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 his ability to analyze and draw conclusions led to the creation of one of the most significant theories in the history of geometry.

From 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 their knowledge when marking fields and creating architectural masterpieces. There was even a service of land surveyors who applied geometric rules when restoring boundaries.

The triangle received 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 Cheops pyramid, 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 was the method of constructing a geometric figure using ordinary 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 is sawn at an angle of 45 degrees and connected to each other.

Important! To obtain the required slope, you will need to borrow a sheet of paper from printed edition and bend it. The bend lines will pass 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 construct a right triangle whose legs are equal to 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. To do this, the right angle point is thrown diagonally and everything is ready.

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 of the necessary tools, to obtain 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.

Each science has its own foundation, on the basis of which all its subsequent development is built. This is, of course, the Pythagorean theorem. From school they teach the formula: “Pythagorean pants are equal in all directions.” Scientifically it sounds a little less eloquent. This theorem is visually represented with sides 3-4-5. This is the wonderful Egyptian Triangle.

Story

The famous Greek mathematician and philosopher Pythagoras of Samos, who gave his name to the theorem, lived 2.5 thousand years ago. The biography of this outstanding scientist has been little studied, but some have survived to this day.

At the request of Thales, in order to study mathematics and astronomy, in 535 BC he went on a long journey to Egypt and Babylon. In Egypt, among the endless expanse of the desert, he saw pyramids, amazing with their huge size and slender geometric shapes. It is worth noting that Pythagoras saw them in a slightly different form than the one in which tourists see now. These were unimaginably huge structures for that time with clear, even edges against the backdrop of smaller adjacent temples for wives, children and other relatives. In addition to their direct purpose (the tomb and guardian of the sacred body of the pharaoh), the pyramids were also built as symbols of the greatness, wealth and power of Egypt.

And so Pythagoras, during a careful study of these structures, noticed a strict pattern in the relationship between the sizes and shapes of the structures. The Cheops pyramid corresponds to the size of the Egyptian triangle; it was considered sacred and had a special magical meaning.

The Pyramid of Cheops is reliable evidence that the knowledge of the proportions of the Egyptian triangle was used by the Egyptians long before the discovery of Pythagoras.

Application

The shape of the triangle is the simplest and most harmonious, it is easy to work with; this will require only the most simple tools - a compass and a ruler.

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.

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.

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.

Anyone who listened carefully to a geometry teacher at school is very familiar with what the Egyptian triangle is. It differs from other types of similar ones with an angle of 90 degrees in its special aspect ratio. When a person first hears the phrase “Egyptian triangle,” pictures of majestic pyramids and pharaohs come to mind. But what does history say?

As is always the case, there are several theories regarding the name "Egyptian Triangle". According to one of them, the famous Pythagorean theorem came to light precisely thanks to this figure. In 535 BC. Pythagoras, following the recommendation of Thales, went to Egypt in order to fill some gaps in his knowledge of mathematics and astronomy. There he drew attention to the peculiarities of the work of Egyptian land surveyors. They performed a construction with a right angle in a very unusual way, the sides of which were interconnected with one another in a 3-4-5 ratio. This mathematical series made it relatively easy to connect the squares of all three sides with one rule. This is how the famous theorem arose. And the Egyptian triangle is precisely the same figure that prompted Pythagoras to a most ingenious solution. According to other historical data, the figure was given its name by the Greeks: at that time they often visited Egypt, where they could be interested in the work of land surveyors. There is a possibility that, as often happens with scientific discoveries, both stories happened at the same time, so it is impossible to say with certainty who came up with the name “Egyptian triangle” first. Its properties are amazing and, of course, are not limited to the aspect ratio alone. Its area and sides are represented by integers. Thanks to this, applying the Pythagorean theorem to it allows us to obtain integer numbers of the squares of the hypotenuse and legs: 9-16-25. Of course, this could be just a coincidence. But how, in this case, can we explain the fact that the Egyptians considered “their” triangle sacred? They believed in his interconnection with the entire Universe.

After information about this unusual geometric figure became publicly available, the world began searching for other similar triangles with integer sides. It was obvious that they existed. But the importance of the question was not simply to perform mathematical calculations, but to test the “sacred” properties. The Egyptians, for all their unusualness, were never considered stupid - scientists still cannot explain how exactly the pyramids were built. And here, suddenly, an ordinary figure was attributed a connection with Nature and the Universe. And, indeed, the found cuneiform contains instructions about a similar triangle with a side whose size is described by a 15-digit number. Currently, the Egyptian triangle, whose angles are 90 (right), 53 and 37 degrees, is found in completely unexpected places. For example, when studying the behavior of molecules of ordinary water, it turned out that the change is accompanied by a restructuring of the spatial configuration of the molecules, in which you can see... that same Egyptian triangle. If we remember that it consists of three atoms, then we can talk about conditional three sides. Of course, we are not talking about a complete coincidence of the famous ratio, but the resulting numbers are very, very close to the required ones. Is this why the Egyptians recognized their “3-4-5” triangle as a symbolic key to natural phenomena and the secrets of the Universe? After all, water, as you know, is the basis of life. Without a doubt, it is too early to draw an end to the study of the famous Egyptian figure. Science never rushes to conclusions, seeking to prove its assumptions. And we can only wait and be amazed at the knowledge

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. The basis of the Cheops pyramid 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 inverse 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.