Formulas of Egyptian geometry. Egyptian triangle and converse of Pythagoras' theorem

Construction using the Egyptian triangle is an ancient method that is still actively used by modern builders. It got its name thanks to ancient Egyptian buildings, although it is known that its history begins long before this period.

But, most likely, the properties of the unique figure were not appreciated in those days until Pythagoras appeared, who was able to analyze and evaluate the graceful forms of the figure.

The Egyptian triangle has been known since ancient times. It has been and remains popular in construction and architecture for many centuries.

It is believed that the great Greek mathematician Pythagoras of Samos created the geometric structure. Thanks to him, today we can use all the properties of geometric construction in the field of structure.

The birth of an idea

The mathematician got the idea after traveling to Africa at the request of Thales, who set the task for Pythagoras to study the mathematics and astronomy of those places. In Egypt, among the endless desert, he encountered majestic buildings that amazed him with their size, grace and beauty.

It should be noted that more than two and a half thousand years ago the pyramids were somewhat different - huge, with clear edges. Having carefully studied the powerful buildings, of which there were quite a few, since next to the giants there were smaller temples built for the children, wives and other relatives of the pharaoh, this gave him an idea.

Thanks to his mathematical abilities, Pythagoras was able to determine the pattern in the shapes of the pyramid, and the ability to analyze and draw conclusions led to the creation of one of the most significant theories in the history of geometry.

From the history

Did they know about geometry and mathematics in ancient Egypt? Of course yes. The life of the Egyptians was closely connected with science. They regularly used knowledge when marking fields, creating architectural masterpieces. There was even a service of land surveyors who applied geometric rules when restoring boundaries.

The triangle got its name thanks to the Hellenes, who often visited Egypt in the 7th-5th centuries. BC. It is believed that the prototype of the figure was the Pyramid of Cheops, characterized by perfect proportions. Her place in history is special. If you look at the cross section, you can see two triangles, whose internal angle is 51 about 50’.

Structure

The task is much easier if you use a protractor or triangle. But, previously only cords and ropes, divided into segments, were used. Thanks to the marks on the rope, it was possible to accurately recreate a rectangular figure. The builders replaced the protractor and square with a rope, for which they marked 12 parts with knots on it and folded a triangle with segments 3,4,5. A right angle was obtained without difficulty. This knowledge helped create many structures, including the pyramids.

It is interesting that before ancient Egypt, they built in this way in China, Babylon, and Mesopotamia.

The properties of the Egyptian triangular figure obey the truth - the square of the hypotenuse is equal to the squares of the two legs. This Pythagorean theorem is familiar to everyone from school. For example, we multiply 5x5 and get a hypotenuse equal to the number 25. The squares of both sides are 16 and 9, which adds up to 25.

Thanks to these properties, the triangle has found application in construction. You can take any part in order to draw a straight line with the condition that its length must be a multiple of five. After this, notice one edge and draw a line from it that is a multiple of four, and from the other a line that is a multiple of three. In this case, each segment must be at least four and three in length. Intersecting, they form one right angle of 90 degrees. Other angles are 53.13 and 36.87 degrees.

What alternatives are there?

How to create a right angle

The best option make a right angle is the use of a square or protractor. This will allow you to find the required proportions with minimal cost. But, the main point of the Egyptian triangle is its versatility due to the ability to create a figure without having anything at hand.

Anything can be useful in this matter, even printed publications. Any book or even magazine always has an aspect ratio that forms a right angle. Printing presses always work precisely so that the roll inserted into the machine is cut at proportional angles.

Ancient engineers came up with many ways to build the Egyptian triangle and always saved resources.

Therefore, the simplest and most widely used method of construction was geometric figure using regular rope. The string was taken and cut into 12 even pieces, from which a figure with proportions of 3,4 and 5 was laid out.

How to create other angles?

The Egyptian Triangle cannot be underestimated in the construction world. Its properties are definitely useful, but without the ability to construct angles of a different degree in construction it is impossible. To form an angle of 45 degrees, you will need a frame or baguette, which are sawn at an angle of 45 degrees and connected to each other.

Important! To get the required slope, you will need to borrow a sheet of paper from the printed publication and bend it. The bend lines will go through the corner. The edges must be connected.

You can get 60 degrees using two 30 degree triangles. Most often used to create decorative elements.

Small tricks

The Egyptian triangle 3x4x5 is relevant for small houses. But what if the house is 12x15?

To do this you need to build right triangle, whose legs are 12 and 15 m. The hypotenuse is found as Square root from the sum of 12x12 and 15x15. As a result, we get 19.2 m. Using something - rope, twine, twine, cable, military cable, we measure 12, 15 and 19.2 m. We make knots in these places and put presses.

Then you need to stretch the triangle in the right place and install 3 support points into which to drive pegs. The fourth point can be obtained without touching the ends of the legs. For this point right angle throw it diagonally and you're done.

For example, there is an area where a right angle is required - for space for a kitchen unit, tile layout and other aspects. It would be nice to take such issues into account when laying, but the reality is different and you don’t always come across smooth walls and right angles. The Egyptian triangle with a ratio of 3:4:5, or, if necessary, 1.5:2:2.5, is useful here.

The thickness of the beacons, errors, bumps on the walls, etc. must be taken into account. The triangle is drawn using a tape measure and chalk. If the markings are small, then you can use a sheet of drywall, since they are cut with the correct angles.

The Egyptian triangle was widely used in construction for as long as 2.5 centuries. And today sometimes it is necessary to use this technique, in the absence necessary tools to get right angles. The properties of this figure are unique, which guarantees precision in architecture and construction, which cannot be avoided. It is easy to work with, its shape is harmonious and beautiful. To this day, inquisitive minds are trying to unravel the mystery of the Egyptian triangle.

Mathematical lifehack from the field of geometry “How to get a triangle with a right angle using a simple rope.”
The Egyptians, 4,000 years ago, used a method to build the pyramids by making a right triangle using a rope divided into 12 equal parts.

The concept of the “Egyptian triangle”.

Why is a triangle with sides 3, 4, 5 called Egyptian?

And the whole point is that the builders of Ancient Egypt pyramids needed a simple and reliable method for constructing a triangle with a right angle. And this is how they implemented it. The rope was divided into twenty equal parts, marking the boundaries between adjacent parts; the ends of the rope were connected. After this, 3 people pulled the rope so that it formed a triangle, and the distances between each two Egyptians pulling the rope were three parts, four parts and five parts respectively. The result was a triangle with a right angle with legs in three and four parts and a hypotenuse in five parts. It is known that the angle between sides of three and four parts was right. As you know, ancient Egyptian surveyors, who in addition to measuring land plots were engaged in construction on the ground, in ancient Egypt they were called harpedonaptes (which literally translates as “pulling ropes”). Harpedonaptes occupied 3rd place in the hierarchy of priests of Ancient Egypt.

Converse Pythagorean theorem.

But what makes a triangle with sides 3, 4, 5 turn out to be rectangular? Most would answer this question that this fact This is a theorem: since three squared plus four squared equals five squared. But he says that if a triangle has a right angle, then the sum of the squares of its 2 sides is equal to the square of the third. Here we are dealing with a theorem inverse to the Pythagorean theorem: if the sum of the squares of 2 sides of a triangle is equal to the square of the third, then the triangle is right-angled.

The practical application outlined goes back to the distant past. Hardly anyone gets right angles using this method today. But nonetheless this method is an excellent mathematical life hack and can be applied by you in any life situation.

The method of determining a right triangle using a rope has moved from the world of practice to the world of ideas, just as much of the material culture of antiquity has entered the spiritual culture of present reality.

Lesson topic

Lesson Objectives

• Get acquainted with new definitions and remember some already studied.
• Deepen your knowledge of geometry, study the history of origin.
• Pin theoretical knowledge students 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

1. Introduction.
2. It's useful to remember.
3. 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. Even existed special service surveyors who quickly, using geometric techniques, restored the boundaries of fields when the water subsided.

It is still unknown what we will call our younger generation, which grows up on computers that allow us not to memorize the multiplication table or perform other elementary mathematical calculations in our heads or geometric constructions. 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.

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.

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. Height, median and bisector isosceles triangle, lowered onto the base, coincide.
• 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.

– a right triangle with an aspect ratio of 3:4:5. Sum specified 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.

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 have direct relation 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 member equal to the sum two previous ones: 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.

Expression " golden ratio", according to some, 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 radius, equal to the segment with a ratio of 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 a ratio of 4. If we now connect the intersection point of the arcs with lines to points B and C, we will obtain a right-angled triangle with an aspect ratio of 3: 4: 5.

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.

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 you consider that at the base of Khafre’s pyramid lies the Egyptian triangle, then we can conclude, people 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 distinctive features 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, that even if there are no corresponding measuring instruments, then 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 visiting Egypt and studying its features 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 car) and consisting of quite large quantity parts, for example, a car wheel includes a disk, rim, tire, sometimes a tube, mounting 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; dealt with problems of aesthetics of technical objects, industrial design, and in his designs attached great importance external forms of machines. Reuleaux is often called the father of kinematics.

Questions

1. What is a triangle?
2. Types of triangles?
3. What is special about the Egyptian triangle?
4. Where is the Egyptian triangle used? > Mathematics 8th grade

The famous mathematician Pythagoras made many different discoveries, but for most people who do not regularly deal with algebra and geometry, he is known for his theorem. The scientist discovered it while in Egypt, where he was captivated by the beauty and elegance of the pyramids, and this, in turn, gave him the idea that a certain pattern could be traced in their forms.

History of discovery

The Egyptian triangle owes its name to the Hellenes, who often visited Egypt in the 7th-5th centuries BC. e., among them was Pythagoras. basis the Pyramid of Cheops is a rectangular polygon, and

The pyramids of Khafre are the so-called Egyptian triangle, which the ancients called sacred. Plutarch wrote that the inhabitants of Egypt correlated nature with this geometric figure: the vertical leg symbolized a man, the base a woman, and the hypotenuse a child. The aspect ratio in it is 3:4:5, and this leads to the Pythagorean theorem, since 3 2 x 4 2 = 5 2. Therefore, the fact that the Egyptian triangle lies at the base of Khafre's pyramid suggests that the famous theorem was known to the inhabitants of the ancient world even before Pythagoras formulated it. A special feature of this figure is also considered to be that, thanks to this aspect ratio, it is the first and simplest of the Heronian triangles, since its sides and area are integer.

Application

The Egyptian triangle has been popular in architecture and construction since ancient times.

It was mainly used when constructing right angles using a cord or rope divided into 12 parts. Using the marks on such a rope, it was possible to very accurately create a rectangular figure, the legs of which would serve as guides for setting the right angle of the structure. It is known that such properties of this geometric figure were used not only in Ancient Egypt, but also, long before that, in China, Babylon and Mesopotamia. The Egyptian triangle was also used to create proportional structures in the Middle Ages.

Angles

The aspect ratio of this triangle is 3:4:5 resulting in it being a right triangle, i.e. one angle is 90 degrees and the other two are 53.13 and 36.87 degrees. A right angle is an angle between sides whose ratio is 3:4.

Proof

With some simple calculations you can prove that the triangle is a right triangle. If we follow the converse theorem to the one created by Pythagoras, i.e., if the sum of the squares of two sides is equal to the square of the third, then it is rectangular, and since its sides lead to the equality 3 2 x 4 2 = 5 2, therefore, it is rectangular.
To summarize, it should be noted that the Egyptian triangle, the properties of which have been known to mankind for many centuries, continues to be used in architecture today. This is not at all surprising, because this method guarantees accuracy, which is very important during construction. In addition, it is very easy to use, which also makes the process much easier. All the advantages of using this method have been tested for centuries and remain popular to this day.