Hypothesis: we believe that the perfection of the shape of the pyramid is due to the mathematical laws embedded in its shape.
Target: having studied the pyramid as a geometric body, give an explanation for the perfection of its shape.
Tasks:
1. Give a mathematical definition of the pyramid.
2. Study the pyramid as a geometric body.
3. Understand what mathematical knowledge the Egyptians laid in their pyramids.
Private questions:
1. What is a pyramid as a geometric body?
2. How can you explain the uniqueness of the shape of the pyramid from a mathematical point of view?
3. What explains the geometric wonders of the pyramid?
4. What explains the perfection of the pyramid shape?
Definition of the pyramid.
PYRAMID (from the Greek pyramis, genus pyramidos) - a polyhedron, the base of which is a polygon, and the other faces are triangles with a common vertex (figure). According to the number of angles of the base, pyramids are distinguished triangular, quadrangular, etc.
PYRAMID - a monumental structure with a geometric pyramid shape (sometimes also stepped or tower-like). The pyramids are called the giant tombs of the ancient Egyptian pharaohs of the 3rd - 2nd millennium BC. e., as well as ancient American pedestals of temples (in Mexico, Guatemala, Honduras, Peru) associated with cosmological cults.
It is possible that the Greek word "pyramid" comes from the Egyptian expression per-em-us, that is, from a term meaning the height of the pyramid. Prominent Russian Egyptologist V. Struve believed that the Greek “puram… j” comes from the ancient Egyptian “p” -mr ”.
From the history. After studying the material in the textbook "Geometry" by the authors of Atanasyan. Butuzov and others, we learned that: A polyhedron composed of n - gon A1A2A3 ... An and n triangles PA1A2, PA2A3, ..., PnA1 is called a pyramid. Polygon A1A2A3 ... An is the base of the pyramid, and triangles PA1A2, PA2A3, ..., PANA1 are the side faces of the pyramid, P is the top of the pyramid, the segments PA1, PA2, ..., Pn are the lateral edges.
However, this definition of a pyramid did not always exist. For example, the ancient Greek mathematician, the author of theoretical treatises on mathematics that have come down to us, Euclid, defines a pyramid as a bodily figure bounded by planes that converge from one plane to one point.
But this definition was criticized already in antiquity. So Heron proposed the following definition of a pyramid: "It is a figure bounded by triangles converging at one point and the base of which is a polygon."
Our group, comparing these definitions, came to the conclusion that they do not have a clear formulation of the concept of “foundation”.
We examined these definitions and found the definition of Adrien Marie Legendre, who in 1794 in his work “Elements of Geometry” defines the pyramid as follows: “The pyramid is a bodily figure, formed by triangles converging at one point and ending on different sides of the flat base ”.
It seems to us that the last definition gives a clear idea of the pyramid, since in it in question that the base is flat. Another definition of a pyramid appeared in a 19th century textbook: "a pyramid is a solid angle intersected by a plane."
The pyramid as a geometric body.
That. A pyramid is a polyhedron, one of the faces of which (base) is a polygon, the other faces (side) are triangles that have one common vertex (apex of the pyramid).
The perpendicular drawn from the top of the pyramid to the plane of the base is called heighth pyramids.
In addition to an arbitrary pyramid, there are correct pyramid, at the base of which is a regular polygon and truncated pyramid.
The figure shows the pyramid PABCD, ABCD is its base, PO is the height.
Full surface area a pyramid is called the sum of the areas of all its faces.
S full = S side + S main, where S side- the sum of the areas of the side faces.
The volume of the pyramid is found by the formula:
V = 1 / 3Sb. h, where Sosn. - base area, h- height.
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Apothem ST - the height of the side face of the regular pyramid.
The area of the side face of a regular pyramid is expressed as follows: S side. = 1 / 2P h, where P is the perimeter of the base, h- the height of the side face (apothem of the regular pyramid). If the pyramid is intersected by plane A'B'C'D 'parallel to the base, then:
1) lateral ribs and height are divided by this plane into proportional parts;
2) in the section, a polygon A'B'C'D 'is obtained, similar to the base;
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A regular triangular pyramid is called tetrahedron .
Truncated pyramid is obtained by cutting off its upper part from the pyramid by a plane parallel to the base (figure ABCDD'C'B'A ').
Truncated pyramid bases- similar polygons ABCD and A`B`C`D`, side faces - trapezoid.
Height truncated pyramid - the distance between the bases.
Truncated volume pyramid is found by the formula:
V = 1/3 h(S + https://pandia.ru/text/78/390/images/image019_2.png "align =" left "width =" 91 "height =" 96 "> The lateral surface area of a regular truncated pyramid is expressed as follows: S side. = ½ (P + P ') h, where P and P 'are the perimeters of the bases, h- the height of the side face (apothem of the correct truncated pyramids
Sections of the pyramid.
The sections of the pyramid by planes passing through its apex are triangles.
The section passing through two non-adjacent lateral edges of the pyramid is called diagonal section.
If the section passes through a point on the side edge and the side of the base, then this side will be its trace on the plane of the base of the pyramid.
A section passing through a point lying on the face of the pyramid, and a given trace of the section on the base plane, then the construction should be carried out as follows:
· Find the point of intersection of the plane of the given face and the trace of the section of the pyramid and designate it;
Build a straight line through set point and the resulting intersection point;
· Repeat these steps for the next faces.
, which corresponds to the ratio of the legs of a right-angled triangle 4: 3. This ratio of legs corresponds to the well-known right-angled triangle with sides 3: 4: 5, which is called the "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 to a "sacred" triangle; they symbolically likened the vertical leg to the husband, the base to the wife, and the hypotenuse to that which is born of both.
For a triangle 3: 4: 5, the equality is true: 32 + 42 = 52, which expresses the Pythagorean theorem. Was 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's hard to find more good 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 amaze distant descendants with the depth of their knowledge, and they achieved this by choosing the “golden” right triangle for the Cheops pyramid, and “sacred” or “Egyptian” for the Khephren pyramid triangle.
Very often in their research, scientists use the properties of pyramids with the proportions of the Golden Section.
In the mathematical encyclopedic dictionary, the following definition of the Golden Section is given - this is harmonic division, division in the extreme and average ratio - dividing the segment AB into two parts in such a way that most of its AC is the average proportional between the entire segment AB and its smaller part CB.
Algebraic Finding of the Golden Ratio of a Segment AB = a is reduced to solving the equation a: x = x: (a - x), whence x is approximately equal to 0.62a. The ratio x can be expressed in fractions 2/3, 3/5, 5/8, 8/13, 13/21 ... = 0.618, where 2, 3, 5, 8, 13, 21 are Fibonacci numbers.
The geometric construction of the Golden Section of the segment AB is carried out as follows: at point B, the perpendicular to AB is restored, the segment BE = 1/2 AB is laid on it, A and E are put off, DE = BE and, finally, AC = HELL, then the equality AB is fulfilled: SV = 2: 3.
The golden ratio is often used in works of art, architecture, and occurs in nature. Striking examples are the sculpture of Apollo Belvedere, Parthenon. During the construction of the Parthenon, the ratio of the height of the building to its length was used and this ratio is 0.618. The objects around us also provide examples of the Golden Ratio, for example, the bindings of many books have a ratio of width to length close to 0.618. Considering the arrangement of leaves on the common stem of plants, you can see that between every two pairs of leaves, the third is located in the place of the Golden Section (slides). Each of us “carries” the Golden Ratio with us “in our hands” - this is the ratio of the phalanges of the fingers.
Through the discovery of several mathematical papyri, Egyptologists have learned a thing or two about ancient Egyptian systems of numbers and measures. The tasks contained in them were solved by scribes. One of the most famous is the Rindi Mathematical Papyrus. By studying these problems, Egyptologists learned how the ancient Egyptians coped with varying amounts arising in calculating the measures of weight, length and volume, in which fractions were often used, and also how they were manipulated with angles.
The ancient Egyptians used a method for calculating angles based on the ratio of the height to the base of a right-angled triangle. They expressed any angle in the language of the gradient. The gradient of the slope was expressed by an integer ratio called "seced". In his book Mathematics in the Time of the Pharaohs, Richard Pillins explains: “The seked of a regular pyramid is the inclination of any of the four triangular faces to the plane of the base, measured by an nth number of horizontal units per one vertical unit of lift. Thus, this unit is equivalent to our modern tilt cotangent. Therefore, the Egyptian word "seked" is related to our modern word"gradient"".
The numerical key to the pyramids lies in the ratio of their height to the base. In practical terms, this is the easiest way to make templates needed to constantly check the correct angle of inclination throughout the construction of the pyramid.
Egyptologists would be happy to convince us that each pharaoh was eager to express his individuality, which is why the different angles of inclination for each pyramid. But there could be another reason. Perhaps they all wished to embody different symbolic associations, hidden in different proportions. However, the angle of Khafre's pyramid (based on a triangle (3: 4: 5) appears in the three problems represented by the pyramids in the Rindi Mathematical Papyrus). So this attitude was well known to the ancient Egyptians.
To be fair to Egyptologists who claim that the ancient Egyptians did not know the 3: 4: 5 triangle, let us say that the length of the hypotenuse 5 was never mentioned. But math problems concerning pyramids are always resolved on the basis of the angle seked - the ratio of the height to the base. Since the length of the hypotenuse was never mentioned, it was concluded that the Egyptians never calculated the length of the third side.
The height to base ratios used in the pyramids of Giza were undoubtedly known to the ancient Egyptians. It is possible that these relationships were chosen arbitrarily for each pyramid. However, this contradicts the importance attached to numerical symbolism in all forms of Egyptian visual arts... It is highly likely that such relationships were significant because they expressed specific religious ideas. In other words, the entire Giza complex was subordinated to a coherent plan designed to reflect a certain divine theme. This would explain why the designers chose different angles for the three pyramids.
In The Mystery of Orion, Bauval and Gilbert presented convincing evidence of the connection of the pyramids of Giza with the constellation Orion, in particular with the stars of Orion's Belt. The same constellation is present in the myth of Isis and Osiris, and there is reason to consider each pyramid as an image of one of the three main deities - Osiris, Isis and Horus.
MIRACLES "GEOMETRIC".
Among the grandiose pyramids of Egypt, a special place is held by Great Pyramid of Pharaoh Cheops (Khufu)... Before proceeding to the analysis of the shape and size of the Cheops pyramid, one should recall 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, equal to four "fingers" (16.6 mm).
Let's analyze the dimensions of the Cheops pyramid (Fig. 2), following the reasoning given in the wonderful book of the Ukrainian scientist Nikolai Vasyutinsky " Golden proportion"(1990).
Most researchers agree that the length of the side of the base of the pyramid, for example, Gf is equal to L= 233.16 m. This value corresponds to almost exactly 500 "cubits". Full compliance with 500 "cubits" will be if the length of the "cubit" is considered equal to 0.4663 m.
Pyramid height ( H) is estimated by researchers differently from 146.6 to 148.2 m. And depending on the accepted height of the pyramid, all the ratios of its geometric elements change. What is the reason for the differences in the estimate of the height of the pyramid? The fact is that, strictly speaking, the Cheops pyramid is truncated. Its upper platform nowadays has a size of about 10 ´ 10 m, and a century ago it was 6 ´ 6 m. Obviously, the top of the pyramid was taken apart, and it does not correspond to the original one.
When evaluating the height of the pyramid, it is necessary to take into account the following physical factor as a "draft" of the structure. Per long time under the influence of colossal pressure (reaching 500 tons per 1 m2 bottom surface) the height of the pyramid has decreased compared to the original height.
What was the initial height of the pyramid? This height can be recreated by finding the basic "geometric idea" of the pyramid.
Figure 2.
In 1837, the English Colonel G. Weisz measured the angle of inclination of the pyramid's faces: it turned out to be equal 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 AS to half of its base CB(Fig. 2), that is AC / CB = H / (L / 2) = 2H / L.
And here the researchers were in for a big surprise! .Png "width =" 25 "height =" 24 "> = 1.272. Comparing this value with the value of tg a= 1.27306, we see that these values are very close to each other. If we take the angle a= 51 ° 50 ", that is, reduce it by only one arc minute, then the value a will become equal to 1.272, that is, coincide with the value. It should be noted that in 1840 G. Weis repeated his measurements and specified that the value of the angle a= 51 ° 50 ".
These measurements led the researchers to the following very interesting hypothesis: the AC / CB = = 1,272!
Consider now a right-angled triangle ABC, in which the ratio of the legs AC / CB= (Fig. 2). If now the lengths of the sides of the rectangle ABC denote through x, y, z, and also take into account that the ratio y/x=, then in accordance with the Pythagorean theorem, the length z can be calculated by the formula:
If you accept x = 1, y= https://pandia.ru/text/78/390/images/image027_1.png "width =" 143 "height =" 27 ">
Figure 3."Golden" right-angled triangle.
Right-angled triangle in which the sides are related as t: golden "right-angled triangle."
Then, if we take as a basis the hypothesis that the main "geometric idea" of the Cheops pyramid is the "golden" right-angled triangle, then from here it is easy to calculate the "design" height of the Cheops pyramid. It is equal to:
H = (L / 2) ´ = 148.28 m.
Let us now deduce some other relations for the Cheops pyramid arising from the "golden" hypothesis. In particular, we find the ratio of the outer area of the pyramid to the area of its base. To do this, take the length of the leg CB per unit, that is: CB= 1. But then the length of the side of the base of the pyramid Gf= 2, and the base area EFGH will be equal SEFGH = 4.
We now calculate the area of the side face of the Cheops pyramid SD... Since the height AB 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 4 t, and the ratio of the total outer area of the pyramid to the area of the base will be equal to the golden ratio! That's what it is - the main geometric mystery of the Cheops pyramid!
The group of "geometric miracles" of the Cheops pyramid includes the real and contrived properties of relations between different dimensions in the pyramid.
As a rule, they are obtained in search of certain "constants", in particular, the number "pi" (Ludolph's number), equal to 3.14159 ...; base of natural logarithms "e" (Napier's number), equal to 2.71828 ...; the number "F", the number of the "golden ratio", equal, for example, 0.618 ... etc.
You can name, for example: 1) Property of Herodotus: (Height) 2 = 0.5 tbsp. main x Apothem; 2) Property of V. Price: Height: 0.5 st. osn = Square root of "Ф"; 3) Property of M. Eyst: Perimeter of the base: 2 Height = "Pi"; in a different interpretation - 2 tbsp. main : Height = "Pi"; 4) Property of G. Ribs: Inscribed circle radius: 0.5 tbsp. main = "F"; 5) Property of K. Kleppisch: (Art. Main.) 2: 2 (art. Main. X Apothem) = (art. Main. U. Apothem) = 2 (art. Main. X Apothem): ((2 art. base X Apothem) + (st. base) 2). Etc. You can think of a lot of such properties, especially if you connect two neighboring pyramids. For example, as "A. Arefiev's Properties", one can mention that the difference between the volumes of the Cheops pyramid and the Chephren pyramid is equal to the doubled volume of the Mikerin pyramid ...
Many interesting provisions, in particular, about the construction of pyramids according to the "golden ratio" are described in the books by D. Hambidge "Dynamic symmetry in architecture" and M. Geek "Aesthetics of proportion in nature and art". Recall that the "golden ratio" is the division of a segment in such a ratio when part A is as many times larger than part B, how many times A is less than the entire segment A + B. The ratio A / B is equal to the number "Ф" == 1.618. .. The use of the "golden ratio" is indicated not only in individual pyramids, but also in the entire complex of pyramids in Giza.
The most curious thing, however, is that one and the same pyramid of Cheops simply "cannot" contain so many wonderful properties. Taking a certain property one by one, it can be "adjusted", but all at once they do not fit - they do not coincide, they contradict each other. Therefore, if, for example, when checking all properties, we initially take the same side of the pyramid base (233 m), then the heights of pyramids with different properties will also be different. In other words, there is a certain "family" of pyramids, outwardly similar to Cheops, but corresponding to different properties. Note that there is nothing particularly miraculous in the "geometric" properties - much arises purely automatically, from the properties of the figure itself. Only something clearly impossible for the ancient Egyptians should be considered a "miracle". This, in particular, includes "cosmic" miracles, in which the measurements of the Cheops pyramid or the pyramid complex at Giza are compared with some astronomical measurements and "even" numbers are indicated: a million times, a billion times less, and so on. Let's consider some "cosmic" relationships.
One of the statements is this: "If we divide the side of the base of the pyramid by the exact length of the year, then we get exactly 10-millionth of the earth's axis." Calculate: Divide 233 by 365, we get 0.638. The radius of the Earth is 6378 km.
Another statement is actually the opposite of the previous one. F. Noetling pointed out that if we use the "Egyptian elbow" invented by him, then the side of the pyramid will correspond to "the most exact duration solar year, expressed with an accuracy of one billionth of a day "- 365.540.903.777.
P. Smith's statement: "The height of the pyramid is exactly one billionth of the distance from the Earth to the Sun." Although an altitude of 146.6 m is usually taken, Smith took it 148.2 m. According to modern radar measurements, the semi-major axis of the earth's orbit is 149.597.870 + 1.6 km. This is the average distance from the Earth to the Sun, but at perihelion it is 5,000,000 kilometers less than at aphelion.
One last curious statement:
"How to explain that the masses of the pyramids of Cheops, Khafre and Mykerinus relate to each other, like the masses of the planets Earth, Venus, Mars?" Let's calculate. The masses of the three pyramids are as follows: Khafre - 0.835; Cheops - 1,000; Mikerin - 0.0915. The ratio of the masses of the three planets: Venus - 0.815; Land - 1,000; Mars - 0.108.
So, in spite of the skepticism, let us note the well-known harmony of the construction of statements: 1) the height of the pyramid, as a line "extending into space" - corresponds to the distance from the Earth to the Sun; 2) the side of the base of the pyramid closest to "the substrate", that is, to the Earth, is responsible for the earth's radius and earthly circulation; 3) the volumes of the pyramid (read - masses) correspond to the ratio of the masses of the planets closest to the Earth. A similar "cipher" can be traced, for example, in the bee language analyzed by Karl von Frisch. However, we will refrain from commenting on this for now.
PYRAMID SHAPE
The famous four-sided shape of the pyramids did not appear immediately. The Scythians made burials in the form of earthen hills - mounds. The Egyptians set up "hills" of stone - pyramids. This happened for the first time after the unification of Upper and Lower Egypt, in the XXVIII century BC, when before the founder Dynasty III Pharaoh Djoser (Zoser) had the task of strengthening the unity of the country.
And here, according to historians, important role in strengthening central government played a "new concept of deification" of the king. Although the royal burials were distinguished by greater splendor, they, in principle, did not differ from the tombs of the court nobles, they were the same structures - mastabas. Above the chamber with the sarcophagus containing the mummy, a rectangular mound of small stones, where then a small building of large stone blocks was erected - "mastaba" (in Arabic - "bench"). In place of the mastab of his predecessor, Sanakht, Pharaoh Djoser built the first pyramid. It was stepwise and was a visible transitional stage from one architectural form to another, from a mastaba to a pyramid.
In this way, the sage and architect Imhotep, who was later considered a wizard and identified by the Greeks with the god Asclepius, "elevated" the pharaoh. As it were, six mastabas were erected in a row. Moreover, the first pyramid occupied an area of 1125 x 115 meters, with an estimated height of 66 meters (according to Egyptian measures - 1000 "palms"). At first, the architect planned to build a mastaba, but not oblong, but square in plan. Later it was expanded, but since the extension was made lower, there were two steps, as it were.
This situation did not satisfy the architect, and on the upper platform of the huge flat mastaba Imhotep put three more, gradually decreasing to the top. The tomb was under the pyramid.
Several more stepped pyramids are known, but later the builders moved on to the construction of the more familiar tetrahedral pyramids for us. Why, however, not three-sided or, say, octahedral? An indirect answer is given by the fact that almost all pyramids are perfectly oriented along the four cardinal directions, and therefore have four sides. Moreover, the pyramid was a "house", a shell of a quadrangular burial chamber.
But what caused the angle of inclination of the edges? In the book "The principle of proportions" a whole chapter is devoted to this: "What could determine the angles of inclination of the pyramids." In particular, it is indicated that "the image to which the great pyramids of the Old Kingdom gravitate is a triangle with a right angle at the top.
In space, it is a semi-octahedron: a pyramid in which the edges and sides of the base are equal, the faces are equilateral triangles. "Certain considerations are given on this matter in the books of Hambage, Geek and others.
What is the advantage of the angle of the semi-octahedron? According to the descriptions of archaeologists and historians, some of the pyramids collapsed under their own weight. What was needed was a "longevity angle", the angle most energetically reliable. Purely empirically, this angle can be taken from the apex angle in a heap of crumbling dry sand. But to get accurate data, you need to use a model. Taking four firmly fixed balls, you need to put the fifth on them and measure the angles of inclination. However, you can make a mistake here, so a theoretical calculation helps out: you should connect the centers of the balls with lines (mentally). At the base, you get a square with a side equal to twice the radius. The square will be just the base of the pyramid, the length of the edges of which will also be equal to twice the radius.
Thus, a dense packing of balls of the 1: 4 type will give us the correct semi-octahedron.
However, why do many pyramids, gravitating towards a similar shape, nevertheless not retain it? The pyramids are probably aging. Contrary to the famous saying:
"Everything in the world is afraid of time, and time is afraid of pyramids", the buildings of the pyramids should grow old, not only external weathering processes can and should take place in them, but also processes of internal "shrinkage", from which the pyramids may become lower. Shrinkage is also possible because, as found out by the works of D. Davidovits, the ancient Egyptians used the technology of making blocks from lime crumb, in other words, from "concrete". It is these processes that could explain the reason for the destruction of the Medum pyramid, located 50 km south of Cairo. It is 4600 years old, the dimensions of the base are 146 x 146 m, the height is 118 m. “Why is it so disfigured?” Asks V. Zamarovsky. “Usual references to the destructive influence of time and“ the use of stone for other buildings ”are not suitable here.
After all, most of its blocks and facing slabs have remained in place to this day, in ruins at its foot. " ...
The shape of the pyramids could also be generated by imitation: some natural patterns, "miraculous perfection", say, some crystals in the form of an octahedron.
Such crystals could be crystals of diamond and gold. Characteristically a large number of"intersecting" signs for concepts such as Pharaoh, Sun, Gold, Diamond. Everywhere - noble, shining (brilliant), great, flawless and so on. The similarities are not accidental.
The solar cult is known to be an important part of religion. Ancient egypt... "No matter how we translate the name of the greatest of the pyramids," says one of the modern manuals - "Khufu's Heaven" or "Khufu Heavenly", it meant that the king is the sun. " If Khufu, in the splendor of his power, imagines himself to be the second sun, then his son Djedef-Ra became the first of the Egyptian kings who began to call himself "the son of Ra", that is, the son of the Sun. The sun was symbolized by almost all peoples by the "solar metal", gold. "Great disk of bright gold" - this is how the Egyptians called our daylight. The Egyptians knew gold perfectly, they knew its native forms, where gold crystals can appear in the form of octahedrons.
As a "sample of forms" the "sun stone" - diamond is also interesting here. The name of the diamond came just from the Arab world, "almas" - the hardest, hardest, indestructible. The ancient Egyptians knew diamond and its properties quite well. According to some authors, they even used bronze pipes with diamond cutters for drilling.
Currently, the main supplier of diamonds is South Africa, but Western Africa is also rich in diamonds. The territory of the Republic of Mali is even called the "Diamond Land" there. Meanwhile, it is on the territory of Mali that the Dogon live, with whom the supporters of the Paleovisite hypothesis pin many hopes (see below). Diamonds could not serve as the reason for the contacts of the ancient Egyptians with this land. However, one way or another, it is possible that it was precisely by copying the octahedrons of diamond and gold crystals that the ancient Egyptians deified thereby "indestructible" like a diamond and "brilliant" like gold pharaohs, the sons of the Sun, comparable only with the most wonderful creations of nature.
Output:
Having studied the pyramid as a geometric body, having become acquainted with its elements and properties, we were convinced of the validity of the opinion about the beauty of the shape of the pyramid.
As a result of our research, we came to the conclusion that the Egyptians, having collected the most valuable mathematical knowledge, embodied it in the pyramid. Therefore, the pyramid is truly the most perfect creation of nature and man.
BIBLIOGRAPHY
"Geometry: Textbook. for 7 - 9 cl. general education. institutions \, etc. - 9th ed. - M .: Education, 1999
History of mathematics at school, M: "Education", 1982
Geometry 10-11 grade, M: "Education", 2000
Peter Tompkins "Secrets great pyramid Cheops ", M:" Tsentropoligraf ", 2005
Internet resources
http: // veka-i-mig. ***** /
http: // tambov. ***** / vjpusk / vjp025 / rabot / 33 / index2.htm
http: // www. ***** / enc / 54373.html
Instructions
In the event that at the base pyramids lies a square, the length of its diagonal is known, as well as the length of the edge of this pyramids, then the height this pyramids can be expressed from the Pythagorean theorem, because the triangle, which is formed by the edge pyramids, and half the diagonal at the base is a right-angled triangle.
The Pythagorean theorem states that the square of the hypotenuse in a rectangular one is equal in magnitude to the sum of the squares of its legs (a² = b² + c²). Edge pyramids- the hypotenuse, one of the legs is half the diagonal of the square. Then the length of the unknown leg (height) is found by the formulas:
b² = a² - c²;
c² = a² - b².
To make both situations as clear and understandable as possible, you can consider a couple.
Example 1: Base area pyramids 46 cm², its volume is 120 cm³. Based on these data, the height pyramids is found like this:
h = 3 * 120/46 = 7.83 cm
Answer: the height of this pyramids will be approximately 7.83 cm
Example 2: Y pyramids, at the base of which lies a polygon - a square, its diagonal is 14 cm, the length of the edge is 15 cm.According to these data, to find the height pyramids, you need to use following formula(which as a consequence of the Pythagorean theorem):
h² = 15² - 14²
h² = 225 - 196 = 29
h = √29 cm
Answer: the height of this pyramids is √29 cm or approximately 5.4 cm
note
If there is a square or other regular polygon at the base of the pyramid, then this pyramid can be called regular. Such a pyramid has a number of properties:
its lateral ribs are equal;
facets of it - isosceles triangles that are equal to each other;
near such a pyramid, you can describe a sphere, as well as inscribe it.
Sources:
- Correct pyramid
A pyramid is a figure at the base of which lies a polygon, while its faces are triangles with a common vertex for all. In typical tasks, it is often required to construct and determine the length of the perpendicular drawn from the vertex pyramids to the plane of its base. The length of this segment is called the height pyramids.
You will need
- - ruler
- - pencil
- - compass
Instructions
To complete it, build a pyramid in accordance with the condition of the problem. For example, to build a regular tetrahedron, you need to draw a figure so that all 6 edges are equal to each other. If you want to build the height quadrangular, then only 4 edges of the base should be equal. Then the edges of the side faces can be constructed unequal with the edges of the polygon. Name the pyramid, marking all the vertices with Latin letters. For example, for pyramids with a triangle at the base, you can choose A, B, C (for the base), S (for the top). If the condition specifies the specific dimensions of the edges, then when constructing the figure, proceed from these values.
To begin with, conditionally select with the help of a compass, touching from the inside all the edges of the polygon. If a pyramid, then a point (call it, for example, H) on the base pyramids, into which the height falls, must correspond to the center of the circle inscribed in the correct base pyramids... The center will correspond to a point equidistant from any other point on the circle. If we connect the vertex pyramids S with the center of the circle H, then the segment SH will be the height pyramids... At the same time, remember that a circle can be inscribed in a quadrilateral, the sums of the opposite sides of which are the same. This applies to the square and rhombus. In this case, point H will lie in the quadrilateral. For any triangle it is possible to inscribe and describe a circle.
To build the height pyramids, use a compass to draw a circle, and then use a ruler to connect its center H to the vertex S. SH is the desired height. If at the bottom pyramids SABC is an irregular figure, then the height will connect the top pyramids with the center of the circle in which the base polygon is inscribed. All vertices of the polygon lie on such a circle. In this case, this segment will be perpendicular to the plane of the base pyramids... You can describe a circle around a quadrilateral if the sum of opposite angles is 180 °. Then the center of such a circle will lie at the intersection of the diagonals of the corresponding
How can you build a pyramid? On surface R let's build some kind of polygon, for example pentagon ABCDE. Out of plane R take point S. Connecting point S with segments with all points of the polygon, we get a pyramid SABCDE (fig.).
Point S is called apex and the polygon ABCDE is basis this pyramid. Thus, a pyramid with vertex S and base ABCDE is the union of all segments, where M ∈ ABCDE.
Triangles SAB, SBC, SCD, SDE, SEA are called side faces pyramids, common sides side faces SA, SB, SC, SD, SE - lateral ribs.
The pyramids are called triangular, quadrangular, n-angular depending on the number of sides of the base. In fig. given images of triangular, quadrangular and hexagonal pyramids.
The plane passing through the top of the pyramid and the diagonal of the base is called diagonal, and the resulting section is diagonal. In fig. 186 one of the diagonal sections of the hexagonal pyramid is shaded.
The segment of the perpendicular drawn through the top of the pyramid to the plane of its base is called the height of the pyramid (the ends of this segment are the top of the pyramid and the base of the perpendicular).
The pyramid is called correct if the base of the pyramid is a regular polygon and the top of the pyramid is projected to its center.
All lateral faces of a regular pyramid are congruent isosceles triangles. In a regular pyramid, all lateral edges are congruent.
The height of the side face of a regular pyramid drawn from its top is called apothem pyramids. All apothems of a regular pyramid are congruent.
If we designate the side of the base through a, and apothem through h, then the area of one side face of the pyramid is 1/2 ah.
The sum of the areas of all the side faces of the pyramid is called lateral surface area pyramids and denoted by S side.
Because side surface the correct pyramid consists of n congruent faces, then
S side. = 1/2 ahn= P h / 2 ,
where P is the perimeter of the base of the pyramid. Hence,
S side. = P h / 2
i.e. the lateral surface area of a regular pyramid is half the product of the base perimeter times the apothem.
The total surface area of the pyramid is calculated by the formula
S = S main + S side. ...
The volume of the pyramid is equal to one third of the product of the area of its base S ocн. to the height H:
V = 1/3 S main N.
The derivation of this and some other formulas will be given in a subsequent chapter.
Let's build the pyramid in a different way. Let a polyhedral angle, for example, a pentahedral, with apex S (fig.) Be given.
Let's draw a plane R so that it intersects all the edges of a given polyhedral angle at different points A, B, C, D, E (Fig.). Then the SABCDE pyramid can be considered as the intersection of a polyhedral angle and a half-space with the boundary R where the vertex S.
Obviously, the number of all faces of the pyramid can be arbitrary, but not less than four. When a plane intersects a triangular angle, a triangular pyramid is obtained, which has four faces. Any triangular pyramid is sometimes called tetrahedron, which means tetrahedron.
Truncated pyramid can be obtained if the pyramid is crossed by a plane parallel to the plane of the base.
In fig. an image of a quadrangular truncated pyramid is given.
Truncated pyramids are also called triangular, quadrangular, n-angular depending on the number of sides of the base. From the construction of the truncated pyramid it follows that it has two bases: upper and lower. The bases of the truncated pyramid are two polygons, the sides of which are parallel in pairs. The side faces of the truncated pyramid are trapeziums.
Height a truncated pyramid is called a perpendicular segment drawn from any point of the upper base to the plane of the lower one.
Regular truncated pyramid is called the part of a regular pyramid, enclosed between the base and the section plane parallel to the base. The height of the side face of a regular truncated pyramid (trapezoid) is called apothem.
It can be proved that a regular truncated pyramid has congruent lateral edges, all lateral edges are congruent, and all apothems are congruent.
If in the correct truncated n-angled pyramid through a and b n designate the lengths of the sides of the upper and lower bases, and through h is the length of the apothem, then the area of each side face of the pyramid is
1 / 2 (a + b n) h
The sum of the areas of all the lateral faces of the pyramid is called the area of its lateral surface and is denoted S side. ... Obviously, for a correct truncated n- angle pyramid
S side. = n 1 / 2 (a + b n) h.
Because na= P and nb n= Р 1 - perimeters of the bases of the truncated pyramid, then
S side. = 1/2 (P + P 1) h,
that is, the lateral surface area of a regular truncated pyramid is equal to half the product of the sum of the perimeters of its bases by the apothem.
Section parallel to the base of the pyramid
Theorem. If the pyramid is crossed by a plane parallel to the base, then:1) side ribs and height are divided into proportional parts;
2) in the section, you get a polygon similar to the base;
3) the cross-sectional and base areas are related as the squares of their distances from the top.
It suffices to prove the theorem for a triangular pyramid.
Since the parallel planes are intersected by the third plane along parallel lines, then (AB) || (A 1 B 1), (BC) || (B 1 C 1), (AC) || (A 1 C 1) (fig.).
Parallel straight lines cut the sides of the corner into proportional parts, and therefore
$$ \ frac (\ left | (SA) \ right |) (\ left | (SA_1) \ right |) = \ frac (\ left | (SB) \ right |) (\ left | (SB_1) \ right | ) = \ frac (\ left | (SC) \ right |) (\ left | (SC_1) \ right |) $$
Therefore, ΔSAB ~ ΔSA 1 B 1 and
$$ \ frac (\ left | (AB) \ right |) (\ left | (A_ (1) B_1) \ right |) = \ frac (\ left | (SB) \ right |) (\ left | (SB_1 ) \ right |) $$
ΔSBC ~ ΔSB 1 C 1 and
$$ \ frac (\ left | (BC) \ right |) (\ left | (B_ (1) C_1) \ right |) = \ frac (\ left | (SB) \ right |) (\ left | (SB_1 ) \ right |) = \ frac (\ left | (SC) \ right |) (\ left | (SC_1) \ right |) $$
Thus,
$$ \ frac (\ left | (AB) \ right |) (\ left | (A_ (1) B_1) \ right |) = \ frac (\ left | (BC) \ right |) (\ left | (B_ (1) C_1) \ right |) = \ frac (\ left | (AC) \ right |) (\ left | (A_ (1) C_1) \ right |) $$
The corresponding angles of triangles ABC and A 1 B 1 C 1 are congruent, like angles with parallel and equally directed sides. That's why
ΔABC ~ ΔA 1 B 1 C 1
The areas of such triangles are referred to as the squares of the corresponding sides:
$$ \ frac (S_ (ABC)) (S_ (A_1 B_1 C_1)) = \ frac (\ left | (AB) \ right | ^ 2) (\ left | (A_ (1) B_1) \ right | ^ 2 ) $$
$$ \ frac (\ left | (AB) \ right |) (\ left | (A_ (1) B_1) \ right |) = \ frac (\ left | (SH) \ right |) (\ left | (SH_1 ) \ right |) $$
Hence,
$$ \ frac (S_ (ABC)) (S_ (A_1 B_1 C_1)) = \ frac (\ left | (SH) \ right | ^ 2) (\ left | (SH_1) \ right | ^ 2) $$
Theorem. If two pyramids with equal heights are dissected at the same distance from the top by planes parallel to the bases, then the cross-sectional areas are proportional to the areas of the bases.
Let (Fig. 84) B and B 1 - the area of the bases of two pyramids, H - the height of each of them, b and b 1 - cross-sectional areas by planes parallel to the bases and removed from the vertices by the same distance h.
According to the previous theorem, we will have:
$$ \ frac (b) (B) = \ frac (h ^ 2) (H ^ 2) \: and \: \ frac (b_1) (B_1) = \ frac (h ^ 2) (H ^ 2) $ $
where
$$ \ frac (b) (B) = \ frac (b_1) (B_1) \: or \: \ frac (b) (b_1) = \ frac (B) (B_1) $$
Consequence. If B = B 1, then b = b 1, i.e. if two pyramids with equal base heights are of equal size, then equal sizes and sections equidistant from the top.
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Introduction
When we meet the word "pyramid", the associative memory takes us to Egypt. If we talk about the early monuments of architecture, then it can be argued that their number is not less than several hundred. An Arab writer of the 13th century said: "Everything in the world is afraid of time, and time is afraid of the pyramids." The pyramids are the only miracle of the seven wonders of the world that has survived to our time, to the era computer technology... However, researchers have still not been able to find clues to all of their mysteries. The more we learn about the pyramids, the more questions we have. The pyramids are of interest to historians, physicists, biologists, physicians, philosophers, etc. They are of great interest and encourage a deeper study of their properties from both mathematical and other points of view (historical, geographical, etc.).
That's why aim our research was the study of the properties of the pyramid from different points of view. As intermediate goals, we have identified: consideration of the properties of the pyramid from the point of view of mathematics, the study of hypotheses about the existence of secrets and mysteries of the pyramid, as well as the possibilities of its application.
Object research in this paper is a pyramid.
Item research: features and properties of the pyramid.
Tasks research:
Study popular science literature on the research topic.
Consider the pyramid as a geometric body.
Determine the properties and features of the pyramid.
Find material confirming the use of the properties of the pyramid in different areas science and technology.
Methods research: analysis, synthesis, analogy, mental modeling.
The expected result of the work there should be structured information about the pyramid, its properties and application possibilities.
Stages of project preparation:
Determination of the project theme, goals and objectives.
Studying and collecting material.
Drawing up a project plan.
Formulation of the expected result of activities on the project, including the assimilation of new material, the formation of knowledge, skills and abilities in substantive activity.
Registration of research results.
Reflection
Pyramid as a geometric body
Consider the origins of the word and term " pyramid". It should be noted right away that the "pyramid" or " pyramid "(English), " piramide "(French, Spanish and Slavic languages), "Pyramide"(German) is a Western term with its origins in ancient Greece. In ancient greek πύραμίς ("NS iramis"And many others. h. Πύραμίδες « pyramides") Has several meanings. The ancient Greeks called “ pyramis»A wheat cake that resembled the shape of Egyptian structures. Later, this word came to mean “a monumental structure with a square area at the base and sloping sides meeting at the top. Etymological Dictionary indicates that the Greek "pyramis" comes from the Egyptian " pimar ". The first written interpretation of the word "pyramid" found in Europe in 1555 and means: "one of the types of ancient buildings of kings." After the discovery of the pyramids in Mexico and with the development of sciences in the 18th century, the pyramid became not only an ancient architectural monument, but also a regular geometric figure with four symmetrical sides (1716). The beginning of the geometry of the pyramid was laid in Ancient Egypt and Babylon, however active development received in Ancient Greece... The first to establish what the volume of the pyramid is, was Democritus, and Eudoxus of Cnidus proved it.
The first definition belongs to the ancient Greek mathematician, the author of theoretical treatises on mathematics that have come down to us, Euclid. In the XII volume of his "Principles" he defines a pyramid as a bodily figure bounded by planes that converge from one plane (base) at one point (apex). But this definition was criticized already in antiquity. So Heron proposed the following definition of a pyramid: "It is a figure bounded by triangles converging at one point and the base of which is a polygon."
There is a definition French mathematician Adrienne Marie Legendre, who in 1794 in his work "Elements of Geometry" defines the pyramid as follows: "The pyramid is a bodily figure formed by triangles converging at one point and ending on different sides of a flat base."
Modern dictionaries interpret the term "pyramid" as follows:
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A polyhedron whose base is a polygon and the other faces are triangles with a common vertex |
Explanatory dictionary of the Russian language, ed. D. N. Ushakova |
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A body bounded by equal triangles composed of vertices at one point and forming their bases with their gon |
Dahl's Explanatory Dictionary |
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A polyhedron whose base is a polygon and the other faces are triangles with a common vertex |
Explanatory dictionary, ed. S. I. Ozhegova and N. Yu. Shvedova |
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A polyhedron whose base is a polygon and whose side faces are triangles that have a common vertex |
T.F. Efremov. New explanatory and derivational dictionary of the Russian language. |
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A polyhedron, one face of which is a polygon and the other faces are triangles with a common vertex |
Dictionary foreign words |
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A geometric body, the base of which is a polygon, and the sides are as many triangles as the base has sides, converging with vertices to one point. |
Dictionary of foreign words of the Russian language |
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A polyhedron, one face of which is some kind of flat polygon, and all other faces are triangles, the bases of which are the sides of the base of the P., and the vertices converge at one point |
F. Brockhaus, I.A. Efron. encyclopedic Dictionary |
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A polyhedron whose base is a polygon and the other faces are triangles with a common vertex |
Modern explanatory dictionary |
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A polyhedron, one of the faces of which is a polygon and the other faces are triangles with a common vertex |
Mathematical encyclopedic dictionary |
Analyzing the definitions of the pyramid, we can conclude that all sources have similar wording:
A pyramid is a polyhedron, the base of which is a polygon, and the other faces are triangles with a common vertex. According to the number of angles of the base, pyramids are distinguished triangular, quadrangular, etc.
Polygon А 1 А 2 А 3 ... Аn is the base of the pyramid, and the triangles RA 1 А 2, PA 2 А 3, ..., PANА 1 are the side faces of the pyramid, P is the top of the pyramid, the segments RA 1, RA 2, ..., PAN are lateral ribs.
The perpendicular drawn from the top of the pyramid to the plane of the base is called height h pyramids.
In addition to an arbitrary pyramid, there is a regular pyramid, at the base of which is a regular polygon and a truncated pyramid.
Square the total surface of a pyramid is called the sum of the areas of all its faces. S total = S side + S main, where S side is the sum of the areas of the side faces.
Volume pyramid is found by the formula: V = 1 / 3S main h, where S main. - base area, h - height.
TO properties of the pyramid relate:
When all the side edges have the same size, then it is easy to describe a circle near the base of the pyramid, while the top of the pyramid will be projected into the center of this circle; lateral ribs form equal angles with the base plane; moreover, the converse is also true, i.e. when the side ribs form with the base plane equal angles, or when a circle can be described near the base of the pyramid and the top of the pyramid will be projected into the center of this circle, it means that all the side edges of the pyramid have the same size.
When the side faces have an angle of inclination to the base plane of the same magnitude, then it is easy to describe a circle near the base of the pyramid, while the top of the pyramid will be projected into the center of this circle; the heights of the side faces are of equal length; the lateral surface area is equal to half the product of the base perimeter and the lateral edge height.
The pyramid is called correct, if its base is a regular polygon, and the vertex is projected to the center of the base. The side faces of a regular pyramid are equal, isosceles triangles (Fig. 2a). Axis a regular pyramid is called a straight line containing its height. Apothem - the height of the side face of a regular pyramid drawn from its top.
Square the side face of a regular pyramid is expressed as follows: S side. = 1 / 2P h, where P is the perimeter of the base, h is the height of the side face (apothem of the regular pyramid). If the pyramid is intersected by the plane A'B'C'D ', parallel to the base, then the lateral edges and the height are divided by this plane into proportional parts; in the section, a polygon A'B'C'D 'is obtained, similar to the base; the cross-sectional and base areas are referred to as the squares of their distances from the apex.
Truncated pyramid is obtained by cutting off the upper part of the pyramid by a plane parallel to the base (Fig. 2b). The bases of the truncated pyramid are similar polygons ABCD and A`B`C`D`, the side faces are trapeziums. The height of the truncated pyramid is the distance between the bases. The volume of the truncated pyramid is found by the formula: V = 1/3 h (S + + S ’), where S and S’ are the areas of the bases ABCD and A’B’C’D’, h is the height.
The bases of a regular truncated n-gon pyramid are regular n-gons. The lateral surface area of a regular truncated pyramid is expressed as follows: S side. = ½ (P + P ’) h, where P and P’ are the perimeters of the bases, h is the height of the side face (apothem of the regular truncated pyramid)
The sections of the pyramid by planes passing through its apex are triangles. The section passing through two non-adjacent side edges of the pyramid is called a diagonal section. If the section passes through a point on the side edge and the side of the base, then this side will be its trace on the plane of the base of the pyramid. A section passing through a point lying on the edge of the pyramid and a given trace of the section on the plane of the base, then the construction should be carried out as follows: find the point of intersection of the plane of this face and the trace of the section of the pyramid and designate it; build a straight line passing through a given point and the resulting intersection point; repeat these steps for the following faces.
Rectangular pyramid - it is a pyramid in which one of the side edges is perpendicular to the base. In this case, this edge will be the height of the pyramid (Figure 2c).
Regular triangular pyramid is a pyramid, the base of which is regular triangle, and the top is projected to the center of the base. A special case of a regular triangular pyramid is tetrahedron... (Figure 2a)
Consider theorems connecting the pyramid with other geometric bodies.
Sphere
A sphere can be described near a pyramid when a polygon lies at the base of the pyramid, around which a circle can be described (a necessary and sufficient condition). The center of the sphere will be the point of intersection of the planes passing through the midpoints of the pyramid's edges perpendicular to them. It follows from this theorem that a sphere can be described both around any triangular and around any regular pyramid; A sphere can be inscribed into a pyramid when the bisector planes of the inner dihedral angles of the pyramid intersect at one point (a necessary and sufficient condition). This point will be the center of the sphere.
Cone
A cone is called inscribed in a pyramid if their tops coincide, and its base is inscribed in the base of the pyramid. Moreover, it is possible to inscribe a cone into a pyramid only when the apothems of the pyramid are equal to each other (a necessary and sufficient condition); A cone is said to be described near the pyramid when their tops coincide, and its base is described near the base of the pyramid. Moreover, it is possible to describe the cone near the pyramid only when all the side edges of the pyramid are equal to each other (a necessary and sufficient condition); The heights of such cones and pyramids are equal to each other.
Cylinder
A cylinder is called inscribed in a pyramid if one of its base coincides with a circle inscribed in the section of the pyramid by a plane parallel to the base, and the other base belongs to the base of the pyramid. A cylinder is said to be described near the pyramid if the top of the pyramid belongs to its one base, and its other base is described near the base of the pyramid. Moreover, it is possible to describe a cylinder near a pyramid only when there is an inscribed polygon at the base of the pyramid (a necessary and sufficient condition).
Very often in their research, scientists use the properties of the pyramid. with proportions of the Golden Ratio... We will consider how the golden ratio ratios were used to construct the pyramids in the next paragraph, and here we will focus on the definition of the golden ratio.
The mathematical encyclopedic dictionary gives the following definition Golden ratio- this is the division of the segment AB into two parts in such a way that most of its AC is the average proportional between the entire segment AB and its smaller part CB.
Algebraically finding the Golden Section of the segment AB = a is reduced to solving the equation a: x = x: (a-x), whence x is approximately equal to 0.62a. The ratio x can be expressed in fractions n / n + 1 = 0,618, where n is the Fibonacci number n.
The golden ratio is often used in works of art, architecture, and occurs in nature. Prominent examples are the sculpture of Apollo Belvedere, the Parthenon. During the construction of the Parthenon, the ratio of the height of the building to its length was used and this ratio is 0.618. The objects around us also provide examples of the Golden Ratio, for example, the bindings of many books also have a ratio of width to length close to 0.618.
Thus, having studied the popular scientific literature on the research problem, we came to the conclusion that a pyramid is a polyhedron, the base of which is a polygon, and the other faces are triangles with a common vertex. We examined the elements and properties of the pyramid, its types and relationship with the proportions of the Golden Section.
2. Features of the pyramid
So in the Big Encyclopedic Dictionary it is written that a pyramid is a monumental structure with a geometric pyramid shape (sometimes stepped or tower-like). The tombs of the ancient Egyptian pharaohs of the 3rd - 2nd millennia BC were called pyramids. e., as well as the pedestals of temples in Central and South America associated with cosmological cults. Among the grandiose pyramids of Egypt, the Great Pyramid of Pharaoh Cheops occupies a special place. Before proceeding to the analysis of the shape and size of the Cheops pyramid, one should recall 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, equal to four "fingers" (16.6 mm).
Most researchers agree that the length of the side of the base of the pyramid, for example, GF is equal to L = 233.16 m. This value corresponds to almost exactly 500 "cubits". Full compliance with 500 "cubits" will be if the length of the "cubit" is considered equal to 0.4663 m.
The height of the pyramid (H) is estimated by researchers differently from 146.6 to 148.2 m. And depending on the accepted height of the pyramid, all ratios of its geometric elements change. What is the reason for the differences in the estimate of the height of the pyramid? The fact is that the Cheops pyramid is truncated. Its top platform today is about 10x10 m, and a century ago it was 6x6 m. Obviously, the top of the pyramid was taken apart, and it does not correspond to the original one. When evaluating the height of the pyramid, it is necessary to take into account such a physical factor as the structural settlement. For a long time, under the influence of colossal pressure (reaching 500 tons per 1 m 2 of the lower surface), the height of the pyramid has decreased compared to its original height. The original height of the pyramid can be recreated if the basic geometric idea is found.
In 1837, English Colonel G. Weisz measured the angle of inclination of the pyramid's faces: 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 AC pyramid to half of its base CB, that is, AC / CB = H / (L / 2) = 2H / L.
And here the researchers were in for a big surprise! The fact is that if we take the square root of the golden ratio, then we get the following result = 1.272. Comparing this value with the value tan a = 1.27306, we see that these values are very close to each other. If we take the angle a = 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 value. It should be noted that in 1840 G. Weis repeated his measurements and specified that the value of the angle a = 51 ° 50 ".
These measurements led the researchers to the following interesting hypothesis: the AC / CB = 1.272 ratio was laid in the basis of the ACB triangle of the Cheops pyramid.
Consider now a right-angled triangle ABC, in which the ratio of legs AC / CB =. If now the lengths of the sides of the rectangle ABC are denoted by x, y, z, and also take into account that the ratio y / x =, then in accordance with the Pythagorean theorem, the length z can be calculated by the formula:
If we take x = 1, y =, then:
A right-angled triangle in which the sides are related as t :: 1 is called a "golden" right-angled triangle.
Then, if we take as a basis the hypothesis that the main "geometric idea" of the Cheops pyramid is the "golden" right-angled triangle, then from here it is easy to calculate the "design" height of the Cheops pyramid. It is equal to:
H = (L / 2) / = 148.28 m.
Let us now deduce some other relations for the Cheops pyramid arising from the "golden" hypothesis. In particular, we find the ratio of the outer area of the pyramid to the area of its base. To do this, let us take the length of the CB leg as one, that is: CB = 1. But then the length of the side of the base of the pyramid is GF = 2, and the area of the base EFGH will be equal to S EFGH = 4.
Let us now calculate the area of the side face of the Cheops pyramid S D. Since the height AB of the triangle AEF is equal to t, the area of the side face will be S D = t. Then the total area of all four side faces of the pyramid will be equal to 4t, and the ratio of the total outer area of the pyramid to the area of the base will be equal to the golden ratio... This is the main geometrical secret of the Cheops pyramid.
And also, during the construction of the Egyptian pyramids, it was found that the square, built at the height of the pyramid, exactly equal to the area each of the side triangles. This is confirmed by the latest measurements.
We know that the ratio between the circumference of a circle and its diameter is a constant, well known to modern mathematicians, schoolchildren - this is the number "Pi" = 3.1416 ... But if we add up the four sides of the base of the Cheops pyramid, we get 931.22 m. this is the number for twice the height of the pyramid (2x148.208), we get 3.1416 ..., that is, the number "Pi". Consequently, the Cheops pyramid is a one-of-a-kind monument that is the material embodiment of the number Pi, which plays an important role in mathematics.
Thus, the presence in the size of the pyramid of the golden section - the ratio of the doubled side of the pyramid to its height - there is a number that is very close in value to the number π. This is undoubtedly a feature too. Although many authors believe that this coincidence is accidental, since the fraction 14/11 is “a good approximation for square root from the ratio of the golden section, and for the ratio of the areas of the square and the circle inscribed in it. "
However, it is wrong to speak here only of the Egyptian pyramids. There are not only Egyptian pyramids, there is a whole network of pyramids on Earth. The main monuments (the Egyptian and Mexican pyramids, Easter Island and the Stonehenge complex in England), at first glance, are haphazardly scattered across our planet. But if the Tibetan pyramid complex is included in the study, then a strict mathematical system of their location on the Earth's surface appears. Against the background of the Himalayan ridge, a pyramidal formation is clearly distinguished - Mount Kailash. The location of the city of Kailash, the Egyptian and Mexican pyramids is very interesting, namely, if you connect the city of Kailash with the Mexican pyramids, then the line connecting them goes to Easter Island. If you connect the city of Kailash with the Egyptian pyramids, then the line of their connection again goes to Easter Island. Outlined exactly one fourth the globe... If we connect the Mexican pyramids and the Egyptian ones, then we will see two equal triangle... If you find their area, then their sum is equal to one fourth of the area of the globe.
Revealed an undeniable connection between the complex of Tibetan pyramids with other structures antiquity - the Egyptian and Mexican pyramids, the colossi of Easter Island and the Stonehenge complex in England. The height of the main pyramid of Tibet - Mount Kailash - is 6714 meters. Distance from Kailash to North Pole equals 6714 kilometers, the distance from Kailash to Stonehenge is 6714 kilometers. If you put off on the globe from the North Pole these 6714 kilometers, then we will get to the so-called Devil's Tower, which looks like a truncated pyramid. And finally, exactly 6714 kilometers from Stonehenge to the Bermuda Triangle.
As a result of these studies, it can be concluded that there is a pyramidal-geographical system on Earth.
Thus, the features include the ratio of the total outer area of the pyramid to the area of the base will be equal to the golden ratio; the presence of the golden section in the size of the pyramid - the ratio of the doubled side of the pyramid to its height - is a number that is very close in value to the number π, i.e. the pyramid of Cheops is a one-of-a-kind monument that represents the material embodiment of the number "Pi"; the existence of a pyramidal-geographical system.
3. Other properties and applications of the pyramid.
Let's consider a practical application of this geometric shape. For example, hologram. First, let's look at what holography is. Holography - a set of technologies for accurate recording, reproduction and reshaping of optical wave fields electromagnetic radiation, a special photographic method in which, using a laser, images of three-dimensional objects that are highly similar to real ones are recorded and then restored. A hologram is a product of holography, a three-dimensional image created using a laser that reproduces an image of a three-dimensional object. With the help of a regular truncated tetrahedral pyramid, you can recreate an image - a hologram. A photo file and a regular truncated tetrahedral pyramid from a translucent material are created. A small indent is made from the bottommost pixel and the middle one relative to the ordinate axis. This point will be the midpoint of the side of the square formed by the section. The photo is multiplied, and its copies are located in the same way relative to the other three sides. A pyramid with a section downward is placed on the square so that it coincides with the square. Monitor generates light wave, each of the four identical photographs, being in the plane that is the projection of the pyramid face, falls on the face itself. As a result, on each of the four faces we have the same images, and since the material from which the pyramid is made has the property of transparency, the waves are refracted, as it were, meeting in the center. As a result, we get the same interference pattern. standing wave, the central axis, or the axis of rotation of which is the height of the regular truncated pyramid. This method also works with video, since the principle of operation remains unchanged.
Considering particular cases, you can see that the pyramid is widely used in everyday life, even in household... The pyramidal shape is often found, primarily in nature: plants, crystals, the methane molecule has the shape of a regular triangular pyramid - a tetrahedron, the unit cell of a diamond crystal is also a tetrahedron, in the center and four vertices of which carbon atoms are located. Pyramids are found at home, children's toys. Buttons, computer keyboards are often similar to a rectangular truncated pyramid. They can be seen in the form of building elements or architectural structures themselves, as translucent roof structures.
Let's consider some more examples of using the term "pyramid"
Ecological pyramids- these are graphic models (usually in the form of triangles), reflecting the number of individuals (pyramid of numbers), the amount of their biomass (pyramid of biomass) or energy contained in them (pyramid of energy) on each trophic level and indicating a decrease in all indicators with an increase in the trophic level
Information pyramid. It reflects a hierarchy of different types of information. The provision of information is built according to the following pyramidal scheme: at the top - the main indicators, by which it is possible to unambiguously track the pace of movement of the enterprise towards the chosen goal. If something is wrong, then you can go to the average level of the pyramid - generalized data. They clarify the picture for each indicator individually or in relation to each other. From this data, you can determine the possible location of the failure or problem. For more complete information you need to refer to the base of the pyramid - detailed description the states of all processes in numerical form. This data helps to identify the cause of the problem so that it can be corrected and avoided in the future.
Bloom's taxonomy. Bloom's taxonomy proposes a pyramid classification of tasks that teachers set for students and, accordingly, learning objectives. She divides educational goals into three spheres: cognitive, affective and psychomotor. Within each separate sphere, in order to move to a higher level, the experience of previous levels, distinguished in this area, is required.
Financial Pyramide- a specific phenomenon economic development... The name “pyramid” clearly illustrates the situation when people “at the bottom” of the pyramid give money to a small top. At the same time, each new participant pays to increase the possibility of his promotion to the top of the pyramid.
Pyramid of needs Maslow reflects one of the most popular and famous theories motivation - hierarchy theory needs. Maslow's needs distributed as it increases, explaining such a construction by the fact that a person cannot experience needs high level, while it needs more primitive things. As the lower needs are satisfied, the needs of a higher level become more and more urgent, but this does not mean at all that the place of the previous need is taken by a new one only when the former is fully satisfied.
Another example of the use of the term "pyramid" is food pyramid - schematic diagram of principles healthy eating developed by nutritionists. The foods that make up the base of the pyramid should be eaten as often as possible, while the foods at the top of the pyramid should be avoided or consumed in limited quantities.
Thus, all of the above shows the variety of uses of the pyramid in our life. Perhaps the pyramid has much more high purpose, and is intended for something more than the practical ways of using it that are now open.
Conclusion
We constantly meet with pyramids in our life - these are ancient Egyptian pyramids and toys that children play with; objects of architecture and design, natural crystals; viruses that can only be seen under an electron microscope. Over the many millennia of their existence, the pyramids have turned into a kind of symbol that personifies the desire of man to reach the pinnacle of knowledge.
In the course of our research, we determined that pyramids are quite common all over the globe.
We studied the popular scientific literature on the research topic, considered various interpretations of the term "pyramid", determined that in the geometrical sense, a pyramid is a polyhedron, the base of which is a polygon, and the other faces are triangles with a common vertex. We studied the types of pyramids (regular, truncated, rectangular), elements (apothem, side faces, side edges, top, height, base, diagonal section) and the properties of geometric pyramids when the side edges are equal and when the side faces are inclined to the base plane at one angle. Considered theorems connecting the pyramid with other geometric bodies (sphere, cone, cylinder).
We attributed to the features of the pyramid:
the ratio of the total outer area of the pyramid to the area of the base will be equal to the golden ratio;
the presence of the golden section in the size of the pyramid - the ratio of the doubled side of the pyramid to its height - is a number that is very close in value to the number π, i.e. the pyramid of Cheops is a one-of-a-kind monument that represents the material embodiment of the number "Pi";
the existence of a pyramidal-geographical system.
We have explored the modern use of this geometric shape. We examined how the pyramid and the hologram are connected, drew attention to the fact that the pyramidal shape is most often found in nature (plants, crystals, methane molecules, the structure of the diamond lattice, etc.). Throughout the research, we met with material confirming the use of the properties of the pyramid in various fields of science and technology, in the everyday life of people, in the analysis of information, in economics and in many other areas. And they came to the conclusion that perhaps the pyramids have a much higher purpose, and are intended for something more than the practical ways of using them that are now open.
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