An important example of a Diophantine equation is given by the Pythagorean theorem, which relates the lengths x and y of the legs of a right triangle to the length z of its hypotenuse:

Of course, you have come across one of the wonderful solutions of this equation in natural numbers, namely the Pythagorean triple of numbers x=3, y=4, z=5. Are there any other triplets?

It turns out that there are infinitely many Pythagorean triples, and all of them were found a long time ago. They can be obtained by well-known formulas, which you will learn about from this paragraph.

If Diophantine equations of the first and second degree have already been solved, then the question of solving equations of higher degrees still remains open, despite the efforts of leading mathematicians. At present, for example, Fermat's famous conjecture that for any integer value n2 the equation

has no solutions in integers.

For solving certain types of Diophantine equations, the so-called complex numbers. What it is? Let the letter i denote some object that satisfies the condition i 2 \u003d -1(it is clear that no real number satisfies this condition). Consider expressions of the form α+iβ, where α and β are real numbers. We will call such expressions complex numbers, having defined the operations of addition and multiplication on them, as well as on binomials, but with the only difference that the expression i 2 everywhere we will replace the number -1:

7.1. Many of the three

Prove that if x0, y0, z0- Pythagorean triple, then triples y 0 , x 0 , z 0 And x 0 k, y 0 k, z 0 k for any value of the natural parameter k are also Pythagorean.

7.2. Private formulas

Check that for any natural values m>n trinity of the form

is Pythagorean. Is it any Pythagorean triple x, y, z can be represented in this form, if you allow to rearrange the numbers x and y in the triple?

7.3. Irreducible triplets

A Pythagorean triple of numbers that do not have a common divisor greater than 1 will be called irreducible. Prove that a Pythagorean triple is irreducible only if any two of the numbers in the triple are coprime.

7.4. Property of irreducible triples

Prove that in any irreducible Pythagorean triple x, y, z the number z and exactly one of the numbers x or y are odd.

7.5. All irreducible triples

Prove that a triple of numbers x, y, z is an irreducible Pythagorean triple if and only if it coincides with the triple up to the order of the first two numbers 2mn, m 2 - n 2, m 2 + n 2, where m>n- coprime natural numbers of different parity.

7.6. General formulas

Prove that all solutions of the equation

in natural numbers are given up to the order of the unknown x and y by the formulas

where m>n and k are natural parameters (in order to avoid duplication of any triples, it is enough to choose numbers of type coprime and, moreover, of different parity).

7.7. First 10 triplets

Find all Pythagorean triples x, y, z satisfying the condition x

7.8. Properties of Pythagorean triplets

Prove that for any Pythagorean triple x, y, z statements are true:

a) at least one of the numbers x or y is a multiple of 3;

b) at least one of the numbers x or y is a multiple of 4;

c) at least one of the numbers x, y or z is a multiple of 5.

7.9. Application of complex numbers

The modulus of a complex number α + iβ called a non-negative number

Check that for any complex numbers α + iβ And γ + iδ property is executed

Using the properties of complex numbers and their moduli, prove that any two integers m and n satisfy the equality

i.e., they give a solution to the equation

integers (compare with Problem 7.5).

7.10. Non-Pythagorean triples

Using the properties of complex numbers and their moduli (see Problem 7.9), find formulas for any integer solutions of the equation:

a) x 2 + y 2 \u003d z 3; b) x 2 + y 2 \u003d z 4.

Solutions

7.1. If x 0 2 + y 0 2 = z 0 2 , then y 0 2 + x 0 2 = z 0 2 , and for any natural value of k we have

Q.E.D.

7.2. From equalities

we conclude that the triple indicated in the problem satisfies the equation x 2 + y 2 = z 2 in natural numbers. However, not every Pythagorean triple x, y, z can be represented in this form; for example, the triple 9, 12, 15 is Pythagorean, but the number 15 cannot be represented as the sum of the squares of any two natural numbers m and n.

7.3. If any two numbers from the Pythagorean triple x, y, z have a common divisor d, then it will also be a divisor of the third number (so, in the case x = x 1 d, y = y 1 d we have z 2 \u003d x 2 + y 2 \u003d (x 1 2 + y 1 2) d 2, whence z 2 is divisible by d 2 and z is divisible by d). Therefore, for a Pythagorean triple to be irreducible, it is necessary that any two of the numbers in the triple be coprime,

7.4. Note that one of the numbers x or y, say x, of an irreducible Pythagorean triple x, y, z is odd because otherwise the numbers x and y would not be coprime (see problem 7.3). If the other number y is also odd, then both numbers

give a remainder of 1 when divided by 4, and the number z 2 \u003d x 2 + y 2 gives a remainder of 2 when divided by 4, that is, it is divisible by 2, but not divisible by 4, which cannot be. Thus, the number y must be even, and the number z must therefore be odd.

7.5. Let the Pythagorean triple x, y, z is irreducible and, for definiteness, the number x is even, while the numbers y, z are odd (see Problem 7.4). Then

where are the numbers are whole. Let us prove that the numbers a and b are coprime. Indeed, if they had a common divisor greater than 1, then the numbers would have the same divisor z = a + b, y = a - b, i.e., the triple would not be irreducible (see Problem 7.3). Now, expanding the numbers a and b into products of prime factors, we notice that any prime factor must be included in the product 4ab = x2 only to an even degree, and if it is included in the expansion of the number a, then it is not included in the expansion of the number b and vice versa. Therefore, any prime factor is included in the expansion of the number a or b separately only to an even degree, which means that these numbers themselves are squares of integers. Let's put then we get the equalities

moreover, the natural parameters m>n are coprime (due to the coprimeness of the numbers a and b) and have different parity (due to the odd number z \u003d m 2 + n 2).

Let now natural numbers m>n of different parity be coprime. Then the troika x \u003d 2mn, y \u003d m 2 - n 2, z \u003d m 2 + n 2, according to Problem 7.2, is Pythagorean. Let us prove that it is irreducible. To do this, it suffices to check that the numbers y and z do not have common divisors (see Problem 7.3). In fact, both of these numbers are odd, since the type numbers have different parities. If the numbers y and z have some simple common divisor (then it must be odd), then each of the numbers and and with them and each of the numbers m and n has the same divisor, which contradicts their mutual simplicity.

7.6. By virtue of the assertions formulated in Problems 7.1 and 7.2, these formulas define only Pythagorean triples. On the other hand, any Pythagorean triple x, y, z after its reduction by the greatest common divisor k, the pair of numbers x and y becomes irreducible (see Problem 7.3) and, therefore, can be represented up to the order of the numbers x and y in the form described in Problem 7.5. Therefore, any Pythagorean triple is given by the indicated formulas for some values ​​of the parameters.

7.7. From inequality z and the formulas of Problem 7.6, we obtain the estimate m 2 i.e. m≤5. Assuming m = 2, n = 1 And k = 1, 2, 3, 4, 5, we get triplets 3, 4, 5; 6, 8, 10; 9, 12, 15; 12,16,20; 15, 20, 25. Assuming m=3, n=2 And k = 1, 2, we get triplets 5, 12, 13; 10, 24, 26. Assuming m = 4, n = 1, 3 And k = 1, we get triplets 8, 15, 17; 7, 24, 25. Finally, assuming m=5, n=2 And k = 1, we get three 20, 21, 29.

"Regional center of education"

Methodical development

Using Pythagorean triples in solving

geometric problems and trigonometric tasks USE

Kaluga, 2016

I Introduction

The Pythagorean theorem is one of the main and, one might even say, the most important theorem of geometry. Its significance lies in the fact that most of the theorems of geometry can be deduced from it or with its help. The Pythagorean theorem is also remarkable in that in itself it is not at all obvious. For example, the properties of an isosceles triangle can be seen directly on the drawing. But no matter how you look at a right triangle, you will never see that there is such a simple ratio between its sides: a2+b2=c2. However, it was not Pythagoras who discovered the theorem that bears his name. It was known even earlier, but perhaps only as a fact derived from measurements. Presumably, Pythagoras knew this, but found proof.

There are an infinite number of natural numbers a, b, c, satisfying the relation a2+b2=c2.. They are called Pythagorean numbers. According to the Pythagorean theorem, such numbers can serve as the lengths of the sides of some right-angled triangle - we will call them Pythagorean triangles.

Objective: to study the possibility and effectiveness of using Pythagorean triples for solving problems of a school mathematics course, USE assignments.

Based on the purpose of the work, the following tasks:

To study the history and classification of Pythagorean triples. Analyze tasks using Pythagorean triples that are available in school textbooks and found in the control and measuring materials of the exam. Evaluate the effectiveness of using Pythagorean triples and their properties for solving problems.

Object of study: Pythagorean triples of numbers.

Subject of study: tasks of the school course of trigonometry and geometry, in which Pythagorean triples are used.

The relevance of research. Pythagorean triples are often used in geometry and trigonometry, knowing them will eliminate errors in calculations and save time.

II. Main part. Solving problems using Pythagorean triples.

2.1. Table of triples of Pythagorean numbers (according to Perelman)

Pythagorean numbers have the form a= m n, , where m and n are some coprime odd numbers.

Pythagorean numbers have a number of interesting features:

One of the "legs" must be a multiple of three.

One of the "legs" must be a multiple of four.

One of the Pythagorean numbers must be a multiple of five.

The book "Entertaining Algebra" contains a table of Pythagorean triples containing numbers up to one hundred, which do not have common factors.

		32+42=52
		52+122=132
		72+242=252
		92+402=412
		112+602=612
		132+842=852
		152+82=172
		212 +202=292
		332+562=652
		392+802=892
		352+122=372
		452+282=532
		552+482=732
		652+722=972
		632+162=652
		772+362=852

2.2. Shustrov's classification of Pythagorean triples.

Shustrov discovered the following pattern: if all Pythagorean triangles are divided into groups, then the following formulas are valid for the odd leg x, even y and hypotenuse z:

x \u003d (2N-1) (2n + 2N-1); y = 2n (n+2N-1); z = 2n (n+2N-1)+(2N-1) 2, where N is the number of the family and n is the ordinal number of the triangle in the family.

Substituting in the formula in place of N and n any positive integers, starting from one, you can get all the main Pythagorean triples of numbers, as well as multiples of a certain type. You can make a table of all Pythagorean triples for each family.

2.3. Planimetry tasks

Consider problems from various textbooks on geometry and find out how often Pythagorean triples are found in these tasks. Trivial problems of finding the third element in the table of Pythagorean triples will not be considered, although they are also found in textbooks. Let us show how to reduce the solution of a problem whose data is not expressed by natural numbers to Pythagorean triples.

Consider tasks from a geometry textbook for grades 7-9.

№ 000. Find the hypotenuse of a right triangle but=, b=.

Solution. Multiply the lengths of the legs by 7, we get two elements from the Pythagorean triple 3 and 4. The missing element is 5, which we divide by 7. Answer.

№ 000. In rectangle ABCD find BC if CD=1.5, AC=2.5.

https://pandia.ru/text/80/406/images/image007_0.gif" width="240" height="139 src=">

Solution. Let's solve right triangle ACD. We multiply the lengths by 2, we get two elements from the Pythagorean triple 3 and 5, the missing element is 4, which we divide by 2. Answer: 2.

When solving the next number, check the ratio a2+b2=c2 it is completely optional, it is enough to use Pythagorean numbers and their properties.

№ 000. Find out if a triangle is right-angled if its sides are expressed by numbers:

a) 6,8,10 (Pythagorean triple 3,4.5) - yes;

One of the legs of a right triangle must be divisible by 4. Answer: no.

c) 9,12,15 (Pythagorean triple 3,4.5) - yes;

d) 10,24,26 (Pythagorean triple 5,12.13) - yes;

One of the Pythagorean numbers must be a multiple of five. Answer: no.

g) 15, 20, 25 (Pythagorean triple 3,4.5) - yes.

Of the thirty-nine tasks in this section (the Pythagorean theorem), twenty-two are solved orally using Pythagorean numbers and knowledge of their properties.

Consider problem #000 (from the "Additional Tasks" section):

Find the area of ​​quadrilateral ABCD where AB=5 cm, BC=13 cm, CD=9 cm, DA=15 cm, AC=12 cm.

The task is to check the ratio a2+b2=c2 and prove that the given quadrilateral consists of two right triangles (the inverse theorem). And the knowledge of Pythagorean triples: 3, 4, 5 and 5, 12, 13, eliminates the need for calculations.

Let's give solutions to several problems from a textbook on geometry for grades 7-9.

Problem 156 (h). The legs of a right triangle are 9 and 40. Find the median drawn to the hypotenuse.

Solution . The median drawn to the hypotenuse is equal to half of it. The Pythagorean triple is 9.40 and 41. Therefore, the median is 20.5.

Problem 156 (i). The sides of the triangle are: but= 13 cm, b= 20 cm and height hс = 12 cm. Find the base from.

Task (KIM USE). Find the radius of a circle inscribed in an acute triangle ABC if the height BH is 12 and it is known that sin A=,sin C \u003d left "\u003e

Solution. We solve rectangular ∆ ASC: sin A=, BH=12, hence AB=13,AK=5 (Pythagorean triple 5,12,13). Solve rectangular ∆ BCH: BH =12, sin C===https://pandia.ru/text/80/406/images/image015_0.gif" width="12" height="13">3=9 (Pythagorean triple 3,4,5).The radius is found by the formula r === 4. Answer.4.

2.4. Pythagorean triples in trigonometry

The main trigonometric identity is a special case of the Pythagorean theorem: sin2a + cos2a = 1; (a/c) 2 + (b/c)2 =1. Therefore, some trigonometric tasks are easily solved orally using Pythagorean triples.

Problems in which it is required to find the values ​​of other trigonometric functions by a given value of a function can be solved without squaring and extracting a square root. All tasks of this type in the school textbook of algebra (10-11) Mordkovich (No. 000-No. 000) can be solved orally, knowing only a few Pythagorean triples: 3,4,5 ; 5,12,13 ; 8,15,17 ; 7,24,25 . Let's consider the solutions of two tasks.

No. 000 a). sin t = 4/5, π/2< t < π.

Solution. Pythagorean triple: 3, 4, 5. Therefore, cos t = -3/5; tg t = -4/3,

No. 000 b). tg t = 2.4, π< t < 3π/2.

Solution. tg t \u003d 2.4 \u003d 24/10 \u003d 12/5. Pythagorean triple 5,12,13. Given the signs, we get sin t = -12/13, cos t = -5/13, ctg t = 5/12.

3. Control and measuring materials of the exam

a) cos (arcsin 3/5)=4/5 (3, 4, 5)

b) sin (arccos 5/13)=12/13 (5, 12, 13)

c) tg (arcsin 0.6)=0.75 (6, 8, 10)

d) ctg (arccos 9/41) = 9/40 (9, 40, 41)

e) 4/3 tg (π–arcsin (–3/5))= 4/3 tg (π+arcsin 3/5)= 4/3 tg arcsin 3/5=4/3 3/4=1

e) check the validity of the equality:

arcsin 4/5 + arcsin 5/13 + arcsin 16/65 = π/2.

Solution. arcsin 4/5 + arcsin 5/13 + arcsin 16/65 = π/2

arcsin 4/5 + arcsin 5/13 = π/2 - arcsin 16/65

sin (arcsin 4/5 + arcsin 5/13) = sin (arccos 16/65)

sin (arcsin 4/5) cos (arcsin 5/13) + cos (arcsin 4/5) sin (arcsin 5/13) = 63/65

4/5 12/13 + 3/5 5/13 = 63/65

III. Conclusion

In geometric problems, one often has to solve right triangles, sometimes several times. After analyzing the tasks of school textbooks and USE materials, we can conclude that triples are mainly used: 3, 4, 5; 5, 12, 13; 7, 24, 25; 9, 40, 41; 8,15,17; which are easy to remember. When solving some trigonometric tasks, the classical solution using trigonometric formulas and a large number of calculations takes time, and knowledge of Pythagorean triples will eliminate errors in calculations and save time for solving more difficult problems on the exam.

Bibliographic list

1. Algebra and the beginnings of analysis. 10-11 grades. At 2 hours. Part 2. A task book for educational institutions / [and others]; ed. . - 8th ed., Sr. - M. : Mnemosyne, 2007. - 315 p. : ill.

2. Perelman algebra. - D.: VAP, 1994. - 200 p.

3. Roganovsky: Proc. For 7-9 cells. with a deep the study of mathematics general education. school from Russian lang. learning, - 3rd ed. - Mn.; Nar. Asveta, 2000. - 574 p.: ill.

4. Mathematics: Reader on history, methodology, didactics. / Comp. . - M.: Publishing house of URAO, 2001. - 384 p.

5. Journal "Mathematics at School" No. 1, 1965.

6. Control and measuring materials of the exam.

7. Geometry, 7-9: Proc. for educational institutions /, etc. - 13th ed. - M .: Education, 2003. – 384 p. : ill.

8. Geometry: Proc. for 10-11 cells. avg. school /, etc. - 2nd ed. - M .: Education, 1993, - 207 p.: ill.

Perelman algebra. - D.: VAP, 1994. - 200 p.

Journal "Mathematics at School" No. 1, 1965.

Geometry, 7-9: Proc. for educational institutions /, etc. - 13th ed. - M .: Education, 2003. – 384 p. : ill.

Roganovsky: Proc. For 7-9 cells. with a deep the study of mathematics general education. school from Russian lang. learning, - 3rd ed. - Mn.; Nar. Asveta, 2000. - 574 p.: ill.

Algebra and the beginnings of analysis. 10-11 grades. At 2 hours. Part 2. A task book for educational institutions / [and others]; ed. . - 8th ed., Sr. - M. : Mnemosyne, 2007. - 315 p. : ill., p.18.

Properties

Since the equation x 2 + y 2 = z 2 homogeneous, when multiplied x , y And z for the same number you get another Pythagorean triple. The Pythagorean triple is called primitive, if it cannot be obtained in this way, that is - relatively prime numbers.

Examples

Some Pythagorean triples (sorted in ascending order of maximum number, primitive ones are highlighted):

(3, 4, 5), (6, 8, 10), (5, 12, 13), (9, 12, 15), (8, 15, 17), (12, 16, 20), (15, 20, 25), (7, 24, 25), (10, 24, 26), (20, 21, 29), (18, 24, 30), (16, 30, 34), (21, 28, 35), (12, 35, 37), (15, 36, 39), (24, 32, 40), (9, 40, 41), (14, 48, 50), (30, 40, 50)…

History

Pythagorean triples have been known for a very long time. In the architecture of ancient Mesopotamian tombstones, an isosceles triangle is found, made up of two rectangular ones with sides of 9, 12 and 15 cubits. The pyramids of Pharaoh Snefru (XXVII century BC) were built using triangles with sides of 20, 21 and 29, as well as 18, 24 and 30 tens of Egyptian cubits.

X All-Russian Symposium on Applied and Industrial Mathematics. St. Petersburg, May 19, 2009

Report: Algorithm for solving Diophantine equations.

The paper considers the method of studying Diophantine equations and presents the solutions solved by this method: - Fermat's great theorem; - search for Pythagorean triplets, etc. http://referats.protoplex.ru/referats_show/6954.html

Links

• E. A. Gorin Powers of prime numbers in Pythagorean triples // Mathematical education. - 2008. - V. 12. - S. 105-125.

Wikimedia Foundation. 2010 .

See what "Pythagorean triples" are in other dictionaries:

In mathematics, Pythagorean numbers (Pythagorean triple) is a tuple of three integers that satisfy the Pythagorean relation: x2 + y2 = z2. Contents 1 Properties ... Wikipedia

Triples of natural numbers such that a triangle whose side lengths are proportional (or equal) to these numbers is right-angled, e.g. triple of numbers: 3, 4, 5… Big Encyclopedic Dictionary

Triples of natural numbers such that a triangle whose side lengths are proportional (or equal) to these numbers is a right triangle. According to the theorem, the inverse of the Pythagorean theorem (see Pythagorean theorem), for this it is enough that they ... ... Great Soviet Encyclopedia

Triplets of positive integers x, y, z satisfying the equation x2+y 2=z2. All solutions of this equation, and consequently, all P. p., are expressed by the formulas x=a 2 b2, y=2ab, z=a2+b2, where a, b are arbitrary positive integers (a>b). P. h ... Mathematical Encyclopedia

Triples of natural numbers such that a triangle, the lengths of the sides to which are proportional (or equal) to these numbers, is rectangular, for example. triple of numbers: 3, 4, 5… Natural science. encyclopedic Dictionary

Triples of natural numbers such that a triangle whose side lengths are proportional (or equal) to these numbers is rectangular, for example, a triple of numbers: 3, 4, 5. * * * PYTHAGORAN NUMBERS PYTHAGORAN NUMBERS, triples of natural numbers such that ... ... encyclopedic Dictionary

In mathematics, a Pythagorean triple is a tuple of three natural numbers satisfying the Pythagorean relation: In this case, the numbers that form a Pythagorean triple are called Pythagorean numbers. Contents 1 Primitive triples ... Wikipedia

The Pythagorean theorem is one of the fundamental theorems of Euclidean geometry, establishing the relationship between the sides of a right triangle. Contents 1 ... Wikipedia

The Pythagorean theorem is one of the fundamental theorems of Euclidean geometry, establishing the relationship between the sides of a right triangle. Contents 1 Statements 2 Proofs ... Wikipedia

This is an equation of the form where P is an integer function (for example, a polynomial with integer coefficients), and the variables take integer values. Named after the ancient Greek mathematician Diophantus. Contents 1 Examples ... Wikipedia

Chervyak Vitaly

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Competition of scientific projects of schoolchildren

Within the framework of the regional scientific and practical conference "Eureka"

Minor Academy of Sciences of students of the Kuban

Study of Pythagorean numbers

Section of mathematics.

Chervyak Vitaliy Gennadievich, Grade 9

MOBU SOSH №14

Korenovsky district

Art. Zhuravskaya

Scientific adviser:

Manko Galina Vasilievna

Mathematic teacher

MOBU SOSH №14

Korenovsk 2011

Chervyak Vitaly Gennadievich

Pythagorean numbers

Annotation.

Research topic:Pythagorean numbers

Research objectives:

Research objectives:

• Identification and development of mathematical abilities;
• Expansion of mathematical representation on the topic;
• Formation of sustainable interest in the subject;
• Development of communicative and general educational skills of independent work, the ability to conduct a discussion, argue, etc.;
• Formation and development of analytical and logical thinking;

Research methods:

• Use of Internet resources;
• Access to reference literature;
• Conducting an experiment;

Output:

• This work can be used in a geometry lesson as an additional material, for conducting elective courses or elective courses in mathematics, as well as in extracurricular work in mathematics;

Chervyak Vitaly Gennadievich

Krasnodar Territory, village Zhuravskaya, MOBU secondary school No. 14, grade 9

Pythagorean numbers

Supervisor: Manko Galina Vasilievna, mathematics teacher, MOBU secondary school No. 14

1. Introduction……………………………………………………………………3
2. Main part

2.1 Historical page………………………………………………………4

2.2 Proof of even and odd legs……….................................................5-6

2.3 Derivation of a pattern for finding

Pythagorean numbers………………………………………………………………7

2.4 Properties of Pythagorean numbers ……………………………………………… 8

3. Conclusion………………………………………………………………………9

4. List of used sources and literature…………………… 10

Applications ................................................. ................................................. ......eleven

Appendix I………………………………………………………………………11

Annex II…………………………………………………………………..13

Chervyak Vitaly Gennadievich

Krasnodar Territory, village Zhuravskaya, MOBU secondary school No. 14, grade 9

Pythagorean numbers

Supervisor: Manko Galina Vasilievna, mathematics teacher, MOBU secondary school No. 14

Introduction

I heard about Pythagoras and his life in the fifth grade at a mathematics lesson, and I was interested in the statement "Pythagorean pants are equal in all directions." When studying the Pythagorean theorem, I became interested in Pythagorean numbers. I putpurpose of the study: learn more about the Pythagorean theorem and "Pythagorean numbers".

Relevance of the topic. The value of the Pythagorean theorem and Pythagorean triples has been proven by many scientists around the world for many centuries. The problem that will be discussed in my work looks quite simple because it is based on a mathematical statement that everyone knows - the Pythagorean theorem: in any right triangle, the square built on the hypotenuse is equal to the sum of the squares built on the legs. Now triples of natural numbers x, y, z, for which x 2 + y 2 = z 2 , commonly calledPythagorean triplets. It turns out that Pythagorean triples were already known in Babylon. Gradually, Greek mathematicians also found them.

The purpose of this work

1. Explore Pythagorean numbers;
2. Understand how Pythagorean numbers are obtained;
3. Find out what properties Pythagorean numbers have;
4. Experimentally build perpendicular lines on the ground using Pythagorean numbers;

In accordance with the purpose of the work, a number of the following tasks :

1. Deeper study of the history of the Pythagorean theorem;

2. Analysis of the universal properties of Pythagorean triples.

3. Analysis of the practical application of Pythagorean triples.

Object of study: Pythagorean triples.

Subject of study: maths .

Research methods: - Use of Internet resources; - Appeal to reference literature; - Conducting an experiment;

Theoretical significance:the role played by the discovery of Pythagorean triples in science; practical application of the discovery of Pythagoras in human life.

Applied valueresearch consists in the analysis of literary sources and the systematization of facts.

Chervyak Vitaly Gennadievich

Krasnodar Territory, village Zhuravskaya, MOBU secondary school No. 14, grade 9

Pythagorean numbers

Supervisor: Manko Galina Vasilievna, mathematics teacher, MOBU secondary school No. 14

From the history of Pythagorean numbers.

• Ancient China:

Chu-pei math book:[ 2]

"If a right angle is decomposed into its component parts, then the line connecting the ends of its sides will be 5, when the base is 3, and the height is 4."

• Ancient Egypt: [2]

Cantor (the largest German historian of mathematics) believes that equality 3² + 4² = 5² was already known to the Egyptians around 2300 BC. e., in the time of the king Amenemhet (according to Papyrus 6619 of the Berlin Museum). According to Kantor harpedonapts, or "rope tensioners", built right angles using right-angled triangles with sides of 3; 4 and 5.

• Babylonia: [ 3 ]

“The merit of the first Greek mathematicians, such as Thales, Pythagoras and the Pythagoreans, is not the discovery of mathematics, but its systematization and justification. In their hands, computational recipes based on vague ideas have become an exact science.

• History of the Pythagorean theorem:,

Although this theorem is associated with the name of Pythagoras, it was known long before him.

In Babylonian texts, she occurs 1200 years before Pythagoras.

Apparently, he was the first to find its proof. In this regard, the following entry was made: "... when he discovered that in a right triangle the hypotenuse corresponds to the legs, he sacrificed a bull made of wheat dough."

Chervyak Vitaly Gennadievich

Krasnodar Territory, village Zhuravskaya, MOBU secondary school No. 14, grade 9

Pythagorean numbers

Supervisor: Manko Galina Vasilievna, mathematics teacher, MOBU secondary school No. 14

Study of Pythagorean numbers.

• Each triangle, the sides are related as 3:4:5, according to the well-known Pythagorean theorem, is right-angled, since

3 2 + 4 2 = 5 2.

• In addition to the numbers 3,4 and 5, there is, as is known, an infinite set of positive integers a, b, and c that satisfy the relation
• A 2 + in 2 = c 2.
• These numbers are calledPythagorean numbers

Pythagorean triples have been known for a very long time. In the architecture of the ancient Forest Potam tombstones, there is an isosceles triangle, composed of two rectangular ones with sides of 9, 12 and 15 cubits. The pyramids of Pharaoh Snefru (XXVII century BC) were built using triangles with sides of 20, 21 and 29, as well as 18, 24 and 30 tens of Egyptian cubits.[ 1 ]

A right triangle with legs 3, 4 and hypotenuse 5 is called the Egyptian triangle. The area of ​​this triangle is equal to the perfect number 6. The perimeter is equal to 12 - a number that was considered a symbol of happiness and prosperity.

With the help of a rope divided by knots into 12 equal parts, the ancient Egyptians built a right triangle and a right angle. A convenient and very accurate method used by land surveyors to draw perpendicular lines on the ground. It is necessary to take a cord and three pegs, the cord is arranged in a triangle so that one side consists of 3 parts, the second of 4 shares and the last of five such shares. The cord will be located in a triangle in which there is a right angle.

This ancient method, apparently used thousands of years ago by the builders of the Egyptian pyramids, is based on the fact that every triangle whose sides are related as 3:4:5, according to the Pythagorean theorem, is a right triangle.

Euclid, Pythagoras, Diophantus and many others were engaged in finding Pythagorean triples.[ 1]

It is clear that if (x, y, z ) is a Pythagorean triple, then for any natural k triple (kx, ky, kz ) will also be a Pythagorean triple. In particular, (6, 8, 10), (9, 12, 15), etc. are Pythagorean triples.

As the numbers increase, Pythagorean triples become rarer and harder to find. The Pythagoreans invented the method of finding

such triples and, using it, proved that there are infinitely many Pythagorean triples.

Triples that do not have common divisors greater than 1 are called simple triples.

Consider some properties of Pythagorean triples.[ 1]

According to the Pythagorean theorem, these numbers can serve as the lengths of some right-angled triangle; therefore, a and b are called “legs”, and c is called the “hypotenuse”.
It is clear that if a, b, c are a triple of Pythagorean numbers, then pa, p, pc, where p is an integer factor, are Pythagorean numbers.
The opposite is also true!
Therefore, we will first study only triples of coprime Pythagorean numbers (the rest are obtained from them by multiplying by an integer factor p).

Let us show that in each of such triples a, b, c, one of the “legs” must be even, and the other odd. Let's argue "on the contrary". If both "legs" a and b are even, then the number a will be even 2 + in 2 , and hence the hypotenuse. But this contradicts the fact that the numbers a, b and c do not have common factors, since three even numbers have a common factor of 2. Thus, at least one of the "legs" a and b is odd.

There remains one more possibility: both "legs" are odd, and the "hypotenuse" is even. It is easy to prove that this cannot be, since if the "legs" have the form 2 x + 1 and 2y + 1, then the sum of their squares is equal to

4x 2 + 4x + 1 + 4y 2 + 4y + 1 = 4 (x 2 + x + y 2 + y) +2, i.e. is a number that, when divided by 4, gives a remainder of 2. Meanwhile, the square of any even number must be divisible by 4 without a remainder.

So the sum of the squares of two odd numbers cannot be the square of an even number; in other words, our three numbers are not Pythagorean.

OUTPUT:

So, from the "legs" a, to one even, and the other odd. So the number a 2 + in 2 odd, which means that the “hypotenuse” c.

Pythagoras found formulas that in modern symbolism can be written like this: a=2n+1, b=2n(n+1), c=2 n 2 +2n+1, where n is an integer.

These numbers are Pythagorean triples.

Chervyak Vitaly Gennadievich

Krasnodar Territory, village Zhuravskaya, MOBU secondary school No. 14, grade 9

Pythagorean numbers

Supervisor: Manko Galina Vasilievna, mathematics teacher, MOBU secondary school No. 14

Derivation of a pattern for finding Pythagorean numbers.

Here are the following Pythagorean triples:

• 3, 4, 5; 9+16=25.
• 5, 12, 13; 25+144=225.
• 7, 24, 25; 49+576=625.
• 8, 15, 17; 64+225=289.
• 9, 40, 41; 81+1600=1681.
• 12, 35, 37; 144+1225=1369.
• 20, 21, 29; 400+441=881

It is easy to see that when multiplying each of the numbers of the Pythagorean triple by 2, 3, 4, 5, etc., we get the following triples.

• 6, 8, 10;
• 9,12,15.
• 12, 16, 20;
• 15, 20, 25;
• 10, 24, 26;
• 18, 24, 30;
• 16, 30, 34;
• 21, 28, 35;
• 15, 36, 39;
• 24, 32, 40;
• 14, 48, 50;
• 30, 40, 50 etc.

They are also Pythagorean numbers/

Chervyak Vitaly Gennadievich

Krasnodar Territory, village Zhuravskaya, MOBU secondary school No. 14, grade 9

Pythagorean numbers

Supervisor: Manko Galina Vasilievna, mathematics teacher, MOBU secondary school No. 14

Properties of Pythagorean numbers.

• When considering Pythagorean numbers, I saw a number of properties:
• 1) One of the Pythagorean numbers must be a multiple of three;
• 2) Another of them must be a multiple of four;
• 3) And the third of the Pythagorean numbers must be a multiple of five;

Chervyak Vitaly Gennadievich

Krasnodar Territory, village Zhuravskaya, MOBU secondary school No. 14, grade 9

Pythagorean numbers

Supervisor: Manko Galina Vasilievna, mathematics teacher, MOBU secondary school No. 14

Conclusion.

Geometry, like other sciences, arose from the needs of practice. The very word "geometry" - Greek, in translation means "surveying".

People very early faced the need to measure land. Already for 3-4 thousand years BC. every piece of fertile land in the valleys of the Nile, Euphrates and Tigris, the rivers of China, was important for people's lives. This required a certain stock of geometric and arithmetic knowledge.

Gradually, people began to measure and study the properties of more complex geometric shapes.

Both in Egypt and in Babylon, colossal temples were built, the construction of which could only be carried out on the basis of preliminary calculations. Aqueducts were also built. All this required drawings and calculations. By this time, special cases of the Pythagorean theorem were well known, they already knew that if we take triangles with sides x, y, z, where x, y, z are such integers that x 2 + y 2 = z 2 , then these triangles will be right-angled.

All this knowledge was directly applied in many spheres of human life.

So until now, the great discovery of the scientist and philosopher of antiquity Pythagoras finds direct application in our life.

Construction of houses, roads, spaceships, cars, machine tools, oil pipelines, airplanes, tunnels, subways and much, much more. Pythagorean triplets find direct application in designing many things that surround us in everyday life.

And the minds of scientists continue to look for new versions of the proofs of the Pythagorean theorem.

• IN As a result of my work, I managed to:
• 1. Learn more about Pythagoras, his life, the Pythagorean brotherhood.
• 2. Get acquainted with the history of the Pythagorean theorem.
• 3. Learn about Pythagorean numbers, their properties, learn how to find them and apply them in practice.

Chervyak Vitaly Gennadievich

Krasnodar Territory, village Zhuravskaya, MOBU secondary school No. 14, grade 9

Pythagorean numbers

Supervisor: Manko Galina Vasilievna, mathematics teacher, MOBU secondary school No. 14

Literature.

1. Entertaining algebra. ME AND. Perelman (p.117-120)
2. www.garshin.ru
3. image.yandex.ru

4. Anosov D.V. A look at mathematics and something from it. – M.: MTsNMO, 2003.

5. Children's encyclopedia. - M .: Publishing house of the Academy of Pedagogical Sciences of the RSFSR, 1959.

6. Stepanova L.L. Selected chapters of elementary number theory. – M.: Prometheus, 2001.

7. V. Sierpinsky Pythagorean triangles. - M.: Uchpedgiz, 1959. S.111

Progress of research Historical page; Pythagorean theorem; Prove that one of the "legs" must be even and the other odd; Derivation of a pattern for finding Pythagorean numbers; Reveal the properties of Pythagorean numbers;

Introduction I heard about Pythagoras and his life in the fifth grade at a mathematics lesson, and I was interested in the statement "Pythagorean pants are equal in all directions." When studying the Pythagorean theorem, I became interested in Pythagorean numbers. I set the goal of the study: to learn more about the Pythagorean theorem and "Pythagorean numbers".

Truth will be eternal, how soon a weak person will know It! And now the theorem of Pythagoras Verne, as in his distant age

From the history of Pythagorean numbers. Ancient China Mathematical book Chu-pei: "If a right angle is decomposed into its component parts, then the line connecting the ends of its sides will be 5 when the base is 3 and the height is 4."

Pythagorean numbers among the ancient Egyptians Kantor (the largest German historian of mathematics) believes that the equality 3 ² + 4 ² = 5² was already known to the Egyptians around 2300 BC. e., during the time of King Amenemhat (according to papyrus 6619 of the Berlin Museum). According to Cantor, the harpedonapts, or "stringers", built right angles using right triangles with sides of 3; 4 and 5.

The Pythagorean theorem in Babylonia “The merit of the first Greek mathematicians, such as Thales, Pythagoras and the Pythagoreans, is not the discovery of mathematics, but its systematization and justification. In their hands, computational recipes based on vague ideas have become an exact science.

Each triangle, the sides are related as 3:4:5, according to the well-known Pythagorean theorem, is rectangular, since 3 2 + 4 2 \u003d 5 2. In addition to the numbers 3,4 and 5, there is, as you know, an infinite set of positive integers a , in and c, satisfying the relation A 2 + in 2 \u003d c 2. These numbers are called Pythagorean numbers

According to the Pythagorean theorem, these numbers can serve as the lengths of some right-angled triangle; therefore, a and b are called “legs”, and c is called the “hypotenuse”. It is clear that if a, b, c are a triple of Pythagorean numbers, then pa, p, pc, where p is an integer factor, are Pythagorean numbers. The opposite is also true! Therefore, we will first study only triples of coprime Pythagorean numbers (the rest are obtained from them by multiplying by an integer factor p)

Output! So, from the numbers a and b, one is even, and the other is odd, which means that the third number is also odd.

Here are the following Pythagorean triples: 3, 4, 5; 9+16=25 . 5, 12, 13; 25+144=169. 7, 24, 25; 49+576=625. 8, 15, 17; 64+225=289. 9, 40, 41; 81+1600=1681. 12, 35, 37; 144+1225=1369. 20, 21, 29; 400+441=841

It is easy to see that when multiplying each of the numbers of the Pythagorean triple by 2, 3, 4, 5, etc., we get the following triples. 6, 8, 10; 9,12,15. 12, 16, 20; 15, 20, 25; 10, 24, 26; 18, 24, 30; 16, 30, 34; 21, 28, 35; 15, 36, 39; 24, 32, 40; 14, 48, 50; 30, 40, 50 etc. They are also Pythagorean numbers

Properties of Pythagorean numbers When considering Pythagorean numbers, I saw a number of properties: 1) One of the Pythagorean numbers must be a multiple of three; 2) one of them must be a multiple of four; 3) And the other of the Pythagorean numbers must be a multiple of five;

Practical application of Pythagorean numbers

Conclusion: As a result of my work, I managed to 1. Learn more about Pythagoras, his life, the Pythagorean brotherhood. 2. Get acquainted with the history of the Pythagorean theorem. 3. Learn about Pythagorean numbers, their properties, learn how to find them. Experimentally-experimentally set aside a right angle using Pythagorean numbers.

» Honored Professor of Mathematics at the University of Warwick, a well-known 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 a right angle and integer sides. In the simplest of them, the longest side has a length of 5, the rest are 3 and 4. There are 5 regular polyhedra in total. A fifth-degree equation cannot be solved with fifth-degree roots - or any other roots. Lattices in the plane and in three-dimensional space do not have a five-lobe rotational symmetry; therefore, such symmetries are also absent in crystals. However, they can be in lattices in four-dimensional space 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) correlates with 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 after Pythagoras, but in fact its history is rather vague. Clay tablets suggest that the ancient Babylonians knew the Pythagorean theorem long before Pythagoras himself; the glory of the discoverer was brought to him by the mathematical cult of the Pythagoreans, whose supporters believed that the universe was based on numerical patterns. Ancient authors attributed to the Pythagoreans - and therefore to Pythagoras - a variety of mathematical theorems, but in fact we have no idea what kind of mathematics Pythagoras himself was engaged in. We don't even know if the Pythagoreans could prove the Pythagorean Theorem, or if they simply believed it was true. Or, more likely, they had convincing data about its truth, which nevertheless would not have been enough for what we consider proof today.

Evidence of Pythagoras

The first known proof of the Pythagorean theorem is found in Euclid's Elements. This is a rather complicated proof using a drawing that Victorian schoolchildren would immediately recognize as "Pythagorean pants"; the drawing really resembles underpants drying on a rope. Literally hundreds of other proofs are known, 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 collect them inside the square. With one laying, 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: the sum of the squares on the other two sides (plus the same four triangles). Now eliminate the triangles

The dissection of Perigal is another puzzle piece of evidence.

// Rice. 35. Dissection of Perigal

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

// Rice. 36. Proof by paving

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

Pythagorean triplets

In number theory, the Pythagorean theorem became the source of a fruitful idea: to find 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 doubled 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 triples, and he gave what might be called a formula for finding them all. Later, Diophantus of Alexandria offered a simple recipe, basically the same as Euclidean.

Take any two natural numbers and calculate:

their double product;

difference of their squares;

the sum of their squares.

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

Take, for example, the numbers 2 and 1. 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 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 in the Diophantine rule that has already been hinted at: having received 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 everything by 5.

If we switch to the language of algebra, the rule takes 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 get 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. Facets converge with each other on lines called edges; edges meet at points called vertices.

The culmination of the Euclidean "Principles" 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;

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 the 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 a dodecahedron and an icosahedron in crystals, although irregular dodecahedrons and icosahedrons sometimes come across there. True dodecahedrons can appear as quasicrystals, which are like crystals in every way, except that their atoms do not form a periodic lattice.

// Rice. 38. Drawings by Haeckel: radiolarians in the form of regular polyhedra

// Rice. 39. Developments of Regular Polyhedra

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

Equation of the fifth degree

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

In general, the equation of the fifth degree looks like this:

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

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

This assumption ultimately turned out to be wrong. Indeed, no such formula exists; at least there is no formula consisting of the coefficients a, b, c, d, e and f, composed using addition, subtraction, multiplication and division, as well as taking roots. Thus, 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 figure them out.

The first sign of a problem was that no matter how hard mathematicians tried to find such a formula, no matter how smart they were, they always failed. For some time, everyone believed that the reasons lie 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, generally any - is 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 for different values ​​of n) for powers of 1, 2, 3, and 4, but not for powers of 5. This is where the obvious pattern ends.

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

Crystallographic limitation

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

The atoms in a crystal form a lattice, that is, a structure that repeats periodically in several independent directions. For example, the pattern on the 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, the wallpaper is a two-dimensional crystal.

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

The order of symmetry of rotation is how many times you can rotate the body to a full circle so that all the details of the picture return to their original positions. For example, a 90° rotation is 4th order rotational symmetry*. The list of possible types of rotational symmetry in the crystal lattice again points to the unusualness of the number 5: it is not there. There are variants with rotational symmetry of 2nd, 3rd, 4th and 6th orders, but no wallpaper pattern has 5th order rotational symmetry. There is also no rotational symmetry of order greater than 6 in crystals, but the first violation of the sequence still occurs at the number 5.

The same happens with crystallographic systems in three-dimensional space. Here the lattice repeats itself in three independent directions. There are 219 different types of symmetry, or 230 if we consider the mirror reflection of the pattern as a separate version of it - moreover, 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 the crystallographic constraint.

In four dimensions, 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

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

Quasicrystals exist in nature. In 1984, Daniel Shechtman discovered that an alloy of aluminum and manganese can form quasi-crystals; Initially, crystallographers greeted his message with some skepticism, but later the discovery was confirmed, and in 2011 Shechtman was awarded the Nobel Prize in Chemistry. In 2009, a team of scientists led by Luca Bindi discovered quasi-crystals 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 various oxygen isotopes in the mineral with 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 the solar system was just emerging, and spent most of its time in the asteroid belt, orbiting the sun, until some kind of disturbance changed its orbit and brought it eventually to Earth.

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