The formula for the height of a triangular pyramid. Pyramid

Definition

Pyramid is a polyhedron composed of a polygon \(A_1A_2...A_n\) and \(n\) triangles with a common vertex \(P\) (not lying in the plane of the polygon) and opposite sides coinciding with the sides of the polygon.
Designation: \(PA_1A_2...A_n\) .
Example: pentagonal pyramid \(PA_1A_2A_3A_4A_5\) .

Triangles \(PA_1A_2, \ PA_2A_3\) etc. called side faces pyramids, segments \(PA_1, PA_2\), etc. - side ribs, polygon \(A_1A_2A_3A_4A_5\) – basis, point \(P\) – summit.

Height Pyramids are a perpendicular dropped from the top of the pyramid to the plane of the base.

A pyramid with a triangle at its base is called tetrahedron.

The pyramid is called correct, if its base is a regular polygon and one of the following conditions is met:

\((a)\) side edges of the pyramid are equal;

\((b)\) the height of the pyramid passes through the center of the circumscribed circle near the base;

\((c)\) side ribs are inclined to the base plane at the same angle.

\((d)\) side faces are inclined to the base plane at the same angle.

regular tetrahedron is a triangular pyramid, all the faces of which are equal equilateral triangles.

Theorem

The conditions \((a), (b), (c), (d)\) are equivalent.

Proof

Draw the height of the pyramid \(PH\) . Let \(\alpha\) be the plane of the base of the pyramid.

1) Let us prove that \((a)\) implies \((b)\) . Let \(PA_1=PA_2=PA_3=...=PA_n\) .

Because \(PH\perp \alpha\) , then \(PH\) is perpendicular to any line lying in this plane, so the triangles are right-angled. So these triangles are equal in common leg \(PH\) and hypotenuse \(PA_1=PA_2=PA_3=...=PA_n\) . So \(A_1H=A_2H=...=A_nH\) . This means that the points \(A_1, A_2, ..., A_n\) are at the same distance from the point \(H\) , therefore, they lie on the same circle with radius \(A_1H\) . This circle, by definition, is circumscribed about the polygon \(A_1A_2...A_n\) .

2) Let us prove that \((b)\) implies \((c)\) .

\(PA_1H, PA_2H, PA_3H,..., PA_nH\) rectangular and equal in two legs. Hence, their angles are also equal, therefore, \(\angle PA_1H=\angle PA_2H=...=\angle PA_nH\).

3) Let us prove that \((c)\) implies \((a)\) .

Similar to the first point, triangles \(PA_1H, PA_2H, PA_3H,..., PA_nH\) rectangular and along the leg and acute angle. This means that their hypotenuses are also equal, that is, \(PA_1=PA_2=PA_3=...=PA_n\) .

4) Let us prove that \((b)\) implies \((d)\) .

Because in a regular polygon, the centers of the circumscribed and inscribed circles coincide (generally speaking, this point is called the center of a regular polygon), then \(H\) is the center of the inscribed circle. Let's draw perpendiculars from the point \(H\) to the sides of the base: \(HK_1, HK_2\), etc. These are the radii of the inscribed circle (by definition). Then, according to the TTP, (\(PH\) is a perpendicular to the plane, \(HK_1, HK_2\), etc. are projections perpendicular to the sides) oblique \(PK_1, PK_2\), etc. perpendicular to the sides \(A_1A_2, A_2A_3\), etc. respectively. So, by definition \(\angle PK_1H, \angle PK_2H\) equal to the angles between the side faces and the base. Because triangles \(PK_1H, PK_2H, ...\) are equal (as right-angled on two legs), then the angles \(\angle PK_1H, \angle PK_2H, ...\) are equal.

5) Let us prove that \((d)\) implies \((b)\) .

Similarly to the fourth point, the triangles \(PK_1H, PK_2H, ...\) are equal (as rectangular along the leg and acute angle), which means that the segments \(HK_1=HK_2=...=HK_n\) are equal. Hence, by definition, \(H\) is the center of a circle inscribed in the base. But since for regular polygons, the centers of the inscribed and circumscribed circles coincide, then \(H\) is the center of the circumscribed circle. Chtd.

Consequence

The side faces of a regular pyramid are equal isosceles triangles.

Definition

The height of the side face of a regular pyramid, drawn from its top, is called apothema.
The apothems of all lateral faces of a regular pyramid are equal to each other and are also medians and bisectors.

Important Notes

1. The height of a regular triangular pyramid falls to the intersection point of the heights (or bisectors, or medians) of the base (the base is a regular triangle).

2. The height of a regular quadrangular pyramid falls to the point of intersection of the diagonals of the base (the base is a square).

3. The height of a regular hexagonal pyramid falls to the point of intersection of the diagonals of the base (the base is a regular hexagon).

4. The height of the pyramid is perpendicular to any straight line lying at the base.

Definition

The pyramid is called rectangular if one of its lateral edges is perpendicular to the plane of the base.

Important Notes

1. For a rectangular pyramid, the edge perpendicular to the base is the height of the pyramid. That is, \(SR\) is the height.

2. Because \(SR\) perpendicular to any line from the base, then \(\triangle SRM, \triangle SRP\) are right triangles.

3. Triangles \(\triangle SRN, \triangle SRK\) are also rectangular.
That is, any triangle formed by this edge and the diagonal coming out of the vertex of this edge, which lies at the base, will be right-angled.

\[(\Large(\text(Volume and surface area of ​​the pyramid)))\]

Theorem

The volume of a pyramid is equal to one third of the product of the area of ​​the base and the height of the pyramid: \

Consequences

Let \(a\) be the side of the base, \(h\) be the height of the pyramid.

1. The volume of a regular triangular pyramid is \(V_(\text(right triangle pyr.))=\dfrac(\sqrt3)(12)a^2h\),

2. The volume of a regular quadrangular pyramid is \(V_(\text(right.four.pyre.))=\dfrac13a^2h\).

3. The volume of a regular hexagonal pyramid is \(V_(\text(right.hex.pyr.))=\dfrac(\sqrt3)(2)a^2h\).

4. The volume of a regular tetrahedron is \(V_(\text(right tetra.))=\dfrac(\sqrt3)(12)a^3\).

Theorem

The area of ​​the lateral surface of a regular pyramid is equal to half the product of the perimeter of the base and the apothem.

\[(\Large(\text(Truncated pyramid)))\]

Definition

Consider an arbitrary pyramid \(PA_1A_2A_3...A_n\) . Let us draw a plane parallel to the base of the pyramid through a certain point lying on the side edge of the pyramid. This plane will divide the pyramid into two polyhedra, one of which is a pyramid (\(PB_1B_2...B_n\) ), and the other is called truncated pyramid(\(A_1A_2...A_nB_1B_2...B_n\) ).

The truncated pyramid has two bases - polygons \(A_1A_2...A_n\) and \(B_1B_2...B_n\) , which are similar to each other.

The height of a truncated pyramid is a perpendicular drawn from some point of the upper base to the plane of the lower base.

Important Notes

1. All side faces of a truncated pyramid are trapezoids.

2. The segment connecting the centers of the bases of a regular truncated pyramid (that is, a pyramid obtained by a section of a regular pyramid) is the height.

When solving problem C2 using the coordinate method, many students face the same problem. They can't calculate point coordinates included in the scalar product formula. The greatest difficulties are pyramids. And if the base points are considered more or less normal, then the tops are a real hell.

Today we will deal with a regular quadrangular pyramid. There is also a triangular pyramid (aka - tetrahedron). This is a more complex design, so a separate lesson will be devoted to it.

Let's start with the definition:

A regular pyramid is one in which:

1. The base is a regular polygon: triangle, square, etc.;
2. The height drawn to the base passes through its center.

In particular, the base of a 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 - just the numbers will be different.

Vertices of a quadrangular pyramid

So, let a regular quadrangular pyramid SABCD be given, where S is the top, the base of 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:

1. The axis OX is directed parallel to the edge AB ;
2. Axis OY - parallel to AD . Since ABCD is a square, AB ⊥ AD ;
3. Finally, the OZ axis is directed upward, perpendicular to the plane ABCD.

Now we consider the coordinates. Additional construction: SH - height drawn to the base. For convenience, we will take out the base of the pyramid in a separate figure. Since the points A , B , C and D lie in the OXY plane, their coordinate is z = 0. We have:

1. A = (0; 0; 0) - coincides with the origin;
2. B = (1; 0; 0) - step by 1 along the OX axis from the origin;
3. C = (1; 1; 0) - step by 1 along the OX axis and by 1 along the OY axis;
4. D = (0; 1; 0) - step only along the OY axis.
5. H \u003d (0.5; 0.5; 0) - the center of the square, the middle of the segment AC.

It remains to find the coordinates of the point S. Note that the x and y coordinates of the points S and H are the same because 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 :

1. AS = AB = 1 by condition;
2. Angle AHS = AHB = 90° since SH is the height and AH ⊥ HB as the diagonals of a square;
3. Side AH - common.

Therefore right triangles ASH and ABH equal one leg and one hypotenuse. So 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, we write down the coordinates of all the 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 triangle AHS:

Triangle AHS- rectangular, and the hypotenuse AS is also a side edge of the original pyramid SABCD . The leg AH is easily considered: AH = 0.5 AC. Find the remaining leg SH according to the Pythagorean theorem. This will be the z coordinate for point S.

A 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 the point S .

We already know the x and y coordinates of this point: x = y = 0.5. This follows from two facts:

1. The projection of the point S onto the OXY plane is the point H;
2. At the same time, the point H is the center of the square ABCD, all sides of which are equal to 1.

It remains to find the coordinate of the point S. Consider triangle AHS. It is rectangular, with the hypotenuse AS = BS = 3, the leg AH is 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 the point S:

Pyramid Concept

Definition 1

A geometric figure formed by a polygon and a point that does not lie in the plane containing this polygon, connected to all the vertices of the polygon, is called a pyramid (Fig. 1).

The polygon from which the pyramid is composed is called the base of the pyramid, the triangles obtained by connecting with the point are the side faces of the pyramid, the sides of the triangles are the sides of the pyramid, and the point common to all triangles is the top of the pyramid.

Types of pyramids

Depending on the number of corners at the base of the pyramid, it can be called triangular, quadrangular, and so on (Fig. 2).

Figure 2.

Another type of pyramid is a regular pyramid.

Let us introduce and prove the property of a regular pyramid.

Theorem 1

All side faces of a regular pyramid are isosceles triangles that are equal to each other.

Proof.

Consider a regular $n-$gonal pyramid with vertex $S$ of height $h=SO$. Let's describe a circle around the base (Fig. 4).

Figure 4

Consider triangle $SOA$. By the Pythagorean theorem, we get

Obviously, any side edge will be defined in this way. Therefore, all side edges are equal to each other, that is, all side faces are isosceles triangles. Let us prove that they are equal to each other. Since the base is a regular polygon, the bases of all side faces are equal to each other. Consequently, all side faces are equal according to the III sign of equality of triangles.

The theorem has been proven.

We now introduce the following definition related to the concept of a regular pyramid.

Definition 3

The apothem of a regular pyramid is the height of its side face.

Obviously, by Theorem 1, all apothems are equal.

Theorem 2

The lateral surface area of ​​a regular pyramid is defined as the product of the semi-perimeter of the base and the apothem.

Proof.

Let us denote the side of the base of the $n-$coal pyramid as $a$, and the apothem as $d$. Therefore, the area of ​​the side face is equal to

Since, by Theorem 1, all sides are equal, then

The theorem has been proven.

Another type of pyramid is the truncated pyramid.

Definition 4

If a plane parallel to its base is drawn through an ordinary pyramid, then the figure formed between this plane and the plane of the base is called a truncated pyramid (Fig. 5).

Figure 5. Truncated pyramid

The lateral faces of the truncated pyramid are trapezoids.

Theorem 3

The area of ​​the lateral surface of a regular truncated pyramid is defined as the product of the sum of the semiperimeters of the bases and the apothem.

Proof.

Let us denote the sides of the bases of the $n-$coal pyramid by $a\ and\ b$, respectively, and the apothem by $d$. Therefore, the area of ​​the side face is equal to

Since all sides are equal, then

The theorem has been proven.

Task example

Example 1

Find the area of ​​the lateral surface of a truncated triangular pyramid if it is obtained from a regular pyramid with base side 4 and apothem 5 by cutting off by a plane passing through the midline of the lateral faces.

Solution.

According to the median line theorem, we obtain that the upper base of the truncated pyramid is equal to $4\cdot \frac(1)(2)=2$, and the apothem is equal to $5\cdot \frac(1)(2)=2.5$.

Then, by Theorem 3, we obtain

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 form.

Tasks:

1. Give a mathematical definition of a 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 the unique shape of the pyramid be explained mathematically?

3. What explains the geometric wonders of the pyramid?

4. What explains the perfection of the shape of the pyramid?

Definition of a pyramid.

PYRAMID (from Greek pyramis, genus n. pyramidos) - a polyhedron, the base of which is a polygon, and the remaining faces are triangles with a common vertex (figure). According to the number of corners of the base, pyramids are triangular, quadrangular, etc.

PYRAMID - a monumental structure that has the geometric shape of a pyramid (sometimes also stepped or tower-shaped). Giant tombs of the ancient Egyptian pharaohs of the 3rd-2nd millennium BC are called pyramids. 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 that meant the height of the pyramid. The 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. Butuzova and others, we learned that: A polyhedron composed of n-gon A1A2A3 ... An and n triangles RA1A2, RA2A3, ..., RAnA1 is called a pyramid. The polygon A1A2A3 ... An is the base of the pyramid, and the triangles RA1A2, RA2A3, ..., PAnA1 are the lateral faces of the pyramid, P is the top of the pyramid, the segments RA1, RA2, ..., RAn are the lateral edges.

However, such a definition of the 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 solid figure bounded by planes that converge from one plane to one point.

But this definition has been criticized already in antiquity. So Heron proposed the following definition of a pyramid: “This 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 studied these definitions and found the definition of Adrien Marie Legendre, who in 1794 in his work “Elements of Geometry” defines the pyramid as follows: “Pyramid is a bodily 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 ​​\u200b\u200bthe 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 whose faces (base) is a polygon, the remaining faces (sides) are triangles that have one common vertex (the top 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 right pyramid, at the base of which is a regular polygon and truncated pyramid.

In the figure - the pyramid PABCD, ABCD - its base, PO - height.

Full surface area A pyramid is called the sum of the areas of all its faces.

Sfull = Sside + Sbase, where Sside is the sum of the areas of the side faces.

pyramid volume is found according to the formula:

V=1/3Sbase h, where Sosn. - base area h- height.

The axis of a regular pyramid is a straight line containing its height.
Apothem ST - the height of the side face of a regular pyramid.

The area of ​​the side face of a regular pyramid is expressed as follows: Sside. =1/2P h, where P is the perimeter of the base, h- the height of the side face (the apothem of a regular pyramid). If the pyramid is crossed by plane A'B'C'D' parallel to the base, then:

1) side edges 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 from the pyramid its upper part by a plane parallel to the base (figure ABCDD'C'B'A').

The bases of the truncated pyramid are similar polygons ABCD and A`B`C`D`, side faces are trapezoids.

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: Sside. = ½(P+P') h, where P and P’ are the perimeters of the bases, h- the height of the side face (the apothem of a regular truncated by feasts

Sections of the pyramid.

Sections of the pyramid by planes passing through its top 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 plane of the base, then the construction should be carried out as follows:

find the intersection point of the plane of the given face and the trace of the pyramid section 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 triangle 4:3. This ratio of the legs corresponds to the well-known right 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 what is born from both.

For a triangle 3:4:5, the equality is true: 32 + 42 = 52, which expresses the Pythagorean theorem. Is 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 impress distant descendants with the depth of their knowledge, and they achieved this by choosing as the "main geometric idea" for the pyramid of Cheops - the "golden" right-angled triangle, and for the pyramid of Khafre - the "sacred" or "Egyptian" triangle.

Very often, in their research, scientists use the properties of pyramids with the proportions of the Golden Section.

The following definition of the Golden Section is given in the mathematical encyclopedic dictionary - this is a harmonic division, division in the extreme and average ratio - 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.

Algebraic finding of the Golden section of a segment AB = a reduces to solving the equation a: x = x: (a - x), whence x is approximately equal to 0.62a. The x ratio can be expressed as 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 \u003d 1/2 AB is laid on it, A and E are connected, DE \u003d BE is postponed and, finally, AC \u003d AD, then the equality AB is fulfilled: CB = 2: 3.

The golden ratio is often used in works of art, architecture, and is found in nature. Vivid 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. Objects around us also provide examples of the Golden Ratio, for example, the bindings of many books have a width to length ratio close to 0.618. Considering the arrangement of leaves on a common stem of plants, one can notice that between every two pairs of leaves, the third is located in the place of the Golden Ratio (slides). Each of us “wears” the Golden Ratio with us “in our hands” - this is the ratio of the phalanges of the fingers.

Thanks to the discovery of several mathematical papyri, Egyptologists have learned something about the ancient Egyptian systems of calculus and measures. The tasks contained in them were solved by scribes. One of the most famous is the Rhind Mathematical Papyrus. By studying these puzzles, Egyptologists learned how the ancient Egyptians dealt with the various quantities that arose when calculating measures of weight, length, and volume, which often used fractions, as well as how they dealt with angles.

The ancient Egyptians used a method of calculating 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 slope gradient was expressed as a ratio of an integer, called "seked". In 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 a nth number of horizontal units per vertical unit of elevation. Thus, this unit of measure is equivalent to our modern cotangent of the angle of inclination. 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, hence the differences in the angles of inclination for each pyramid. But there could be another reason. Perhaps they all wanted 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 presented by the pyramids in the Rhind 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's say that the length of the hypotenuse 5 was never mentioned. But mathematical problems concerning the pyramids are always solved on the basis of the seked angle - 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 no doubt known to the ancient Egyptians. It is possible that these ratios for each pyramid were chosen arbitrarily. However, this contradicts the importance attached to numerical symbolism in all types of Egyptian fine art. It is very likely that such relationships were of significant importance, since they expressed specific religious ideas. In other words, the whole complex of Giza was subject to a coherent design, designed to reflect some kind of divine theme. This would explain why the designers chose different angles for the three pyramids.

In The Secret of Orion, Bauval and Gilbert presented convincing evidence of the connection of the pyramids of Giza with the constellation of 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 occupied by Great Pyramid of Pharaoh Cheops (Khufu). Before proceeding to the analysis of the shape and size of the pyramid of Cheops, we should remember 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, was equal to four "fingers" (16.6 mm).

Let's analyze the size of the Cheops pyramid (Fig. 2), following the reasoning given in the wonderful book of the Ukrainian scientist Nikolai Vasyutinskiy "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\u003d 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 pyramid of Cheops is truncated. Its upper platform today has a size of approximately 10 ´ 10 m, and a century ago it was 6 ´ 6 m. It is obvious that the top of the pyramid was dismantled, and it does not correspond to the original one.

Estimating 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 decreased compared to its original height.

What was the original height of the pyramid? This height can be recreated if you find the basic "geometric idea" of the pyramid.

Figure 2.

In 1837, the English colonel G. Wise measured the angle of inclination of the faces of the pyramid: 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 pyramid AC to half of its base CB(Fig.2), i.e. 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 tg value a= 1.27306, we see that these values ​​are very close to each other. If we take the angle a\u003d 51 ° 50", that is, to reduce it by only one arc minute, then the value a will become equal to 1.272, that is, it will coincide with the value of . It should be noted that in 1840 G. Wise repeated his measurements and clarified that the value of the angle a=51°50".

These measurements led researchers to the following very interesting hypothesis: the triangle ASV of the pyramid of Cheops was based on the relation AC / CB = = 1,272!

Consider now a right triangle ABC, in which the ratio of legs AC / CB= (Fig.2). If now the lengths of the sides of the rectangle ABC denote 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 accept x = 1, y= https://pandia.ru/text/78/390/images/image027_1.png" width="143" height="27">

Figure 3"Golden" right triangle.

A right triangle in which the sides are related as t:golden" right 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 \u003d (L / 2) ´ \u003d 148.28 m.

Let us now derive some other relations for the pyramid of Cheops, which follow 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, we 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 area of ​​the base EFGH will be equal to SEFGH = 4.

Let us now calculate the area of ​​the side face of the Cheops pyramid SD. Because 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 external area of ​​the pyramid to the base area will be equal to the golden ratio! That's what it is - the main geometric secret of the pyramid of Cheops!

The group of "geometric wonders" of the pyramid of Cheops includes the real and contrived properties of the relationship between the various dimensions in the pyramid.

As a rule, they are obtained in search of some "constant", in particular, the number "pi" (Ludolf number), equal to 3.14159...; bases 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 ... etc..

You can name, for example: 1) Property of Herodotus: (Height) 2 \u003d 0.5 st. main x Apothem; 2) Property of V. Price: Height: 0.5 st. osn \u003d Square root of "Ф"; 3) Property of M. Eist: Perimeter of the base: 2 Height = "Pi"; in a different interpretation - 2 tbsp. main : Height = "Pi"; 4) G. Reber's property: Radius of the inscribed circle: 0.5 st. main = "F"; 5) Property of K. Kleppish: (St. main.) 2: 2 (st. main. x Apothem) \u003d (st. main. W. Apothem) \u003d 2 (st. main. x Apothem) : ((2 st. main X Apothem) + (st. main) 2). Etc. You can come up with a lot of such properties, especially if you connect two adjacent pyramids. For example, as "Properties of A. Arefiev" it can be mentioned that the difference between the volumes of the pyramid of Cheops and the pyramid of Khafre is equal to twice the volume of the pyramid of Menkaure...

Many interesting provisions, in particular, on the construction of pyramids according to the "golden section" are set out in the books of D. Hambidge "Dynamic Symmetry in Architecture" and M. Geek "Aesthetics of Proportion in Nature and Art". Recall that the "golden section" is the division of the segment in such a ratio, when part A is as many times greater 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 section" is indicated not only in individual pyramids, but in the entire pyramid complex 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, you can "adjust" it, but all at once they do not fit - they do not coincide, they contradict each other. Therefore, if, for example, when checking all properties, one and the same side of the base of the pyramid (233 m) is initially taken, 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 those of 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. A "miracle" should be considered only something obviously impossible for the ancient Egyptians. This includes, in particular, "cosmic" miracles, in which the measurements of the Cheops pyramid or the Giza pyramid complex 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" relations.

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 you use the "Egyptian elbow" invented by him, then the side of the pyramid will correspond to "the most accurate duration of the solar year, expressed to the nearest 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 the height of 146.6 m is usually taken, Smith took it as 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.

Last curious statement:

"How to explain that the masses of the pyramids of Cheops, Khafre and Menkaure are related to each other, like the masses of the planets Earth, Venus, Mars?" Let's calculate. The masses of the three pyramids are related as: Khafre - 0.835; Cheops - 1,000; Mikerin - 0.0915. The ratios of the masses of the three planets: Venus - 0.815; Land - 1,000; Mars - 0.108.

So, despite the skepticism, let's note the well-known harmony of the construction of statements: 1) the height of the pyramid, as a line "going 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 earth's 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 bee language, analyzed by Karl von Frisch. However, we refrain from commenting on this for now.

SHAPE OF THE PYRAMIDS

The famous tetrahedral shape of the pyramids did not appear immediately. The Scythians made burials in the form of earthen hills - barrows. The Egyptians built "hills" of stone - pyramids. This happened for the first time after the unification of Upper and Lower Egypt, in the 28th century BC, when the founder of the III dynasty, Pharaoh Djoser (Zoser), faced the task of strengthening the unity of the country.

And here, according to historians, the "new concept of deification" of the tsar played an important role in strengthening the central power. Although the royal burials were distinguished by greater splendor, they did not differ in principle from the tombs of 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 a small building of large stone blocks was then placed - "mastaba" (in Arabic - "bench"). On the site of the mastaba of his predecessor, Sanakht, Pharaoh Djoser erected the first pyramid. It was stepped and was a visible transitional stage from one architectural form to another, from a mastaba to a pyramid.

In this way, the pharaoh was "raised" by the sage and architect Imhotep, who was later considered a magician and identified by the Greeks with the god Asclepius. 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, two steps were formed, as it were.

This situation did not satisfy the architect, and on the top platform of a huge flat mastaba, Imhotep placed three more, gradually decreasing towards the top. The tomb was under the pyramid.

Several more stepped pyramids are known, but later the builders moved on to building more familiar tetrahedral pyramids. Why, however, not triangular or, say, octagonal? An indirect answer is given by the fact that almost all the pyramids are perfectly oriented to the four cardinal points, and therefore have four sides. In addition, 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 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 Hambidge, Geek and others.

What is the advantage of the angle of the semioctahedron? According to the descriptions of archaeologists and historians, some pyramids collapsed under their own weight. What was needed was a "durability angle", an angle that was the most energetically reliable. Purely empirically, this angle can be taken from the vertex angle in a pile of crumbling dry sand. But to get accurate data, you need to use the model. Taking four firmly fixed balls, you need to put the fifth one on them and measure the angles of inclination. However, here you can make a mistake, therefore, 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 a regular semi-octahedron.

However, why do many pyramids, gravitating towards a similar form, nevertheless do not retain it? Probably the pyramids are getting old. Contrary to the famous saying:

"Everything in the world is afraid of time, and time is afraid of the pyramids", the buildings of the pyramids must age, they can and should take place not only the processes of external weathering, but also the 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 chips, 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 mutilated?” asks V. Zamarovsky. “The usual references to the destructive effects of time and “the use of stone for other buildings” do not fit here.

After all, most of its blocks and facing slabs still remain in place, in the ruins at its foot. "As we will see, a number of provisions make one think even that the famous pyramid of Cheops also" shrunken ". In any case, on 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 diamond and gold crystals. A large number of "intersecting" signs for such concepts as Pharaoh, Sun, Gold, Diamond is characteristic. Everywhere - noble, brilliant (brilliant), great, flawless and so on. The similarities are not accidental.

The solar cult, as you know, was an important part of the religion of ancient Egypt. “No matter how we translate the name of the greatest of the pyramids,” one of the modern textbooks says, “Sky Khufu” or “Sky Khufu”, it meant that the king is the sun. If Khufu, in the brilliance of his power, imagined himself to be a second sun, then his son Jedef-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 as "solar metal", gold. "The big disk of bright gold" - so the Egyptians called our daylight. The Egyptians knew gold very well, they knew its native forms, where gold crystals can appear in the form of octahedrons.

As a "sample of forms" the "sun stone" - a 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 the diamond and its properties are quite good. According to some authors, they even used bronze pipes with diamond cutters for drilling.

South Africa is now the main supplier of diamonds, but West 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 paleovisit hypothesis pin many hopes (see below). Diamonds could not be the reason for the contacts of the ancient Egyptians with this region. 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 the pharaohs, “indestructible” like diamond and “brilliant” like gold, the sons of the Sun, comparable only with the most wonderful creations of nature.

Conclusion:

Having studied the pyramid as a geometric body, getting 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 a pyramid. Therefore, the pyramid is truly the most perfect creation of nature and man.

BIBLIOGRAPHY

"Geometry: Proc. for 7 - 9 cells. general education institutions \, etc. - 9th ed. - M .: Education, 1999

History of mathematics at school, M: "Enlightenment", 1982

Geometry grade 10-11, M: "Enlightenment", 2000

Peter Tompkins "Secrets of the Great Pyramid of Cheops", M: "Centropoligraph", 2005

Internet resources

http://veka-i-mig. *****/

http://tambov. *****/vjpusk/vjp025/rabot/33/index2.htm

http://www. *****/enc/54373.html