Definition. Side edge- this is a triangle in which one angle lies at the top of the pyramid, and the opposite side coincides with the side of the base (polygon).
Definition. Side ribs- This common aspects side edges. A pyramid has as many edges as the angles of a polygon.
Definition. Pyramid height- this is a perpendicular lowered from the top to the base of the pyramid.
Definition. Apothem- this is a perpendicular to the side face of the pyramid, lowered from the top of the pyramid to the side of the base.
Definition. Diagonal section- this is a section of a 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 regular polygon, and the height drops to the center of the base.
Volume and surface area of the pyramid
Formula. Volume of the pyramid through base area and height:
Properties of the pyramid
If all the side edges are equal, then a circle can be drawn around the base of the pyramid, and the center of the base coincides with the center of the circle. Also, a perpendicular dropped from the top passes through the center of the base (circle).
If all the side edges are equal, then they are inclined to the plane of the base at the same angles.
The lateral ribs are equal when they form with the plane of the base equal angles or if a circle can be described around the base of the pyramid.
If side faces are inclined to the plane of the base at one angle, then a circle can be inscribed at the base of the pyramid, and the top of the pyramid is projected at its center.
If the side faces are inclined to the plane of the base 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 are inclined at equal angles 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 intersection point of the perpendiculars that pass through the middle of the edges.
8. You can fit a sphere into a pyramid. The center of the inscribed sphere will be the point of intersection 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 plane 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 between the pyramid and the sphere
A sphere can be described around a pyramid when at the base of the pyramid there is a polyhedron around which a circle can be described (necessary and sufficient condition). The center of the sphere will be the intersection point of planes passing perpendicularly through the midpoints of the side edges of the pyramid.
Around any triangular or regular pyramid you can always describe the sphere.
A sphere can be inscribed in a pyramid if the bisector planes of the internal dihedral angles 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 said to be inscribed in a pyramid if their vertices coincide and the base of the cone is inscribed in the base of the pyramid.
A cone can be inscribed in a pyramid if the apothems of the pyramid are equal to each other.
A cone is said to be circumscribed around a pyramid if their vertices 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 lateral edges of the pyramid are equal to each other.
Relationship between a pyramid and 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 angle .
The segment connecting the vertex of a tetrahedron with the center of the opposite face is called median of the tetrahedron(GM).
Bimedian called a segment connecting the midpoints of opposite edges that do not touch (KL).
All bimedians and medians of a tetrahedron intersect at one point (S). In this case, the bimedians are divided in half, and the medians are divided in a ratio of 3:1 starting from the top.
Definition. Acute angled pyramid- this is a pyramid in which the apothem more than a half length of the side of the base.
Definition. Obtuse pyramid- a pyramid in which the apothem is less than half the length of the side of the base.
Definition. Regular tetrahedron- a tetrahedron with all four sides - equilateral triangles. It is one of the five regular polygons. IN regular tetrahedron all dihedral angles (between faces) and trihedral angles (at the vertex) are equal.
Definition. Rectangular tetrahedron is called a tetrahedron in which there is a right angle between three edges at the apex (the edges are perpendicular). Three faces form rectangular triangular angle and the faces are right triangles and the base arbitrary triangle. The apothem of any face is equal to half the side of the base on which the apothem falls.
Definition. Isohedral tetrahedron is called a tetrahedron whose side faces are equal to each other, and the base is a regular triangle. Such a tetrahedron has faces 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 called a polyhedron whose base is a star.
- apothem- the height of the side face of a regular pyramid, which is drawn from its vertex (in addition, the apothem is the length of the perpendicular, which is lowered from the middle of the regular polygon to one of its sides);
- side faces (ASB, BSC, CSD, DSA) - triangles that meet at the vertex;
- lateral ribs ( AS , B.S. , C.S. , D.S. ) — common sides of the side faces;
- top of the pyramid (t. S) - a point that connects the side ribs and which does not lie in the plane of the base;
- height ( SO ) - a perpendicular segment drawn through the top of the pyramid to the plane of its base (the ends of such a segment will be the top of the pyramid and the base of the perpendicular);
- diagonal section pyramids- a section of the pyramid that passes through the top and the diagonal of the base;
- base (ABCD) - a polygon that does not belong to the vertex of the pyramid.
Properties of the pyramid.
1. When all side ribs have the same size, Then:
- it is easy to describe a circle near the base of the pyramid, and the top of the pyramid will be projected into the center of this circle;
- the side ribs form equal angles with the plane of the base;
- Moreover, the opposite is also true, i.e. when the side ribs form equal angles with the plane of the base, or when a circle can be described around the base of the pyramid and the top of the pyramid will be projected into the center of this circle, it means that all the side edges of the pyramid are the same size.
2. When the side faces have an angle of inclination to the plane of the base of the same value, then:
- it is easy to describe a circle near the base of the pyramid, and the top of the pyramid will be projected into the center of this circle;
- the heights of the side faces are equal length;
- the area of the side surface is equal to ½ the product of the perimeter of the base and the height of the side face.
3. A sphere can be described around a pyramid if at the base of the pyramid there is a polygon 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 that pass through the middles of the edges of the pyramid perpendicular to them. From this theorem we conclude that a sphere can be described both around any triangular and around any regular pyramid.
4. A sphere can be inscribed into a pyramid if the bisector planes of the internal dihedral angles of the pyramid intersect at the 1st point (a necessary and sufficient condition). This point will become the center of the sphere.
The simplest pyramid.
Based on the number of angles, the base of the pyramid is divided into triangular, quadrangular, and so on.
There will be a pyramid triangular, quadrangular, and so on, when the base of the pyramid is a triangle, a quadrangle, and so on. A triangular pyramid is a tetrahedron - a tetrahedron. Quadrangular - pentagonal and so on.
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 and give it a definition. Let's consider what a regular pyramid is and what properties it has. Then we prove the theorem about the lateral surface of a regular pyramid.
In this lesson we will get acquainted with the concept of a pyramid and give it a definition.
Consider a polygon A 1 A 2...A n, which lies in the α plane, and the point P, which does not lie in the α plane (Fig. 1). Let's connect the dots 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 and so on.
Definition. Polyhedron RA 1 A 2 ...A n, made up of n-square A 1 A 2...A n And n triangles RA 1 A 2, RA 2 A 3 …RA n A n-1 is called n-coal 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- side rib.
AB- base rib.
From point R let's drop the perpendicular RN to the base plane ABCD. The perpendicular drawn 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 area of the base:
S full = S side + S main
A pyramid is called correct if:
- its base is a regular polygon;
- the segment connecting the top of the pyramid to the center of the base is its height.
Explanation using 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 quadrilateral, that is, a square. Dot ABOUT, the point of intersection of the diagonals, is the center of the square. Means, RO is the height of the pyramid.
Rice. 3
Explanation: in the correct n In a triangle, the center of the inscribed circle and the center of the circumcircle coincide. This center is called the center of the polygon. Sometimes they say that the vertex is projected into the center.
The height of the lateral face of a regular pyramid drawn from its vertex is called apothem and is designated h a.
1. all lateral edges of a regular pyramid are equal;
2. The side faces are equal isosceles triangles.
We will give a proof of these properties using the example of a regular quadrangular pyramid.
Given: PABCD- regular quadrangular pyramid,
ABCD- square,
RO- height of the pyramid.
Prove:
1. RA = PB = RS = PD
2.∆ABP = ∆BCP =∆CDP =∆DAP See Fig. 4.
Rice. 4
Proof.
RO- height of the pyramid. That is, straight RO perpendicular to the plane ABC, and therefore direct JSC, VO, SO And DO lying in it. So triangles ROA, ROV, ROS, ROD- rectangular.
Consider a square ABCD. From the properties of a square it follows that AO = VO = CO = DO.
Then the right triangles ROA, ROV, ROS, ROD leg RO- general and legs JSC, VO, SO And DO are equal, which means that these triangles are equal on two sides. From the equality of triangles follows the equality of segments, RA = PB = RS = PD. Point 1 has been proven.
Segments AB And Sun are equal because they are sides of the same square, RA = PB = RS. So triangles AVR And VSR - isosceles and equal on three sides.
In a similar way we find that triangles ABP, VCP, CDP, DAP are isosceles and equal, as required to be proved in paragraph 2.
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:
To prove this, let’s choose a regular triangular pyramid.
Given: RAVS- correct triangular pyramid.
AB = BC = AC.
RO- height.
Prove: 
. See Fig. 5.
Rice. 5
Proof.
RAVS- regular triangular pyramid. That is AB= AC = BC. Let ABOUT- 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). A triangular pyramid has three side faces: RAV, RVS, RSA. This means that the area of the lateral surface of the pyramid is:
S side = 3S RAW
The theorem has been proven.
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 lateral surface of the pyramid.
Given: regular quadrangular pyramid ABCD,
ABCD- square,
r= 3 m,
RO- height of the pyramid,
RO= 4 m.
Find: S side. See Fig. 6.
Rice. 6
Solution.
According to the proven theorem, .
Let's first find the side of the base 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 M- middle of the side DC. Because ABOUT- middle BD, That 
(m).
Triangle DPC- isosceles. M- middle DC. That is, RM- median, and therefore height in the triangle DPC. Then RM- apothem of the pyramid.
RO- height of the pyramid. Then, straight RO perpendicular to the plane ABC, and therefore direct OM, lying in it. Let's find the apothem RM from a right triangle ROM.
Now we can find lateral surface pyramids:
Answer: 60 m2.
The radius of the circle circumscribed around the base of a regular triangular pyramid is equal to m. The lateral surface area is 18 m 2. Find the length of the apothem.
Given: ABCP- regular triangular pyramid,
AB = BC = SA,
R= m,
S side = 18 m2.
Find: . See Fig. 7.
Rice. 7
Solution.
In a right triangle ABC The radius of the circumscribed circle is given. Let's find a side AB this triangle using the law of sines.
Knowing the side regular triangle(m), let's find its perimeter.
By the theorem on the lateral surface area of a regular pyramid, where h a- apothem of the pyramid. Then:
Answer: 4 m.
So, we looked at what a pyramid is, what a regular pyramid is, and we proved the theorem about the lateral surface of a regular pyramid. In the next lesson we will get acquainted with the truncated pyramid.
Bibliography
- Geometry. Grades 10-11: textbook for students educational institutions(basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th ed., rev. and additional - M.: Mnemosyne, 2008. - 288 p.: ill.
- Geometry. 10-11 grade: Textbook for general education educational institutions/ Sharygin I.F. - M.: Bustard, 1999. - 208 p.: ill.
- Geometry. Grade 10: Textbook for general education institutions with in-depth and specialized study mathematics /E. V. Potoskuev, L. I. Zvalich. - 6th ed., stereotype. - M.: Bustard, 008. - 233 p.: ill.
- Internet portal "Yaklass" ()
- Internet portal "Festival pedagogical ideas"First of September" ()
- 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 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. Build linear angle dihedral angle at the base of the pyramid.
Hypothesis: we believe that the perfection of the pyramid shape is due to mathematical laws, embedded in its form.
Target: having studied the pyramid as geometric body, to explain the perfection of its form.
Tasks:
1. Give mathematical definition pyramid.
2. Study the pyramid as a geometric body.
3. Understand what mathematical knowledge the Egyptians laid it 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 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 a pyramid.
PYRAMID (from Greek pyramis, gen. pyramidos) - a polyhedron whose base is a polygon, and the remaining faces are triangles having a common vertex (drawing). Based on the number of corners of the base, pyramids are classified as triangular, quadrangular, etc.
PYRAMID - a monumental building with geometric shape pyramids (sometimes also stepped or tower-shaped). Pyramids are the name given to the giant tombs of the ancient Egyptian pharaohs of the 3rd-2nd millennium BC. e., as well as ancient American temple pedestals (in Mexico, Guatemala, Honduras, Peru), associated with cosmological cults.
It's possible that Greek word“Pyramid” comes from the Egyptian expression per-em-us, i.e., from a term meaning the height of the pyramid. The outstanding 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 a n-gon A1A2A3 ... An and n triangles PA1A2, PA2A3, ..., PAnA1 is called a pyramid. Polygon A1A2A3...An is the base of the pyramid, and triangles PA1A2, PA2A3,..., PAnA1 are the side faces of the pyramid, P is the top of the pyramid, segments PA1, PA2,..., PAn are the side 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 has come down to us, Euclid, defines a pyramid as a solid figure limited by planes that converge from one plane to one point.
But this definition was criticized already in ancient times. So Heron suggested following definition pyramid: “This is a figure bounded by triangles converging at one point and the base of which is a polygon.”
Our group, having compared 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 a pyramid as follows: “A pyramid is a solid figure formed by triangles converging at one point and ending at different sides flat base."
It seems to us that the last definition gives a clear idea of the pyramid, since it we're talking about 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 vertex 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 the arbitrary pyramid, there are correct pyramid at the base of which is a regular polygon and truncated pyramid.
In the figure there is a pyramid PABCD, ABCD is its base, PO is its height.
Area full surface pyramid is the sum of the areas of all its faces.
Sfull = Sside + Smain, Where Side– the sum of the areas of the side faces.
Volume of the pyramid is found by the formula:
V=1/3Sbas. h, where Sbas. - base area, h- height.
 |
|
Apothem ST is the height of the side face of a regular pyramid.
The area of the lateral face of a regular pyramid is expressed as follows: Sside. =1/2P h, where P is the perimeter of the base, h- height of the side face (apothem of a regular pyramid). If the pyramid is intersected by the plane A’B’C’D’, parallel to the base, then:
1) the side ribs and height are divided by this plane into proportional parts;
2) in cross-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">
Bases of a truncated pyramid – 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- height of the side face (apothem of a regular truncated pirami
Sections of a pyramid.
Sections of a pyramid by planes passing through its apex are triangles.
A section passing through two non-adjacent lateral edges of a pyramid is called diagonal section.
If the section passes through a point on the side edge and the side of the base, then its trace to the plane of the base of the pyramid will be this side.
A section passing through a point lying on the face of the pyramid and a given section trace on the base plane, then the construction should be carried out as follows:
· find the point of intersection of the plane of a given face and the trace of the section of the pyramid and designate it;
construct a straight line passing through 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 that which is born from both.
For a triangle 3:4:5, the equality is true: 32 + 42 = 52, which expresses the Pythagorean theorem. Isn’t this theorem they wanted to perpetuate? Egyptian priests, building a pyramid based on a 3:4:5 triangle? It is difficult to find a more successful example to illustrate the Pythagorean theorem, which was known to the Egyptians long before its discovery by Pythagoras.
Thus, the brilliant creators Egyptian pyramids sought to amaze distant descendants with the depth of their knowledge, and they achieved this by choosing the “golden” right triangle as the “main geometric idea” for the Cheops pyramid, and the “sacred” or “Egyptian” triangle for the Khafre pyramid.
Very often in their research, scientists use the properties of pyramids with Golden Ratio proportions.
In mathematics encyclopedic dictionary The following definition of the Golden Section is given - this is a harmonic division, division in extreme and average ratio - dividing the segment AB into two parts in such a way that its larger part AC is the average proportional between the entire segment AB and its smaller part NE.
Algebraic determination of the Golden section of a segment AB = a reduces to solving the equation a: x = x: (a – x), from which x is approximately equal to 0.62a. The ratio x 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, a perpendicular to AB is restored, the segment BE = 1/2 AB is laid out on it, A and E are connected, DE = BE is laid off and, finally, AC = AD, then the equality AB is satisfied: CB = 2:3.
Golden ratio often used in works of art, architecture, and 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 the common stem of plants, you can notice that between every two pairs of leaves the third is located at the Golden Ratio (slides). Each of us “carries” 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 calculation and measurement. The tasks contained in them were solved by scribes. One of the most famous is the Rhind Mathematical Papyrus. By studying these problems, Egyptologists learned how the ancient Egyptians dealt with in different quantities, which arose in the calculation of measures of weight, length and volume, which often used fractions, and 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 a gradient. The slope gradient was expressed as a whole number ratio called "seced". In Mathematics in the Age 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 the nth number of horizontal units per vertical unit of rise. Thus, this unit of measurement is equivalent to our modern cotangent of the angle of inclination. Therefore, the Egyptian word "seced" is related to our modern word"gradient"".
The numerical key to the pyramids lies in the ratio of their height to the base. IN in practical terms- this is the easiest way to make the templates needed for constant checking the correct angle of inclination throughout the construction of the pyramid.
Egyptologists would be happy to convince us that each pharaoh longed 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 were not aware of the 3:4:5 triangle, the length of the hypotenuse 5 was never mentioned. But math problems questions concerning pyramids are always decided on the basis of the second 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 Giza pyramids 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 number symbolism in all types of Egyptian visual arts. It is very likely that such relationships were significant because they expressed specific religious ideas. In other words, the entire Giza complex was subordinated to a coherent design designed to reflect a certain divine theme. This would explain why the designers chose different angles the inclination of the three pyramids.
In The Mystery of Orion, Bauval and Gilbert presented convincing evidence linking the pyramids of Giza with the constellation Orion, in particular with the stars of Orion's Belt. The same constellation is present in the myth of Isis and Osiris, and there is reason to view each pyramid as a representation of one of the three main deities - Osiris, Isis and Horus.
"GEOMETRICAL" MIRACLES.
Among grand pyramids Egypt special place takes Great Pyramid of Pharaoh Cheops (Khufu). Before we begin to analyze the shape and size of the Cheops pyramid, we should remember what system of measures the Egyptians used. The Egyptians had three units of length: a "cubit" (466 mm), which was equal to seven "palms" (66.5 mm), which, in turn, was equal to four "fingers" (16.6 mm).
Let us analyze the dimensions of the Cheops pyramid (Fig. 2), following the arguments given in the wonderful book of the Ukrainian scientist Nikolai Vasyutinsky " Golden ratio" (1990).
Most researchers agree that the length of the side of the base of the pyramid, for example, GF equal to L= 233.16 m. This value corresponds almost exactly to 500 “elbows”. Full compliance with 500 “elbows” will occur if the length of the “elbow” is considered equal to 0.4663 m.
Height of the pyramid ( H) is estimated by researchers variously from 146.6 to 148.2 m. And depending on the accepted height of the pyramid, all its ratios change geometric elements. What is the reason for the differences in estimates of the height of the pyramid? The fact is that, strictly speaking, the Cheops pyramid is truncated. Its upper platform today measures approximately 10 ´ 10 m, but a century ago it was 6 ´ 6 m. Obviously, the top of the pyramid was dismantled, and it does not correspond to the original one.
When assessing the height of the pyramid, it is necessary to take into account this physical factor, as a “draft” of the structure. Behind 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 by finding 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 a= 51°51". This value is still recognized by most researchers today. Specified value 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 its base C.B.(Fig.2), that is A.C. / C.B. = 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= 51°50", that is, reduce it by just one arc minute, then the value a will become equal to 1.272, that is, it will coincide with the value. 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 the researchers to the following very interesting hypothesis: the triangle ACB of the Cheops pyramid was based on the relation AC / C.B. = = 1,272!
Consider now the right triangle ABC, in which the ratio of the legs A.C. / C.B.= (Fig. 2). If now the lengths of the sides of the rectangle ABC designate 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 using the formula:
If we 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 a “golden” right triangle, then from here we can easily calculate the “design” height of the Cheops pyramid. It is equal to:
H = (L/2) ´ = 148.28 m.
Let us now derive some other relations for the Cheops pyramid, which follow from the “golden” hypothesis. In particular, we will 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 C.B. per unit, that is: C.B.= 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 SEFGH = 4.
Let us now calculate the area of the side face of the Cheops pyramid SD. Because the height AB triangle AEF equal to t, then the area of the side face will be equal to SD = t. Then the total area of all four lateral 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!
To the group " geometric wonders"The Cheops pyramid can be attributed to the real and fictitious properties of the relationships between different dimensions in the pyramid.
As a rule, they are obtained in search of certain “constants”, in particular, the number “pi” (Ludolfo’s number), equal to 3.14159...; grounds natural logarithms"e" (Neper's number), equal to 2.71828...; the number "F", the number of the "golden section", equal to, for example, 0.618... etc.
You can name, for example: 1) Property of Herodotus: (Height)2 = 0.5 art. basic x Apothem; 2) Property of V. Price: Height: 0.5 art. base = Square root of "F"; 3) Property of M. Eist: Perimeter of the base: 2 Height = "Pi"; in a different interpretation - 2 tbsp. basic : Height = "Pi"; 4) Property of G. Edge: Radius of the inscribed circle: 0.5 art. basic = "F"; 5) Property of K. Kleppisch: (Art. main.)2: 2(Art. main. x Apothem) = (Art. main. W. Apothema) = 2(Art. main. x Apothem) : ((2 art. base X Apothem) + (art. base)2). Etc. You can come up with many such properties, especially if you connect two adjacent pyramids. For example, as “Properties of A. Arefyev” it can be mentioned that the difference in the volumes of the pyramid of Cheops and the pyramid of Khafre is equal to twice the volume of the pyramid of Mikerin...
Many interesting provisions In particular, the construction of pyramids according to the “golden ratio” is described in the books by D. Hambidge “Dynamic symmetry in architecture” and M. Gick “Aesthetics of proportion in nature and art.” Let us recall that the “golden ratio” is the division of a segment in such a ratio that part A is as many times greater than part B, how many times A is smaller than the entire segment A + B. The ratio A/B is equal to the number “F” == 1.618. .. The use of the “golden ratio” is indicated not only in individual pyramids, but also in the entire complex of pyramids at Giza.
The most curious thing, however, is that one and the same Cheops pyramid simply “cannot” contain so many wonderful properties. Taking a certain property one by one, it can be “fitted”, but all of them do not fit at once - they do not coincide, they contradict each other. Therefore, if, for example, when checking all properties, we initially take the same side of the base of the pyramid (233 m), then the heights of pyramids with different properties will also be different. In other words, there is a certain “family” of pyramids that are externally similar to Cheops, but correspond 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 only be considered something that was clearly impossible for the ancient Egyptians. This, in particular, includes “cosmic” miracles, in which the measurements of the Cheops pyramid or the pyramid complex at Giza are compared with some astronomical measurements and “even” numbers are indicated: a million times less, a billion times less, and so on. Let's consider some "cosmic" relationships.
One of the statements is: "if you divide the side of the base of the pyramid by exact length year, we get exactly the 10-millionth part 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 cubit” he himself invented, then the side of the pyramid will correspond to “the most accurate duration 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 usually taken is 146.6 m, Smith took it as 148.2 m. According to modern radar measurements, the semimajor axis 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 interesting statement:
“How can we explain that the masses of the pyramids of Cheops, Khafre and Mykerinus relate to each other, like the masses of the planets Earth, Venus, Mars?” Let's calculate. The masses of the three pyramids are: Khafre - 0.835; Cheops - 1,000; Mikerin - 0.0915. The ratios of the masses of the three planets: Venus - 0.815; Earth - 1,000; Mars - 0.108.
So, despite skepticism, we note the well-known harmony of the construction of statements: 1) the height of the pyramid, like 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 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 matter for now.
PYRAMID SHAPE
The famous tetrahedral shape of the pyramids did not arise immediately. The Scythians made burials in the form of earthen hills - mounds. The Egyptians built "hills" of stone - pyramids. This first happened after the unification of Upper and Lower Egypt, in the 28th century BC, when before the founder III dynasty Pharaoh Djoser (Zoser) was tasked with strengthening the unity of the country.
And here, according to historians, important role in strengthening central government played" new concept"deification" of the king. Although the royal burials were distinguished by greater splendor, they in principle did not differ 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 it was then placed a small building made of large stone blocks - "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 mastaba to pyramid.
In this way, the sage and architect Imhotep, who was later considered a wizard and identified by the Greeks with the god Asclepius, “raised” 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 standards - 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, it seemed like there were two steps.
This situation did not satisfy the architect, and on the upper platform of the huge flat mastaba, Imhotep placed three more, gradually decreasing towards the top. The tomb was located under the pyramid.
Several more are known step pyramids, but later the builders switched to building tetrahedral pyramids that are more familiar to us. Why, however, not triangular or, say, octagonal? An indirect answer is given by the fact that almost all pyramids are perfectly oriented along the four cardinal directions, and therefore have four sides. In addition, the pyramid was a “house”, the shell of a quadrangular burial chamber.
But what determined the angle of inclination of the faces? In the book “The Principle of Proportions” an entire chapter is devoted to this: “What could have determined the angles of inclination of the pyramids.” In particular, it is indicated that “the image to which the great pyramids gravitate Ancient kingdom- a triangle with a right angle at the vertex.
In space it is a semi-octahedron: a pyramid in which the edges and sides of the base are equal, the edges are equilateral triangles." Certain considerations are given on this subject in the books of Hambidge, Gick and others.
What is the advantage of the semi-octahedron angle? According to descriptions by 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 a model. Taking four firmly fixed balls, you need to place a fifth one on them and measure the angles of inclination. However, you can make a mistake here, so a theoretical calculation helps: you should connect the centers of the balls with lines (mentally). The base will be 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 close packing of balls like 1:4 will give us a regular semi-octahedron.
However, why do many pyramids, gravitating towards similar form, nevertheless do not save 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 must age, not only processes of external weathering can and should occur in them, but also processes of internal “shrinkage,” from which the pyramids may become lower. Shrinkage is also possible because, as revealed by the work of D. Davidovits, the ancient Egyptians used the technology of making blocks from lime chips, in other words, from “concrete”. It is precisely similar 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. “The usual references to the destructive effects 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 us think that the famous pyramid of Cheops also "shrivelled." In any case, in all ancient images the pyramids are pointed ...
The shape of the pyramids could also have been generated by imitation: some natural samples, “miracle perfection,” say, some crystals in the form of an octahedron.
Similar crystals could be diamond and gold crystals. Characteristic a large number of"overlapping" signs for such concepts as Pharaoh, Sun, Gold, Diamond. Everywhere - noble, brilliant (brilliant), great, impeccable, and so on. The similarities are not accidental.
The solar cult, as is known, was important part religion Ancient Egypt. “No matter how we translate the name of the greatest of the pyramids,” notes one of modern aids- “The firmament of Khufu” or “The firmament of Khufu”, it meant that the king is the sun.” If Khufu, in the brilliance of his power, imagined himself to be the second sun, then his son Djedef-Ra became the first of the Egyptian kings to call himself “the son of Ra ", that is, the son of the Sun. The Sun of almost all peoples was symbolized by the "solar metal", gold. "A large disk of bright gold" - that’s what the Egyptians called our daylight. The Egyptians knew gold perfectly, they knew its native forms, where gold crystals can appear in the form of octahedrons.
How "sample forms" is interesting here and " sun stone" - diamond. The name of the diamond came precisely from Arab world, "almas" - the hardest, most hardened, indestructible. The ancient Egyptians knew diamond and its properties quite well. According to some authors, they even used bronze tubes with diamond cutters for drilling.
Currently the main supplier of diamonds is South Africa, but Western Africa is also rich in diamonds. The territory of the Republic of Mali is even called the “Diamond Land”. Meanwhile, it is on the territory of Mali that the Dogon live, with whom supporters of the paleo-visit hypothesis pin many hopes (see below). Diamonds could not have been 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 to the most wonderful creations nature.
Conclusion:
Having studied the pyramid as a geometric body, becoming 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: Textbook. for 7 – 9 grades. general education institutions\, etc. - 9th ed. - M.: Education, 1999
History of mathematics in school, M: “Prosveshchenie”, 1982.
Geometry 10-11 grades, M: “Enlightenment”, 2000
Peter Tompkins "Secrets" great pyramid Cheops", M: "Tsentropoligraf", 2005.
Internet resources
http://veka-i-mig. *****/
http://tambov. *****/vjpusk/vjp025/rabot/33/index2.htm
http://www. *****/enc/54373.html
When solving Problem C2 using the coordinate method, many students face the same problem. They can't calculate coordinates of points included in the formula dot product. The greatest difficulties arise pyramids. And if the base points are considered more or less normal, then the tops are a real hell.
Today we will work on a regular quadrangular pyramid. There is also a triangular pyramid (aka - tetrahedron). It's more complex design, so a separate lesson will be devoted to it.
First, let's remember the definition:
A regular pyramid is one that:
- The base is a regular polygon: triangle, square, etc.;
- An altitude 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 calculations for a pyramid in which all edges are 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, where S is the vertex, the base ABCD is a square. All edges are equal to 1. You need to enter a coordinate system and find the coordinates of all points. We have:
We introduce a coordinate system with origin at point A:
- The OX axis is directed parallel to the edge AB;
- OY axis is parallel to AD. Since ABCD is a square, AB ⊥ AD;
- Finally, we direct the OZ axis upward, perpendicular to the ABCD plane.
Now we calculate the coordinates. Additional construction: SH - height drawn to the base. For convenience, we will place the base of the pyramid in a separate drawing. Since points A, B, C and D lie in the OXY plane, their coordinate is 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) - the center of the square, the middle of the segment AC.
It remains to find the coordinates of point S. Note that the x and y coordinates of points S and H are the same, since they lie on a straight line, parallel axis OZ. It remains to find the z coordinate for 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.
Hence, right triangles ASH and ABH equal one leg and one hypotenuse each. This means 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
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 also a side edge of the original pyramid SABCD. Leg AH is easily calculated: AH = 0.5 AC. We will find the remaining leg SH according to 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 rib 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 point S onto the OXY plane is point H;
- At the same time, point H is the center of a square ABCD, all sides of which are equal to 1.
It remains to find the coordinate of point S. Consider triangle AHS. It is rectangular, with the hypotenuse AS = BS = 3, the leg AH being 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:
