One and 0 Pythagorean pants. Professor Stewart's incredible numbers. The forces were equal

Almost equal Club of 12 chairs Almost equal IRONIC PROSE Death came to one poor man. And he was rather deaf. So normal, but slightly deaf... And he saw poorly. I saw almost nothing. - Oh, we have guests! Please pass. Death says: “Wait to rejoice,”

» by Emeritus Professor of Mathematics at the University of Warwick, famous popularizer of science Ian Stewart, dedicated to the role of numbers in the history of mankind and the relevance of their study in our time.

Pythagorean hypotenuse

Pythagorean triangles have right angles and integer sides. The simplest of them has a longest side of length 5, the others - 3 and 4. There are 5 regular polyhedra in total. A fifth degree equation cannot be solved using fifth roots - or any other roots. Lattices on a plane and in three-dimensional space do not have five-lobed rotational symmetry, so such symmetries are absent in crystals. However, they can be found in lattices in four dimensions and in interesting structures known as quasicrystals.

Hypotenuse of the smallest Pythagorean triple

The Pythagorean theorem states that the longest side of a right triangle (the notorious hypotenuse) is related to the other two sides of this triangle in a very simple and beautiful way: the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Traditionally, we call this theorem by the name of Pythagoras, but in fact its history is quite vague. Clay tablets suggest that the ancient Babylonians knew the Pythagorean theorem long before Pythagoras himself; The fame of the discoverer was brought to him by the mathematical cult of the Pythagoreans, whose supporters believed that the Universe was based on numerical laws. Ancient authors attributed a variety of mathematical theorems to the Pythagoreans - and therefore to Pythagoras, but in fact we have no idea what kind of mathematics Pythagoras himself was involved in. We don't even know if the Pythagoreans could prove the Pythagorean Theorem or if they simply believed it to be true. Or, most likely, they had convincing evidence of its truth, which nevertheless would not be enough for what we consider evidence today.

Proofs of Pythagoras

The first known proof of the Pythagorean theorem is found in Euclid's Elements. This is a fairly complex proof using a drawing that Victorian schoolchildren would immediately recognize as “Pythagorean trousers”; The drawing really does resemble underpants drying on a line. There are literally hundreds of other proofs, most of which make the assertion more obvious.

// Rice. 33. Pythagorean pants

One of the simplest proofs is a kind of mathematical puzzle. Take any right triangle, make four copies of it and assemble them inside the square. In one arrangement we see a square on the hypotenuse; with the other - squares on the other two sides of the triangle. It is clear that the areas in both cases are equal.

// Rice. 34. Left: square on the hypotenuse (plus four triangles). Right: sum of the squares on the other two sides (plus the same four triangles). Now eliminate the triangles

Perigal's dissection is another puzzle proof.

// Rice. 35. Perigal's dissection

There is also a proof of the theorem using arranging squares on a plane. Perhaps this is how the Pythagoreans or their unknown predecessors discovered this theorem. If you look at how the skew square overlaps two other squares, you can see how to cut a large square into pieces and then put them together into two smaller squares. You can also see right triangles, the sides of which give the dimensions of the three squares involved.

// Rice. 36. Proof by paving

There is interesting evidence using similar triangles in trigonometry. At least fifty different proofs are known.

Pythagorean triples

In number theory, the Pythagorean theorem became the source of a fruitful idea: finding integer solutions to algebraic equations. A Pythagorean triple is a set of integers a, b and c such that

Geometrically, such a triple defines a right triangle with integer sides.

The smallest hypotenuse of a Pythagorean triple is 5.

The other two sides of this triangle are 3 and 4. Here

32 + 42 = 9 + 16 = 25 = 52.

The next largest hypotenuse is 10 because

62 + 82 = 36 + 64 = 100 = 102.

However, this is essentially the same triangle with double sides. The next largest and truly different hypotenuse is 13, for which

52 + 122 = 25 + 144 = 169 = 132.

Euclid knew that there were an infinite number of different variations of Pythagorean triplets, and he gave what might be called a formula for finding them all. Later, Diophantus of Alexandria proposed a simple recipe, basically identical to Euclidean.

Take any two natural numbers and calculate:

their double product;

the difference of their squares;

the sum of their squares.

The three resulting numbers will be the sides of the Pythagorean triangle.

Let's take, for example, the numbers 2 and 1. Let's calculate:

double product: 2 × 2 × 1 = 4;

difference of squares: 22 - 12 = 3;

sum of squares: 22 + 12 = 5,

and we got the famous 3-4-5 triangle. If we take the numbers 3 and 2 instead, we get:

double product: 2 × 3 × 2 = 12;

difference of squares: 32 - 22 = 5;

sum of squares: 32 + 22 = 13,

and we get the next most famous triangle 5 - 12 - 13. Let's try to take the numbers 42 and 23 and get:

double product: 2 × 42 × 23 = 1932;

difference of squares: 422 - 232 = 1235;

sum of squares: 422 + 232 = 2293,

no one has ever heard of the triangle 1235–1932–2293.

But these numbers work too:

12352 + 19322 = 1525225 + 3732624 = 5257849 = 22932.

There is another feature of the Diophantine rule that has already been hinted at: given three numbers, we can take another arbitrary number and multiply them all by it. Thus, a 3–4–5 triangle can be turned into a 6–8–10 triangle by multiplying all sides by 2, or into a 15–20–25 triangle by multiplying all by 5.

If we switch to the language of algebra, the rule takes on the following form: let u, v and k be natural numbers. Then a right triangle with sides

2kuv and k (u2 - v2) has a hypotenuse

There are other ways of presenting the main idea, but they all boil down to the one described above. This method allows you to obtain all Pythagorean triples.

Regular polyhedra

There are exactly five regular polyhedra. A regular polyhedron (or polyhedron) is a three-dimensional figure with a finite number of flat faces. The faces meet each other on lines called edges; the edges meet at points called vertices.

The culmination of Euclidean's Principia is the proof that there can be only five regular polyhedra, that is, polyhedra in which each face is a regular polygon ( equal sides, equal angles), all faces are identical and all vertices are surrounded by an equal number of equally spaced faces. Here are five regular polyhedra:

tetrahedron with four triangular faces, four vertices and six edges;

cube, or hexahedron, with 6 square faces, 8 vertices and 12 edges;

octahedron with 8 triangular faces, 6 vertices and 12 edges;

dodecahedron with 12 pentagonal faces, 20 vertices and 30 edges;

An icosahedron with 20 triangular faces, 12 vertices and 30 edges.

// Rice. 37. Five regular polyhedra

Regular polyhedra can also be found in nature. In 1904, Ernst Haeckel published drawings of tiny organisms known as radiolarians; many of them are shaped like those same five regular polyhedra. Perhaps, however, he slightly corrected nature, and the drawings do not fully reflect the shape of specific living beings. The first three structures are also observed in crystals. You will not find dodecahedrons and icosahedrons in crystals, although irregular dodecahedrons and icosahedrons are sometimes found there. True dodecahedrons can occur as quasicrystals, which are similar to crystals in every way except that their atoms do not form a periodic lattice.

// Rice. 38. Haeckel’s drawings: radiolarians in the form of regular polyhedra

// Rice. 39. Developments of regular polyhedra

It can be interesting to make models of regular polyhedra from paper by first cutting out a set of interconnected faces - this is called developing a polyhedron; the development is folded along the edges and the corresponding edges are glued together. It is useful to add an additional glue pad to one of the ribs of each such pair, as shown in Fig. 39. If there is no such platform, you can use adhesive tape.

Fifth degree equation

There is no algebraic formula for solving 5th degree equations.

IN general view The fifth degree equation looks like this:

ax5 + bx4 + cx3 + dx2 + ex + f = 0.

The problem is to find a formula for solutions to such an equation (it can have up to five solutions). Experience with quadratic and cubic equations, as well as equations of the fourth degree, suggests that such a formula should also exist for equations of the fifth degree, and, in theory, roots of the fifth, third and second degrees should appear in it. Again, we can safely assume that such a formula, if it exists, will be very, very complex.

This assumption ultimately turned out to be wrong. In fact, no such formula exists; at least there is no formula consisting of the coefficients a, b, c, d, e and f, made using addition, subtraction, multiplication and division, and taking roots. So there is something very special about the number 5. The reasons for this unusual behavior of the five are very deep, and it took a lot of time to understand them.

The first sign of trouble was that no matter how hard mathematicians tried to find such a formula, no matter how smart they were, they invariably failed. For some time, everyone believed that the reasons lay in the incredible complexity of the formula. It was believed that no one simply could understand this algebra properly. However, over time, some mathematicians began to doubt that such a formula even existed, and in 1823 Niels Hendrik Abel was able to prove the opposite. There is no such formula. Shortly thereafter, Évariste Galois found a way to determine whether an equation of one degree or another—5th, 6th, 7th, any kind—was solvable using this kind of formula.

The conclusion from all this is simple: the number 5 is special. You can solve algebraic equations (using nth roots degrees for different values ​​of n) for powers 1, 2, 3 and 4, but not for the 5th power. This is where the obvious pattern ends.

No one is surprised that equations of degrees greater than 5 behave even worse; in particular, the same difficulty is associated with them: there are no general formulas for solving them. This does not mean that the equations have no solutions; This also does not mean that it is impossible to find very precise numerical values ​​for these solutions. It's all about the limitations of traditional algebra tools. This is reminiscent of the impossibility of trisection of an angle using a ruler and compass. The answer exists, but the methods listed are insufficient and do not allow us to determine what it is.

Crystallographic limitation

Crystals in two and three dimensions do not have 5-ray rotational symmetry.

Atoms in a crystal form a lattice, that is, a structure that periodically repeats itself in several independent directions. For example, the pattern on wallpaper is repeated along the length of the roll; in addition, it is usually repeated in the horizontal direction, sometimes with a shift from one piece of wallpaper to the next. Essentially, wallpaper is a two-dimensional crystal.

There are 17 varieties of wallpaper patterns on a plane (see Chapter 17). They differ in types of symmetry, that is, in ways to rigidly move the pattern so that it lies exactly on itself in its original position. Types of symmetry include, in particular, various variants of rotational symmetry, where the pattern should be rotated by a certain angle around a certain point - the center of symmetry.

The order of rotational symmetry is how many times the body can be rotated in a full circle so that all the details of the pattern return to their original positions. For example, a 90° rotation is 4th order rotation symmetry*. The list of possible types of rotational symmetry in a crystal lattice again points to the unusualness of the number 5: it is not there. There are options with 2nd, 3rd, 4th and 6th order rotation symmetry, but none of the wallpaper designs have 5th order rotation symmetry. Rotation symmetry of order greater than 6 also does not exist in crystals, but the first violation of the sequence still occurs at number 5.

The same thing happens with crystallographic systems in three-dimensional space. Here the lattice repeats itself in three independent directions. There are 219 various types symmetry, or 230, if we consider the mirror reflection of the drawing as a separate variant of it - despite the fact that in this case there is no mirror symmetry. Again, rotational symmetries of orders 2, 3, 4, and 6 are observed, but not 5. This fact is called crystallographic confinement.

In four-dimensional space, lattices with 5th order symmetry exist; In general, for lattices of sufficiently high dimension, any predetermined order of rotational symmetry is possible.

// Rice. 40. Crystal lattice of table salt. Dark balls represent sodium atoms, light balls represent chlorine atoms

Quasicrystals

Although 5th order rotational symmetry is not possible in 2D or 3D lattices, it can exist in slightly less regular structures known as quasicrystals. Using Kepler's sketches, Roger Penrose discovered flat systems with more general type fivefold symmetry. They are called quasicrystals.

Quasicrystals exist in nature. In 1984, Daniel Shechtman discovered that an alloy of aluminum and manganese could form quasicrystals; Initially, crystallographers greeted his report with some skepticism, but the discovery was later confirmed, and in 2011 Shechtman was awarded the Nobel Prize in Chemistry. In 2009, a team of scientists led by Luca Bindi discovered quasicrystals in a mineral from the Russian Koryak Highlands - a compound of aluminum, copper and iron. Today this mineral is called icosahedrite. By measuring the content of different oxygen isotopes in the mineral using a mass spectrometer, scientists showed that this mineral did not originate on Earth. It formed about 4.5 billion years ago, at a time when solar system was just in its infancy, and spent most of its time in the asteroid belt, orbiting the Sun, until some disturbance changed its orbit and eventually brought it to Earth.

// Rice. 41. Left: one of two quasicrystalline lattices with exact fivefold symmetry. Right: Atomic model of an icosahedral aluminum-palladium-manganese quasicrystal

Everyone has known the Pythagorean theorem since school. An outstanding mathematician proved a great hypothesis, which is currently used by many people. The rule goes like this: the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. For many decades, not a single mathematician has been able to argue this rule. After all, Pythagoras took a long time to achieve his goal, so that as a result the drawings would take place in everyday life.

1. A small verse to this theorem, which was invented shortly after the proof, directly proves the properties of the hypothesis: “Pythagorean pants are equal in all directions.” This two-line line is etched in the memory of many people - to this day the poem is remembered when doing calculations.
2. This theorem was called “Pythagorean Pants” due to the fact that when drawn in the middle, a right-angled triangle was obtained, with squares on each side. In appearance, this drawing resembled pants - hence the name of the hypothesis.
3. Pythagoras was proud of the developed theorem, because this hypothesis differs from similar ones in the maximum amount of evidence. Important: the equation was included in the Guinness Book of Records due to 370 true proofs.
4. The hypothesis was proven by a huge number of mathematicians and professors from different countries in many ways. The English mathematician Jones soon announced the hypothesis and proved it using a differential equation.
5. At present, no one knows the proof of the theorem by Pythagoras himself.. The facts about the proofs of a mathematician are not known to anyone today. It is believed that Euclid's proof of drawings is Pythagoras' proof. However, some scientists argue with this statement: many believe that Euclid independently proved the theorem, without the help of the creator of the hypothesis.
6. Today's scientists have discovered that great mathematician was not the first to discover this hypothesis. The equation was known long before its discovery by Pythagoras. This mathematician was only able to reunite the hypothesis.
7. Pythagoras did not give the equation the name “Pythagorean Theorem”. This name stuck after the “loud two-liner.” The mathematician only wanted the whole world to know and use his efforts and discoveries.
8. Moritz Cantor, the great mathematician, found and saw notes with drawings on ancient papyrus. Soon after this, Cantor realized that this theorem had been known to the Egyptians as early as 2300 BC. Only then no one took advantage of it or tried to prove it.
9. Current scientists believe that the hypothesis was known back in the 8th century BC. Indian scientists of that time discovered an approximate calculation of the hypotenuse of a triangle endowed with right angles. True, at that time no one was able to prove the equation for sure using approximate calculations.
10. The great mathematician Bartel van der Waerden, after proving the hypothesis, concluded an important conclusion: “The merit of the Greek mathematician is considered not to be the discovery of direction and geometry, but only its justification. Pythagoras had in his hands calculating formulas that were based on assumptions, inaccurate calculations and vague ideas. However, an outstanding scientist managed to turn it into an exact science.”
11. The famous poet said that on the day of the discovery of his drawing he erected a glorious sacrifice for the bulls. It was after the discovery of the hypothesis that rumors began to spread that the sacrifice of a hundred bulls “went to wander through the pages of books and publications.” To this day, wits joke that since then all the bulls have been afraid of the new discovery.
12. Proof that it was not Pythagoras who came up with the poem about pants in order to prove the drawings he put forward: During the life of the great mathematician there were no pants yet. They were invented several decades later.
13. Pekka, Leibniz and several other scientists tried to prove the previously known theorem, but no one succeeded.
14. The name of the drawings “Pythagorean theorem” means “persuasion by speech”. This is how the word Pythagoras is translated, which the mathematician took as a pseudonym.
15. Pythagoras' reflections on his own rule: the secret of everything on earth lies in numbers. After all, the mathematician, relying on his own hypothesis, studied the properties of numbers, identified evenness and oddness, and created proportions.

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Jarg. school Joking. The Pythagorean theorem, which establishes the relationship between the areas of squares built on the hypotenuse and legs of a right triangle. BTS, 835… Large dictionary of Russian sayings

Pythagorean pants- A comic name for the Pythagorean theorem, which arose due to the fact that the squares built on the sides of a rectangle and diverging in different directions resemble the cut of pants. I loved geometry... and on the entrance exam to the university I even received an... Phraseological Dictionary of the Russian Literary Language

Pythagorean pants- A humorous name for the Pythagorean theorem, which establishes the relationship between the areas of squares built on the hypotenuse and legs of a right triangle, which looks like the cut of pants in the pictures... Dictionary of many expressions

Monk: about a gifted man Wed. This is undoubtedly a sage. In ancient times, he probably would have invented Pythagorean pants... Saltykov. Variegated letters. Pythagorean pants (geom.): in a rectangle, the square of the hypotenuse is equal to the squares of the legs (teaching ... ... Michelson's Large Explanatory and Phraseological Dictionary

Pythagorean pants are equal on all sides- The number of buttons is known. Why is the dick tight? (rudely) about pants and the male genital organ. Pythagorean pants are equal on all sides. To prove this, it is necessary to remove and show 1) about the Pythagorean theorem; 2) about wide pants... Live speech. Dictionary of colloquial expressions

Pythagorean pants (invent) monk. about a gifted person. Wed. This is undoubtedly a sage. In ancient times, he probably would have invented Pythagorean pants... Saltykov. Motley letters. Pythagorean trousers (geom.): in a rectangle there is a square of the hypotenuse... ... Michelson's Large Explanatory and Phraseological Dictionary (original spelling)

Pythagorean pants are equal in all directions- A humorous proof of the Pythagorean theorem; also as a joke about a friend's baggy trousers... Dictionary of folk phraseology

Adj., rude...

PYTHAGOREAN PANTS ARE EQUAL ON ALL SIDES (THE NUMBER OF BUTTONS IS KNOWN. WHY IS IT TIGHT? / TO PROVE THIS, YOU HAVE TO TAKE IT OFF AND SHOW)- adverb, rude... Explanatory dictionary of modern colloquial phraseological units and proverbs

Noun, plural, used compare often Morphology: pl. What? pants, (no) what? pants, what? pants, (see) what? pants, what? pants, what about? about pants 1. Pants are a piece of clothing that has two short or long legs and covers the lower part... ... Dmitriev's Explanatory Dictionary

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