lecturer at the Berlin Royal Academy of Technology, who later became its president. The first, in 1875, to develop and outline the basic principles of the structure and kinematics of mechanisms; He dealt with the problems of aesthetics of technical objects, industrial design, and in his designs attached great importance to the external forms of machines. Reuleaux is often called the father of kinematics.

Questions

What is a triangle?
Types of triangles?
What is special about the Egyptian triangle?
Where is the Egyptian triangle used? > Mathematics 8th grade

Lesson topic

Lesson Objectives

Get acquainted with new definitions and remember some already studied.
Deepen your knowledge of geometry, study the history of origin.
To consolidate students' theoretical knowledge about triangles in practical activities.
Introduce students to the Egyptian triangle and its use in construction.
Learn to apply the properties of shapes when solving problems.
Developmental – to develop students’ attention, perseverance, perseverance, logical thinking, mathematical speech.
Educational - through the lesson, cultivate an attentive attitude towards each other, instill the ability to listen to comrades, mutual assistance, and independence.

Lesson Objectives

Test students' problem-solving skills.

Lesson Plan

Introduction.
It's useful to remember.
Toegon.

introduction

Useful to remember

Triangle

Triangle- the simplest polygon having 3 vertices (angles) and 3 sides; part of the plane bounded by three points and three segments connecting these points in pairs.

Three points in space that do not lie on the same straight line correspond to one and only one plane.
Any polygon can be divided into triangles - this process is called triangulation.
There is a section of mathematics entirely devoted to the study of the laws of triangles - Trigonometry.

Types of Triangles

By type of angles

Since the sum of the angles of a triangle is 180°, at least two angles in the triangle must be acute (less than 90°). The following types of triangles are distinguished:

If all the angles of a triangle are acute, then the triangle is called acute;
If one of the angles of a triangle is obtuse (more than 90°), then the triangle is called obtuse;
If one of the angles of a triangle is right (equal to 90°), then the triangle is called right-angled. The two sides that form a right angle are called legs, and the side opposite the right angle is called the hypotenuse.

According to the number of equal sides

A scalene triangle is one in which the lengths of the three sides are pairwise different.
An isosceles triangle is one in which two sides are equal. These sides are called lateral, the third side is called the base. In an isosceles triangle, the base angles are equal. The altitude, median and bisector of an isosceles triangle lowered to the base are the same.
An equilateral triangle is one in which all three sides are equal. In an equilateral triangle, all angles are equal to 60°, and the centers of the inscribed and circumscribed circles coincide.

Q.E.D.

The Egyptian triangle was used in the architecture of the Middle Ages to construct proportionality schemes and to construct right angles by surveyors and architects. The Egyptian triangle is the simplest (and first known) of the Heronian triangles - triangles with integer sides and areas.

The Egyptian Triangle - a mystery of antiquity

Each of you knows that Pythagoras was a great mathematician who made invaluable contributions to the development of algebra and geometry, but he gained even more fame thanks to his theorem.

History of discovery

The Egyptian triangle received its name thanks to the Hellenes and Pythagoras, who were frequent guests in Egypt. And this happened approximately in the 7th-5th centuries BC. e.

The famous pyramid of Cheops is actually a rectangular polygon, but the pyramid of Khafre is considered to be the sacred Egyptian triangle.

And already from the topic you have studied, you are well aware that the aspect ratio of this figure is 3: 4: 5 and, therefore, that this leads us to the Pythagorean theorem, since 32 + 42 = 52.

The main feature of the Egyptian triangle was most likely its peculiar aspect ratio, which was the first and simplest of the Heronian triangles, since both the sides and its area were integers.

Features of the Egyptian Triangle

Now let's take a closer look at the distinctive features of the Egyptian triangle:

First, as we have already said, all its sides and area consist of integers;

Secondly, by the Pythagorean theorem we know that the sum of the squares of the legs is equal to the square of the hypotenuse;

Fourthly, as we also already know, even if there are no appropriate measuring instruments, this triangle can be easily constructed using a simple rope.

Application of the Egyptian triangle

In ancient centuries, the Egyptian triangle was very popular in architecture and construction. It was especially necessary if a rope or cord was used to build a right angle.

Triangular Reuleaux wheels

In July 2001, an innovative patent was received for the wheel with the following wording: “a round device used for transporting goods.” This patent was issued to John Kao, a lawyer from Melbourne, who wanted to show the imperfections of Australian patent law.
In 2009, the French company Michelin developed a mass-produced car wheel, the Active Wheel, with built-in electric motors that drive the wheel, spring, shock absorber and brake. Thus, these wheels make the following vehicle systems unnecessary: engine, clutch, gearbox, differential, drive and drive shafts.
In 1959, the American A. Sfredd received a patent for a square wheel. It easily walked through snow, sand, mud, and overcame holes. Contrary to fears, the car on such wheels did not “limp” and reached speeds of up to 60 km/h.

Questions

What is a triangle?
Types of triangles?
What is special about the Egyptian triangle?
Where is the Egyptian triangle used? > Mathematics 8th grade

In the field of geometry, the Egyptians knew exact formulas for the area of a rectangle, triangle, trapezoid and sphere, and could calculate the volumes of a parallelepiped, cylinder and pyramids.

The area of an arbitrary quadrilateral with sides a, b, c, d was calculated approximately as; this rough formula gives acceptable accuracy if the figure is close to a rectangle.

The Egyptians assumed that (error less than 1%).

The formula for the area of a circle with diameter d was:

Another error is contained in the Akmim papyrus: the author believes that if the radius of circle A is the arithmetic mean of the radii of the other two circles B and C, then the area of circle A is the arithmetic mean of the areas of circles B and C.

Calculation of the volume of a truncated pyramid: let us have a regular truncated pyramid with the side of the lower base a, the upper one b and the height h; then the volume was calculated using the original but accurate formula:

Egyptian triangle

Egyptian triangle

An Egyptian triangle is a right triangle with an aspect ratio of 3:4:5. A feature of the triangle, known since antiquity, is that with such a ratio of the sides, the Pythagorean theorem gives whole squares of both the legs and the hypotenuse, that is, 9:16:25. The sum of these numbers (3+4+5=12) has been used since ancient times as a unit of multiplicity when constructing right angles using a rope marked with knots at 3/12 and 7/12 of its length.

The name of a triangle with this aspect ratio was given by the Hellenes. In the 7th - 5th centuries BC. e. Greek philosophers and public figures actively visited Egypt. For example, Pythagoras in 535 BC. e. at the insistence of Thales, he went to Egypt to study astronomy and mathematics - and, apparently, it was the attempt to generalize the ratio of squares characteristic of the Egyptian triangle to any right triangles that led Pythagoras to the formulation and proof of his famous theorem.

Volume of a truncated cone

Reconstruction of a water clock based on drawings from Oxyrhynchus

An ancient papyrus scroll found at Oxyrhynchus suggests that the Egyptians could calculate the volume of a truncated cone. They used this knowledge to build water clocks. For example, it is known that under Amenhotep III a water clock was built at Karnak.

There is no information about the earlier development of mathematics in Egypt. About the later, up to the Hellenistic era - too. After the accession of the Ptolemies, an extremely fruitful synthesis of Egyptian and Greek cultures began.

There are certain canons in mathematics that were, so to speak, the foundation or foundation of all subsequent development of modern mathematics. One of these canons can rightfully be considered the Pythagorean theorem.

Who hasn’t known the funny formulation of the Pythagorean theorem since school days: “Pythagorean pants are equal in all directions.” Well, yes, it sounds correct like this: “the square of the hypotenuse is equal to the sum of the squares of the legs,” but it’s much better remembered about the pants.

This is most clearly seen in a triangle with sides 3-4-5. But if you carefully study the use of such a triangle in ancient history, you will notice one interesting thing and it is called nothing else but.

This same philosopher and mathematician Pythagoras of Samos from Greece, after whom this theorem is named, lived approximately 2.5 thousand years ago. Well, of course, the biography of Pythagoras that has reached our time is not entirely reliable, but, nevertheless, it is known that Pythagoras traveled a lot in the countries of the East. Including he was in Egypt and Babylon. In Southern Italy, Pythagoras founded his famous “Pythagorean School,” which played a very important role in both the scientific and political life of ancient Greece. Since then, according to the legends of Plutarch, Proclus and other famous mathematicians of that time, it was believed that this theorem was not known before Pythagoras and that is why it was named after him.

But history says that this is not so. Let us turn to where Pythagoras visited and what he saw before formulating his theorem. Africa, Egypt. An endless and monotonous ocean of sand, almost no vegetation. Rare bushes of plants, barely noticeable camel tracks. Hot desert. The sun even seems dim, as if covered with this ubiquitous fine sand.

And suddenly, like a mirage, like a vision, the strict outlines of pyramids appear on the horizon, amazing in their ideal geometric shapes, directed toward the scorching sun. They are amazing with their enormous size and perfection of their forms.

Most likely, Pythagoras saw them in a different form than how they look now. These were shining polished masses with clear edges against the backdrop of multi-columned adjacent temples. Next to the majestic royal pyramids there were smaller pyramids: the wives and relatives of the pharaohs.

The power of the pharaohs of Ancient Egypt was unquestionable. Pharaohs were considered deities and were given divine honors. Pharaoh-god was the arbiter of the fate of the people and their patron. Even after death, the cult of the pharaoh was of enormous importance. The dead pharaoh was preserved for centuries, and giant pyramids were built to preserve the pharaoh's body. The grandeur, architecture and size of these pyramids are still amazing. No wonder these buildings were considered one of the seven wonders of the world.

Initially, the purpose of the pyramids was not only as tombs of the pharaohs. It is believed that they were built as attributes of the power, greatness, and wealth of Egypt. These are cultural monuments of that time, repositories of the history of the country and information about the life of the pharaoh and his people, a collection of household items of that time. In addition, it is clear that the pyramids had a certain “scientific content”. Their orientation on the ground, their shape, size and every detail, every element was so carefully thought out that they had to demonstrate the high level of knowledge of the creators of the pyramids. It is obvious that they were built to last for millennia, “forever.” And it’s not for nothing that the Arabic proverb says: “Everything in the world is afraid of time, and time is afraid of the pyramids.”

With his analytical mind, Pythagoras could not help but notice a certain pattern in the shapes and geometric dimensions of the pyramids. Most likely, this prompted Pythagoras to analyze these dimensions, which he later expressed in his famous theorem, which is now the basis for modern geometry.

Among the many pyramids that have survived to this day, the Pyramid of Cheops occupies a special place. If we consider the geometric model of this pyramid and restore its original shape, it is obvious that its cross section consists of two triangles with an internal angle equal to 51°50".

Now the pyramid is truncated, but this is the destruction of time, and if we geometrically restore it to its original form, it turns out that the sides of these triangles are equal: base CB = 116.58 m, height AC = 148.28 m.

The ratio of the legs y/x = 148.28/116.58 = 1.272. And this is the tangent of the angle 51 degrees 50 min. It turns out that the basis of the triangle ACB of the Cheops pyramid was the ratio AC/CB = 1.272. This right triangle is called a "golden" right triangle.

It turns out that the main “geometric idea” of the Cheops pyramid is a “golden” right triangle. But the pyramid of Khafre is special in this regard. The angle of inclination of the side faces of this pyramid is 53°12, at which the ratio of the legs of the right triangle is 4:3. Such a triangle is called the “sacred” or “Egyptian” triangle. According to many famous historians, the “Egyptian” triangle in ancient times was given a special magical meaning. So Plutarch wrote that the Egyptians compared the nature of the Universe with the “sacred” triangle: symbolically they likened the vertical leg to the husband, the base to the wife, and the hypotenuse to that which is born from both.

For an Egyptian triangle with sides 3:4:5, the equality is true: 32 + 42 = 52, and this is the famous Pythagorean theorem. Involuntarily, the question arises: was it not this ratio that the Egyptian priests wanted to perpetuate by building a pyramid based on the triangle 3:4:5. The Pyramid of Khafre is a clear confirmation that the famous theorem was known to the Egyptians long before its discovery by Pythagoras.

It is not known how this came to the ancient Egyptians, whether it was the merit of their scientists, or a gift from outside, it is not excluded that it was a gift from an extraterrestrial civilization, but the use of such a triangle gave the Egyptian builders a very significant and, at the same time, simple opportunity when constructing such huge structures must maintain exact geometric dimensions. After all, the properties of this triangle are such that its angle between the legs is equal to 90 degrees. That is, the use of such an element makes it possible to ensure precise perpendicularity of the mating elements and, naturally, of the entire structure, which is confirmed by the architecture of ancient Egypt.

Getting a right angle without the necessary tools is not easy. But if you use this triangle, everything turns out to be quite simple. You need to take an ordinary rope, divide it into 12 equal parts, and from them make a triangle, the sides of which will be equal to 3, 4 and 5 parts. The angle between sides of length 3 and 4 turns out to be a right angle. This is the Egyptian Pythagorean Triangle.

In many historical writings there are traces that the unique properties of the “Egyptian triangle” were known and widely used many centuries before Pythagoras and not only in Egypt, but also far beyond its borders: in Mesopotamia, in ancient China, in Babylon.

The famous ancient Egyptian proverb “Do as it is done,” which has survived to this day, suggests that the Egyptians themselves, who erected these construction masterpieces, were simple performers and did not have any special knowledge, and all the secrets were hidden from the uninitiated. After all, the construction work was led by priests - members of a special privileged closed caste. They were keepers of ancient knowledge that was kept secret. But the inquisitive mind of the great thinker Pythagoras managed to unravel one of these secrets.

People's minds are always haunted by various mysteries, and this will probably always be the case. , although known to mankind since time immemorial, is still one of the not fully solved mysteries.

After all, no matter what you say, the shape of the Egyptian triangle is simple and at the same time harmonious, in its own way it is even beautiful. And it's quite easy to work with. To do this, you can use the simplest tools - a ruler and a compass. Using this simple element and its symmetrical display, you can get beautiful, harmonious figures. This is the Maltese cross, and the middle section of the Pyramid of Khafre, and a fractal series of decreasing - increasing, in size, Egyptian triangles in accordance with the rule of the golden section. This is an amazing wealth of harmonious proportions.

There are still many inquisitive people in the world who, like madmen, are inventing a perpetual motion machine, looking for the squaring of the circle, the philosopher’s stone and the book of the dead. Most likely, their efforts are in vain, but even in the case of the Egyptian Triangle, it is clear that there are still many “simple secrets” on earth.

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