The problem of square and circle is contemporary with the problem of Tyrus, which was studied by the Greeks.The famous Archimedes reduced the problem to the following form: given that the radius of a circle is r, the circumference is
, area is
。Therefore, if a right triangle can be made, the lengths of both sides of the right triangle are the perimeter of the known circle
And radius
, the area of the triangle is:
Equal to the area of a known circle.By thisright triangleIt is not difficult to make a square of the same area.But how to make the sides of this right triangle.That is, how to do oneline segmentArchimedes could not solve the problem by making its length equal to the circumference of a known circle.
In 2000, despite the fact thatThe problem of turning a circle into a squareHowever, some special curves have been found.Greeceantiphon (430 BC) proposed to solve this problemExhaustion method"Is the embryonic form of modern limit theory.It means that the circle is inscribed firstsquare(or regular hexagon), and then double the number of sides each time to get the inscribed 8, 16, 32,... polygons. He believes that the "last" regular polygon must becircumferenceCoincident, so the circle can be turned into square.Although the conclusion is wrong, it provides the requirementsCircular areaThe approximate method ofPiThe forerunner of the method, and ChinaLiu HuiOfCyclotomyCoincidentally, the exhaustion methodscientific methodThe establishment of has a direct impact.
Is a tall cylinder, rolling one circle on the plane, and the area of the rectangle obtained is just the area of the circle, so the area of the rectangle obtained
, and then convert the rectangle to equal productsquareOK.
It has been proved thatDrawing with ruler and gaugeUnder the condition of, there is no solution to this problem.
history
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In the 5th century BC, ancient Greek philosophersAnaxagoraBecause the sun is a big fireball, notApolloGod, guilty of "blasphemy" and was put into prison.In the court, Anaxagora complained: "There is no Apollo, the sun god! That shining ball is just a hot stone, probablypeloponnesian peninsulaSo big;Moreover, the moon that gave off clear light at that night, which was as bright as a big mirror, did not give off light itself, but only because it was illuminated by the sun. "As a result, he was sentenced to death.
In the days waiting for implementation, Anaxagora could not sleep at night.The round moon shone into the cell through the square iron window, and he became interested in the iron window and the round moon.He kept changing his observation position, and then he saw the circle ratiosquareBig, later I saw that the square is bigger than the circle.Finally, he said, "Well, even if the area of the two figures is the same."
Anaxagora made "a square so that its area is equal to the knownCircular area”As aDrawing with ruler and gaugeQuestion.At first, he thought that the problem was easy to solve. Who expected that he would spend all his time and get nothing.
Through good friends and politiciansPericlesAnaxagora was released from prison after multiple rescue efforts.He published the problems he thought of in prison. Many mathematicians were very interested in this problem and wanted to solve it, but none of them succeeded.This is the famous problem of "turning a circle into a square".
Figure 1. Hippocratic proof
Hippocrates 2000 years ago proved the area of the crescent, that is, Figure 1:
(semicircle
)
S (sector
）, so (crescent
)
(Triangle
)。
Triangles can be squared easily, so can crescents.His method is simple and clever, which makes people full of hope.untilLindermanProvedPiyesTranscendental numberLater, I realized that it was impossible.
Problem description
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Round to SquareThe complete description of the problem is[1]：
Given a circle, can you make a square in a finite number of times so that its area is equal to the area of the circle through the five basic steps described above
If the radius of a circle is set as the unit length, the essence of the problem of turning a circle into a square is to make the length
A line segment that is multiple of the unit length.
Impossibility proof
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After the three major problems of ruler and gauge drawing were proposed, there were manyPlane geometryBut before the 19th century, there was no complete solution.No one can give a solution to the problem of turning a circle into a square, but among those who begin to doubt its possibility, no one can prove that such a solution must not exist.Until the 19th century,GaloisandAbelOnly when the group theory was initiated to discuss the solution of the polynomial equation with rational coefficients, people realized the essence of these three problems.
1. The workability of ruler and gauge and the number of gauge
Studying variousDrawing with ruler and gaugeWhen the problem arises, mathematicians have noticed that whether a ruler can be used to make a specific figure or target is essentially whether it can make a consistent length.After introducing rectangular coordinate system and analytic geometry, length can be interpreted as coordinates.For example, to make a circle is actually to make the position (coordinate) of the center of the circle and the length of the radius.Making a specific intersection point or line is actually to find out their coordinates, slope and intercept.For this reason, mathematicians introduced the concept of ruler gauge workability.Suppose there are two known points on the plane
and
, with
Is unit length, ray
by
-The positive axis direction can establish a standard for the planeRectangular coordinate systemThe points in the plane can be represented by abscissa and ordinate, and the whole plane can be equivalent to.
Let E be a non emptysubset。If a straight line passes through
Two different points in
-Ruler can be used for short
-Yes.Similarly, if the center of a circle and a point on the circle are
The element in is
-Can be done.Further, if a certain point in the
It's some two
-The intersection point of a line or circle that can be made (line line, line circle and circle circle) is called a point
yes
-Can be done.This definition is based on five basic steps, including five basic methods to obtain new elements from known conditions in ruler and gauge drawing.If all
-The collective mark of points that can be made by ruler and gauge
, then when E contains more than two points,
Must be
True subset of.From a point set
At first, the points that can be made in one step form a set
The point that can be made after two steps is
,... and so on, the point set that can be made after n steps is
。All the point sets made from the E energy gauge are:
Another concept related to the workability of ruler and gauge is the rule number.Let H be a slave set
At first, the ruler can be used as a set of points: then the regular number is defined as the number represented by the abscissa and ordinate of the point in H.
It can be proved that,Rational number setIs a collection of all regular numbers
A subset of, and
It is also a subset of the set of real numbers.In addition, in order to discuss the problem in the complex number set, we will also consider the plane as a complex plane and define a complex number
Yes (multiple) regular number if and only if the point
The problem of turning a circle into a square means that when the unit length 1 is known
Length of.This is equivalent to making from 1
。However, the number z that can be made with ruler and gauge has correspondingMinimum polynomial。That is to say, there is a polynomial m with rational coefficient, so that
However, in 1882, Linderman and others proved that for Pi
For example, such a polynomial does not exist.Mathematicians call such numbersTranscendental number, and the number of corresponding polynomials is calledAlgebraic number。All regular numbers are algebraic numbers, and
No, it means that it is impossible to turn a circle into a square when drawing with a ruler.
Linderman proof
OfTranscendenceIt is called Linderman -Weierstrass theoremConclusion.Lindmann Weierstrass theorem shows that if several algebraic numbers
