In this lesson, everyone will be able to study the topic "Rectangular box". At the beginning of the lesson, we will repeat what an arbitrary and straight parallelepipeds are, recall the properties of their opposite faces and diagonals of the parallelepiped. Then we will consider what a cuboid is and discuss its main properties.
Topic: Perpendicularity of lines and planes
Lesson: Cuboid
A surface composed of two equal parallelograms ABCD and A 1 B 1 C 1 D 1 and four parallelograms ABB 1 A 1, BCC 1 B 1, CDD 1 C 1, DAA 1 D 1 is called parallelepiped(Fig. 1).
Rice. 1 Parallelepiped
That is: we have two equal parallelograms ABCD and A 1 B 1 C 1 D 1 (bases), they lie in parallel planes so that the side edges AA 1, BB 1, DD 1, CC 1 are parallel. Thus, a surface composed of parallelograms is called parallelepiped.
Thus, the surface of a parallelepiped is the sum of all the parallelograms that make up the parallelepiped.
1. Opposite faces of a parallelepiped are parallel and equal.
(the figures are equal, that is, they can be combined by overlay)
For example:
ABCD \u003d A 1 B 1 C 1 D 1 (equal parallelograms by definition),
AA 1 B 1 B \u003d DD 1 C 1 C (since AA 1 B 1 B and DD 1 C 1 C are opposite faces of the parallelepiped),
AA 1 D 1 D \u003d BB 1 C 1 C (since AA 1 D 1 D and BB 1 C 1 C are opposite faces of the parallelepiped).
2. The diagonals of the parallelepiped intersect at one point and bisect that point.
The diagonals of the parallelepiped AC 1, B 1 D, A 1 C, D 1 B intersect at one point O, and each diagonal is divided in half by this point (Fig. 2).
Rice. 2 The diagonals of the parallelepiped intersect and bisect the intersection point.
3. There are three quadruples of equal and parallel edges of the parallelepiped: 1 - AB, A 1 B 1, D 1 C 1, DC, 2 - AD, A 1 D 1, B 1 C 1, BC, 3 - AA 1, BB 1, SS 1, DD 1.
Definition. A parallelepiped is called straight if its lateral edges are perpendicular to the bases.
Let the side edge AA 1 be perpendicular to the base (Fig. 3). This means that the line AA 1 is perpendicular to the lines AD and AB, which lie in the plane of the base. And, therefore, rectangles lie in the side faces. And the bases are arbitrary parallelograms. Denote, ∠BAD = φ, the angle φ can be any.
Rice. 3 Right box
So, a right box is a box in which the side edges are perpendicular to the bases of the box.
Definition. The parallelepiped is called rectangular, if its lateral edges are perpendicular to the base. The bases are rectangles.
The parallelepiped АВСДА 1 В 1 С 1 D 1 is rectangular (Fig. 4) if:
1. AA 1 ⊥ ABCD (lateral edge is perpendicular to the plane of the base, that is, a straight parallelepiped).
2. ∠BAD = 90°, i.e., the base is a rectangle.
Rice. 4 Cuboid
A rectangular box has all the properties of an arbitrary box. But there are additional properties that are derived from the definition of a cuboid.
So, cuboid is a parallelepiped whose lateral edges are perpendicular to the base. The base of a cuboid is a rectangle.
1. In a cuboid, all six faces are rectangles.
ABCD and A 1 B 1 C 1 D 1 are rectangles by definition.
2. Lateral ribs are perpendicular to the base. This means that all the side faces of a cuboid are rectangles.
3. All dihedral angles of a cuboid are right angles.
Consider, for example, the dihedral angle of a rectangular parallelepiped with an edge AB, i.e., the dihedral angle between the planes ABB 1 and ABC.
AB is an edge, point A 1 lies in one plane - in the plane ABB 1, and point D in the other - in the plane A 1 B 1 C 1 D 1. Then the considered dihedral angle can also be denoted as follows: ∠А 1 АВD.
Take point A on edge AB. AA 1 is perpendicular to the edge AB in the plane ABB-1, AD is perpendicular to the edge AB in the plane ABC. Hence, ∠A 1 AD is the linear angle of the given dihedral angle. ∠A 1 AD \u003d 90 °, which means that the dihedral angle at the edge AB is 90 °.
∠(ABB 1, ABC) = ∠(AB) = ∠A 1 ABD= ∠A 1 AD = 90°.
It is proved similarly that any dihedral angles of a rectangular parallelepiped are right.
The square of the diagonal of a cuboid is equal to the sum of the squares of its three dimensions.
Note. The lengths of the three edges emanating from the same vertex of the cuboid are the measurements of the cuboid. They are sometimes called length, width, height.
Given: ABCDA 1 B 1 C 1 D 1 - a rectangular parallelepiped (Fig. 5).
Prove: .
Rice. 5 Cuboid
Proof:
The line CC 1 is perpendicular to the plane ABC, and hence to the line AC. So triangle CC 1 A is a right triangle. According to the Pythagorean theorem:
Consider a right triangle ABC. According to the Pythagorean theorem:
But BC and AD are opposite sides of the rectangle. So BC = AD. Then:
As , a , then. Since CC 1 = AA 1, then what was required to be proved.
The diagonals of a rectangular parallelepiped are equal.
Let us designate the dimensions of the parallelepiped ABC as a, b, c (see Fig. 6), then AC 1 = CA 1 = B 1 D = DB 1 =
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parallelepiped, parallelepiped photo
Parallelepiped(ancient Greek παραλληλ-επίπεδον from other Greek παρ-άλληλος - “parallel” and other Greek ἐπί-πεδον - “plane”) - a prism, the base of which is a parallelogram, or (equivalently) a polyhedron, which has six faces and each of them - parallelogram.
- 1 Types of box
- 2 Basic elements
- 3 Properties
- 4 Basic formulas
- 4.1 Right box
- 4.2 Cuboid
- 4.3 Cube
- 4.4 Arbitrary box
- 5 mathematical analysis
- 6 Notes
- 7 Links
Types of box
cuboidThere are several types of parallelepipeds:
- A cuboid is a cuboid whose faces are all rectangles.
- An oblique box is a box whose side faces are not perpendicular to the bases.
Main elements
Two faces of a parallelepiped that do not have a common edge are called opposite, and those that have a common edge are called adjacent. Two vertices of a parallelepiped that do not belong to the same face are called opposite. The segment connecting opposite vertices is called the diagonal of the parallelepiped. The lengths of three edges of a cuboid that have a common vertex are called its dimensions.
Properties
- The parallelepiped is symmetrical about the midpoint of its diagonal.
- Any segment with ends belonging to the surface of the parallelepiped and passing through the middle of its diagonal is divided by it in half; in particular, all the diagonals of the parallelepiped intersect at one point and bisect it.
- Opposite faces of a parallelepiped are parallel and equal.
- The square of the length of the diagonal of a cuboid is equal to the sum of the squares of its three dimensions.
Basic Formulas
Right parallelepiped
The area of the lateral surface Sb \u003d Po * h, where Ro is the perimeter of the base, h is the height
The total surface area Sp \u003d Sb + 2So, where So is the area of \u200b\u200bthe base
Volume V=So*h
cuboid
Main article: cuboidThe area of the side surface Sb=2c(a+b), where a, b are the sides of the base, c is the side edge of the rectangular parallelepiped
Total surface area Sp=2(ab+bc+ac)
Volume V=abc, where a, b, c - measurements of a rectangular parallelepiped.
Cube
Surface area:
Volume: , where is the edge of the cube.
Arbitrary box
The volume and ratios in a skew box are often defined using vector algebra. The volume of a parallelepiped is equal to the absolute value of the mixed product of three vectors defined by the three sides of the parallelepiped coming from one vertex. The ratio between the lengths of the sides of the parallelepiped and the angles between them gives the statement that the Gram determinant of these three vectors is equal to the square of their mixed product:215.
In mathematical analysis
In mathematical analysis, an n-dimensional rectangular parallelepiped is understood as a set of points of the form
Notes
- Dvoretsky's Ancient Greek-Russian Dictionary "παραλληλ-επίπεδον"
- Gusyatnikov P.B., Reznichenko S.V. Vector algebra in examples and problems. - M.: Higher school, 1985. - 232 p.
Links
Wiktionary has an article "parallelepiped"- cuboid
- Parallelepiped, educational film
| Polyhedra | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| correct (Platonic Solids) |
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| correct non-convex |
Stellated dodecahedron Stellated icosidodecahedron Stellated icosahedron Stellated polyhedron Stellated octahedron | ||||||||
| convex |
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| formulas, theorems theories |
Aleksandrov's theorem on convex polytopes Bleecker's theorem Cauchy's theorem on polytopes Lindelöf's theorem on a polytope Minkowski's theorem on polytopes Sabitov's theorem Euler's theorem on polytopes Schläfli's formula |
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| Other |
Orthocentric tetrahedron Isohedral tetrahedron Rectangular parallelepiped Polyhedron group Dodecahedrons Solid angle Unit cube Flexible polyhedron Development Schläfli symbol Johnson polytope |
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cuboid, cuboid dalgamel, cuboid zurag, cuboid and parallelogram, cuboid made of cardboard, cuboid picture, cuboid volume, cuboid definition, cuboid formula, cuboid photo
Box Information About
Parallelogram means plane in Greek. A parallelepiped is a prism whose base is a parallelogram. There are five types of parallelogram: oblique, straight and rectangular parallelepiped. The cube and the rhombohedron also belong to the parallelepiped and are its variety.
Before moving on to the basic concepts, let's give some definitions:
- The diagonal of a parallelepiped is a segment that unites the vertices of the parallelepiped that are opposite each other.
- If two faces have a common edge, then we can call them adjacent edges. If there is no common edge, then the faces are called opposite.
- Two vertices that do not lie on the same face are called opposite.
What are the properties of a parallelepiped?
- The faces of a parallelepiped lying on opposite sides are parallel to each other and equal to each other.
- If you draw diagonals from one vertex to another, then the intersection point of these diagonals will divide them in half.
- The sides of a parallelepiped lying at the same angle to the base will be equal. In other words, the angles of the codirectional sides will be equal to each other.
What are the types of parallelepiped?
Now let's figure out what parallelepipeds are. As mentioned above, there are several types of this figure: a straight, rectangular, oblique parallelepiped, as well as a cube and a rhombohedron. How do they differ from each other? It's all about the planes that form them and the angles that they form.
Let's take a closer look at each of the listed types of parallelepiped.
- As the name implies, an inclined box has inclined faces, namely, those faces that are not at an angle of 90 degrees with respect to the base.
- But for a right parallelepiped, the angle between the base and the face is just ninety degrees. It is for this reason that this type of parallelepiped has such a name.
- If all the faces of the parallelepiped are the same squares, then this figure can be considered a cube.
- The rectangular parallelepiped got its name because of the planes that form it. If they are all rectangles (including the base), then it is a cuboid. This type of parallelepiped is not so common. In Greek, rhombohedron means face or base. This is the name of a three-dimensional figure, in which the faces are rhombuses.
Basic formulas for a parallelepiped
The volume of a parallelepiped is equal to the product of the area of the base and its height perpendicular to the base.
The area of the lateral surface will be equal to the product of the perimeter of the base and the height.
Knowing the basic definitions and formulas, you can calculate the base area and volume. You can choose the base of your choice. However, as a rule, a rectangle is used as the base.
Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.
If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. During the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells of a flying arrow:
A flying arrow is motionless, since at each moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow is at rest at different points in space, which, in fact, is movement. There is another point to be noted here. From one photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.
Wednesday, July 4, 2018
Very well the differences between set and multiset are described in Wikipedia. We look.
As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.
Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase "mind me, I'm in the house", or rather "mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.
First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will frantically recall physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms for each coin is unique ...
And now I have the most interesting question: where is the boundary beyond which elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science here is not even close.
Look here. We select football stadiums with the same field area. The area of the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but they are shamans for that, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics, the task sounds like this: "Find the sum of graphic symbols representing any number." Mathematicians cannot solve this problem, but shamans can do it elementarily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let's say we have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number to a number graphic symbol. This is not a mathematical operation.
2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic characters to numbers. This is not a mathematical operation.
4. Add up the resulting numbers. Now that's mathematics.
The sum of the digits of the number 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that is not all.
From the point of view of mathematics, it does not matter in which number system we write the number. So, in different number systems, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With a large number of 12345, I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have already done that. Let's look at the result.
As you can see, in different number systems, the sum of the digits of the same number is different. This result has nothing to do with mathematics. It's like finding the area of a rectangle in meters and centimeters would give you completely different results.
Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement of numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical action does not depend on the value of the number, the unit of measure used, and on who performs this action.
Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?
Female... A halo on top and an arrow down is male.
If you have such a work of design art flashing before your eyes several times a day,
Then it is not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.
1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.
