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, to explain 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 pyramid shape 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). 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 for "pyramid" comes from the Egyptian expression per-em-us, that is, from the 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. Having studied 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 the triangles PA1A2, PA2A3, ..., PANA1 are the lateral faces of the pyramid, P is the top of the pyramid, the segments PA1, PA2, ..., PAN 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: "A pyramid is a solid figure formed by triangles converging at one point and ending on different sides of a flat base."
It seems to us that the last definition gives a clear idea of the pyramid, since it refers to the fact 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."
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 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;
https://pandia.ru/text/78/390/images/image017_1.png "width =" 287 "height =" 151 ">
Truncated pyramid bases- similar polygons ABCD and A`B`C`D`, side faces - trapeziums.
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 passing through a given 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 the 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 is difficult to find a better example to illustrate the Pythagorean theorem, which was known to the Egyptians long before its discovery by Pythagoras.
Thus, the ingenious creators of the Egyptian pyramids sought to amaze distant descendants with the depth of their knowledge, and they achieved this by choosing the "golden" right triangle for the Cheops pyramid, and the "sacred" or "Egyptian" one 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. 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 equal to 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 puzzles, Egyptologists learned how the ancient Egyptians dealt with the varying amounts that came up when calculating the measures of weight, length, and volume, in which fractions were often used, and how they dealt with angles.
The ancient Egyptians used a method to calculate angles based on the ratio of the height to the base of a right triangle. They expressed any angle in the language of the gradient. The gradient of the slope was expressed by an integer ratio called "seked". In the 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. Hence, the Egyptian word "seked" is akin 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 the 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 mathematical problems related to pyramids are always solved on the basis of an angle seked - the ratio of height to 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 for each pyramid were chosen arbitrarily. 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. This 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 occupied 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 "The 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 almost exactly to 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 such a physical factor as the "draft" of the structure. For a long time, under the influence of colossal pressure (reaching 500 tons per 1 m2 of the lower surface), the height of the pyramid has decreased compared to its 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 the relationship between different dimensions in the pyramid.
As a rule, they are obtained in search of some "constants", in particular, the number "pi" (Ludolph's number), equal to 3.14159 ...; the base of natural logarithms "e" (Napier's number), equal to 2.71828 ...; the number "F", the number of the "golden section", equal, for example, 0.618 ... and so on.
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 "F"; 3) Property of M. Eyst: Base perimeter: 2 Height = "Pi"; in a different interpretation - 2 tbsp. main : Height = "Pi"; 4) Property of G. Ribs: Radius of the inscribed circle: 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 the "Properties of A. Arefiev", one can mention that the difference between the volumes of the Cheops pyramid and the Khafre 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 set forth 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 in 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, 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 of a solar year, expressed with an accuracy of one billionth day" - 365.540.903.777.
The statement of P. Smith: "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 can one explain that the masses of the pyramids of Cheops, Khephren and Mikerin 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 the founder of the III dynasty, Pharaoh Djoser (Zoser), was faced with the task of strengthening the unity of the country.
And here, according to historians, an important role in strengthening the central government was played by the "new concept of deification" of the tsar. 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 hill of small stones was poured, 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. It was as if 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 triangular or, say, octahedral? An indirect answer is given by the fact that almost all pyramids are perfectly oriented along the four cardinal directions, therefore they 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 faces? 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 subject 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", an angle that was 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. "As we will see, a number of provisions even make one think about the fact that the famous pyramid of Cheops has also" dried up. "In any case, in all ancient images, the pyramids are pointed ...
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. There is 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 the religion of 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. "Big disk of bright gold" - this is how the Egyptians called our daylight. The Egyptians knew gold perfectly, 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, South Africa is the main supplier of diamonds, 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 a reason for the contacts of the ancient Egyptians with this region. However, one way or another, it is possible that it was 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 pyramid shape.
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 of the Great Pyramid of Cheops", M: "Tsentropoligraf", 2005
Internet resources
http: // veka-i-mig. ***** /
http: // tambov. ***** / vjpusk / vjp025 / rabot / 33 / index2.htm
http: // www. ***** / enc / 54373.html
Definition. Side edge is a triangle, one corner of which lies at the top of the pyramid, and the opposite side coincides with the side of the base (polygon).
Definition. Side ribs are the common sides of the side faces. The pyramid has as many edges as the corners of the polygon.
Definition. Pyramid height- this is a perpendicular, lowered from the top to the base of the pyramid.
Definition. Apothem is the perpendicular to the side face of the pyramid, lowered from the top of the pyramid to the side of the base.
Definition. Diagonal section is a section of the pyramid by a plane passing through the top of the pyramid and the diagonal of the base.
Definition. Correct pyramid is a pyramid in which the base is a regular polygon, and the height drops to the center of the base.
Volume and surface area of the pyramid
Formula. The volume of the pyramid through the base area and height:
Pyramid properties
If all side edges are equal, then a circle can be described around the base of the pyramid, and the center of the base coincides with the center of the circle. Also, the perpendicular dropped from the top passes through the center of the base (circle).
If all side edges are equal, then they are inclined to the plane of the base at the same angles.
The side edges are equal when they form equal angles with the base plane or if a circle can be described around the base of the pyramid.
If the side faces are inclined to the base plane at one angle, then a circle can be inscribed into the base of the pyramid, and the top of the pyramid is projected into its center.
If the side faces are inclined to the base plane at the same angle, then the apothems of the side faces are equal.
Properties of a regular pyramid
1. The top of the pyramid is equidistant from all corners of the base.
2. All side edges are equal.
3. All side ribs slope at the same angle to the base.
4. The apothems of all lateral faces are equal.
5. The areas of all side faces are equal.
6. All faces have the same dihedral (flat) angles.
7. A sphere can be described around the pyramid. The center of the circumscribed sphere will be the point of intersection of the perpendiculars that pass through the middle of the edges.
8. A sphere can be inscribed in the pyramid. The center of the inscribed sphere will be the intersection point of the bisectors emanating from the angle between the edge and the base.
9. If the center of the inscribed sphere coincides with the center of the circumscribed sphere, then the sum of the flat angles at the vertex is equal to π or vice versa, one angle is equal to π / n, where n is the number of angles at the base of the pyramid.
The connection of the pyramid with the sphere
A sphere can be described around the pyramid when a polyhedron 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 perpendicularly through the midpoints of the side edges of the pyramid.
A sphere can always be described around any triangular or regular pyramid.
A sphere can be inscribed into a pyramid if the bisector planes of the inner dihedral corners of the pyramid intersect at one point (a necessary and sufficient condition). This point will be the center of the sphere.
Connection of a pyramid with a cone
A cone is called inscribed in a pyramid if their tops coincide and the base of the cone is inscribed in the base of the pyramid.
A cone can be inscribed into a pyramid if the apothems of the pyramid are equal to each other.
A cone is called circumscribed around a pyramid if their tops coincide, and the base of the cone is circumscribed around the base of the pyramid.
A cone can be described around a pyramid if all the side edges of the pyramid are equal to each other.
Connection of a pyramid with a cylinder
A pyramid is called inscribed in a cylinder if the top of the pyramid lies on one base of the cylinder, and the base of the pyramid is inscribed in another base of the cylinder.
A cylinder can be described around a pyramid if a circle can be described around the base of the pyramid.
A tetrahedron has four faces and four vertices and six edges, where any two edges do not have common vertices but do not touch.
Each vertex consists of three faces and edges that form triangular corner.
The segment connecting the vertex of the tetrahedron with the center of the opposite face is called median tetrahedron(GM).
Bimedian is the segment connecting the midpoints of opposite edges that are not in contact (KL).
All bimedians and medians of the tetrahedron meet at one point (S). In this case, the bimedians are divided in half, and the medians in the ratio of 3: 1, starting from the top.
Definition. Acute-angled pyramid- this is a pyramid in which the apothem is more than half the length of the side of the base.
Definition. Obtuse pyramid- this is a pyramid in which the apothem is less than half the length of the side of the base.
Definition. Regular tetrahedron- a tetrahedron in which all four faces are equilateral triangles. It is one of five regular polygons. In a regular tetrahedron, all dihedral angles (between faces) and trihedral angles (at the vertex) are equal.
Definition. Rectangular tetrahedron is called a tetrahedron with a right angle between three edges at the vertex (the edges are perpendicular). Three faces form rectangular triangular corner and the faces are right-angled triangles, and the base is an arbitrary triangle. The apothem of any facet is equal to half of the side of the base on which the apothem falls.
Definition. Isohedral tetrahedron called a tetrahedron in which the side faces are equal to each other, and the base is a regular triangle. For such a tetrahedron, the faces are isosceles triangles.
Definition. Orthocentric tetrahedron is called a tetrahedron in which all the heights (perpendiculars) that are lowered from the top to the opposite face intersect at one point.
Definition. Star pyramid is called a polyhedron whose base is a star.
This video tutorial will help users get an idea of the Pyramid theme. Correct pyramid. In this lesson we will get acquainted with the concept of a pyramid, we will give it a definition. Let's consider what a regular pyramid is and what properties it has. Then we prove the theorem on the lateral surface of a regular pyramid.
In this lesson we will get acquainted with the concept of a pyramid, we will give it a definition.
Consider a polygon A 1 A 2...A n, which lies in the plane α, and the point P, which does not lie in the plane α (Fig. 1). Let's connect the point P with peaks A 1, A 2, A 3, … A n... We get n triangles: A 1 A 2 R, A 2 A 3 R etc.
Definition... Polyhedron RA 1 A 2 ... A n composed of n-gonal A 1 A 2...A n and n triangles RA 1 A 2, RA 2 A 3 …PA n А n-1 is called n-gonal pyramid. Rice. 1.
Rice. 1
Consider a quadrangular pyramid PABCD(fig. 2).
R- the top of the pyramid.
ABCD- the base of the pyramid.
RA- lateral rib.
AB- the edge of the base.
From point R omit the perpendicular NS on the plane of the base ABCD... The drawn perpendicular is the height of the pyramid.
Rice. 2
The full surface of the pyramid consists of the lateral surface, that is, the area of all lateral faces, and the base area:
S full = S side + S main
A pyramid is called correct if:
- its base is a regular polygon;
- the line segment connecting the top of the pyramid with the center of the base is its height.
Explanation on the example of a regular quadrangular pyramid
Consider a regular quadrangular pyramid PABCD(fig. 3).
R- the top of the pyramid. Base of the pyramid ABCD- a regular quadrangle, that is, a square. Point O, the intersection point of the diagonals, is the center of the square. Means, RO is the height of the pyramid.
Rice. 3
Explanation: in the correct n-gon, the center of the inscribed circle and the center of the circumcircle coincide. This center is called the center of the polygon. It is sometimes said that the top is projected to the center.
The height of the side face of a regular pyramid drawn from its top is called apothem and denoted h a.
1. all lateral edges of a regular pyramid are equal;
2. the side faces are equal isosceles triangles.
The proof of these properties is given by the example of a regular quadrangular pyramid.
Given: PABSD- regular quadrangular pyramid,
ABCD- square,
RO- the height of the pyramid.
Prove:
1. PA = PB = PC = PD
2.∆АВР = ∆ВCP = ∆СDP = ∆DAP See Fig. 4.
Rice. 4
Proof.
RO- the height of the pyramid. That is, straight RO perpendicular to the plane ABC and hence direct AO, VO, SO and DO lying in it. So the triangles ROA, ROV, ROS, POD- rectangular.
Consider a square ABCD... It follows from the properties of the square that AO = BO = CO = DO.
Then right triangles have ROA, ROV, ROS, POD leg RO- general and legs AO, VO, SO and DO are equal, which means that these triangles are equal in two legs. The equality of the triangles implies the equality of the segments, PA = PB = PC = PD. Item 1 is proved.
Segments AB and Sun are equal, since they are sides of one square, PA = PB = RS... So the triangles ABP and HRV - isosceles and equal on three sides.
Similarly, we find that the triangles ATS, BCP, CDP, DAP are isosceles and equal, as required to prove in paragraph 2.
The lateral surface area of a regular pyramid is equal to half the product of the base perimeter times the apothem:
For the proof, we choose a regular triangular pyramid.
Given: RAVS- regular triangular pyramid.
AB = BC = AC.
RO- height.
Prove: 
... See Fig. 5.
Rice. 5
Proof.
RAVS- regular triangular pyramid. That is AB= AC = BC... Let be O- the center of the triangle ABC, then RO is the height of the pyramid. At the base of the pyramid lies an equilateral triangle ABC... notice, that 
.
Triangles RAV, RVS, RSA- equal isosceles triangles (by property). The triangular pyramid has three side faces: RAV, RVS, RSA... This means that the area of the side surface of the pyramid is equal to:
S side = 3S RAV
The theorem is proved.
The radius of a circle inscribed at the base of a regular quadrangular pyramid is 3 m, the height of the pyramid is 4 m. Find the area of the side surface of the pyramid.
Given: regular quadrangular pyramid ABCD,
ABCD- square,
r= 3 m,
RO- the height of the pyramid,
RO= 4 m.
Find: S side. See Fig. 6.
Rice. 6
Solution.
By the proved theorem,.
Let's find the side of the base first AB... We know that the radius of a circle inscribed at the base of a regular quadrangular pyramid is 3 m.
Then, m.
Find the perimeter of the square ABCD with a side of 6 m:
Consider a triangle BCD... Let be M- middle of the side DC... Because O- middle BD, then 
(m).
Triangle DPC- isosceles. M- middle DC... That is, RM- the median, and hence the height in the triangle DPC... Then RM- the apothem of the pyramid.
RO- the height of the pyramid. Then, straight RO perpendicular to the plane ABC, and hence the straight line OM lying in it. Find apothem RM from a right triangle ROM.
Now we can find the side surface of the pyramid:
Answer: 60 m 2.
The radius of a circle circumscribed about the base of a regular triangular pyramid is m. The lateral surface area is 18 m 2. Find the length of the apothem.
Given: ABCP- regular triangular pyramid,
AB = BC = CA,
R= m,
S side = 18 m 2.
Find:. See Fig. 7.
Rice. 7
Solution.
In a regular triangle ABC the radius of the circumscribed circle is given. Let's find a side AB this triangle using the sine theorem.
Knowing the side of a regular triangle (m), we find its perimeter.
By the theorem on the lateral surface area of a regular pyramid, where h a- the apothem of the pyramid. Then:
Answer: 4 m.
So, we examined what a pyramid is, what a regular pyramid is, and proved the theorem on the lateral surface of a regular pyramid. In the next lesson, we will get acquainted with the truncated pyramid.
Bibliography
- Geometry. Grade 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th ed., Rev. and add. - M .: Mnemosina, 2008 .-- 288 p .: ill.
- Geometry. Grade 10-11: Textbook for general educational institutions / Sharygin I.F. - M .: Bustard, 1999. - 208 p .: ill.
- Geometry. Grade 10: Textbook for educational institutions with in-depth and specialized study of mathematics / E. V. Potoskuev, L. I. Zvalich. - 6th ed., Stereotype. - M .: Bustard, 008 .-- 233 p .: ill.
- Internet portal "Yaklass" ()
- Internet portal "Festival of pedagogical ideas" September 1st "()
- Internet portal "Slideshare.net" ()
Homework
- Can a regular polygon be the base of an irregular pyramid?
- Prove that disjoint edges of a regular pyramid are perpendicular.
- Find the value of the dihedral angle at the side of the base of a regular quadrangular pyramid if the apothem of the pyramid is equal to the side of its base.
- RAVS- regular triangular pyramid. Construct the linear angle of the dihedral at the base of the pyramid.
Here you can find basic information about the pyramids and related formulas and concepts. All of them are studied with a mathematics tutor in preparation for the exam.
Consider a plane, a polygon 
lying in it and a point S not lying in it. Connect S to all the vertices of the polygon. The resulting polyhedron is called a pyramid. The line segments are called side ribs. 
The polygon is called the base, and the point S is called the top of the pyramid. Depending on the number n, the pyramid is called triangular (n = 3), quadrangular (n = 4), ptyagonal (n = 5), and so on. An alternative name for the triangular pyramid is tetrahedron... The height of the pyramid is called the perpendicular, lowered from its top to the plane of the base. 
A pyramid is called correct if a regular polygon, and the base of the height of the pyramid (base of the perpendicular) is its center.
Tutor comment:
Do not confuse the concept of "regular pyramid" and "correct tetrahedron". In a regular pyramid, the side edges are not necessarily equal to the edges of the base, but in a regular tetrahedron, all 6 edges of the edges are equal. This is his definition. It is easy to prove that the equality implies the coincidence of the center P of the polygon with the base of the height, so a regular tetrahedron is a regular pyramid.
What is Apothema?
The apothem of a pyramid is the height of its lateral face. If the pyramid is correct, then all its apothems are equal. The converse is not true. 
Tutor in mathematics about his terminology: work with pyramids is 80% built through two types of triangles:
1) Containing apothem SK and height SP
2) Containing a lateral edge SA and its projection PA 
To simplify references to these triangles, it is more convenient for a math tutor to call the first of them apothemic, and second costal... Unfortunately, you will not find this terminology in any of the textbooks, and the teacher has to enter it unilaterally.
The formula for the volume of a pyramid:
1) 
, where is the area of the base of the pyramid, and is the height of the pyramid
2), where is the radius of the inscribed sphere, and is the area of the full surface of the pyramid.
3) 
, where MN is the distance of any two crossing edges, and is the area of the parallelogram formed by the midpoints of the four remaining edges.
Pyramid height base property:
Point P (see figure) coincides with the center of the inscribed circle at the base of the pyramid if one of the following conditions is met:
1) All apothems are equal
2) All side faces are equally inclined towards the base
3) All apothems are equally inclined to the height of the pyramid
4) The height of the pyramid is equally inclined to all side faces
Math Tutor Commentary: Note that all points have one common property: one way or another, side faces are involved everywhere (apothems are their elements). Therefore, the tutor may offer a less accurate, but more convenient for memorization formulation: the point P coincides with the center of the inscribed circle at the base of the pyramid, if there is any equal information about its lateral faces. To prove it, it suffices to show that all apothemic triangles are equal.
Point P coincides with the center of a circle described near the base of the pyramid, if one of three conditions is true:
1) All side edges are equal
2) All side ribs are equally inclined towards the base
3) All side ribs are equally inclined to height
When solving problem C2 by the coordinate method, many students face the same problem. They cannot calculate point coordinates included in the dot product formula. The greatest difficulties are caused pyramids... And if the base points are considered more or less normal, then the tops are a real hell.
Today we will tackle a regular quadrangular pyramid. There is also a triangular pyramid (it is - tetrahedron). This is a more complex construction, so a separate lesson will be devoted to it.
First, let's remember the definition:
A regular pyramid is a pyramid with:
- The base is a regular polygon: triangle, square, etc .;
- The height drawn to the base passes through its center.
In particular, the base of the quadrangular pyramid is square... Just like Cheops, only a little smaller.
Below are the calculations for a pyramid with all edges equal to 1. If this is not the case in your problem, the calculations do not change - the numbers will simply be different.
The tops of the quadrangular pyramid
So, let a regular quadrangular pyramid SABCD be given, where S is a vertex, base ABCD is a square. All edges are equal to 1. It is required to enter a coordinate system and find the coordinates of all points. We have:
We introduce a coordinate system with the origin at point A:
- The OX axis is directed parallel to the AB edge;
- The OY axis is parallel to AD. Since ABCD is a square, AB ⊥ AD;
- Finally, point the OZ axis up, perpendicular to the ABCD plane.
Now we calculate the coordinates. Additional construction: SH - the height drawn to the base. For convenience, let's take the base of the pyramid into a separate drawing. Since points A, B, C and D lie in the plane OXY, their coordinate z = 0. We have:
- A = (0; 0; 0) - coincides with the origin;
- B = (1; 0; 0) - step by 1 along the OX axis from the origin;
- C = (1; 1; 0) - step by 1 along the OX axis and by 1 along the OY axis;
- D = (0; 1; 0) - step only along the OY axis.
- H = (0.5; 0.5; 0) - center of the square, midpoint of segment AC.
It remains to find the coordinates of the point S. Note that the x and y coordinates of points S and H coincide, since they lie on a straight line parallel to the OZ axis. It remains to find the z coordinate for the point S.
Consider triangles ASH and ABH:
- AS = AB = 1 by condition;
- Angle AHS = AHB = 90 °, since SH is the height, and AH ⊥ HB as the diagonals of the square;
- Side AH is common.
Therefore, right-angled triangles ASH and ABH are equal one leg and one hypotenuse. Hence, SH = BH = 0.5 · BD. But BD is the diagonal of a square with side 1. Therefore, we have:
Total coordinates of point S:
In conclusion, let's write out the coordinates of all vertices of a regular rectangular pyramid:
What to do when the ribs are different
But what if the side edges of the pyramid are not equal to the edges of the base? In this case, consider the triangle AHS:
Triangle AHS - rectangular, and the hypotenuse AS is at the same time the lateral edge of the original pyramid SABCD. The AH leg is easily calculated: AH = 0.5 · AC. Find the remaining leg SH by the Pythagorean theorem... This will be the z coordinate for point S.
Task. Given a regular quadrangular pyramid SABCD, at the base of which lies a square with side 1. Side edge BS = 3. Find the coordinates of point S.
We already know the x and y coordinates of this point: x = y = 0.5. This follows from two facts:
- The projection of the S point onto the OXY plane is the H point;
- At the same time, point H is the center of the square ABCD, all sides of which are equal to 1.
It remains to find the coordinate of point S. Consider the triangle AHS. It is rectangular, with the hypotenuse AS = BS = 3, the leg AH - half the diagonal. For further calculations, we need its length:
Pythagorean theorem for triangle AHS: AH 2 + SH 2 = AS 2. We have:
So, the coordinates of point S:
