Legend has it that around 400 BC, an epidemic prevailed in Athens, ancient Greece. In order to eliminate the disaster, people turned to Apollo, the sun god, for help. Apollo demanded that the volume of the cube altar in front of his temple must be expanded by 1 times, otherwise the epidemic would continue to prevail. People were puzzled and had to ask Plato, the greatest scholar at that time. Plato felt powerless. This is one of the three major geometric problems in ancient Greece. Expressed in mathematical language is: given a cube, find a cube so that its volume is twice that of the known cube. Two other famous problems are bisecting any angle and turning a circle into a square. The three major geometric problems in ancient Greece are both fascinating and very difficult. The beauty of the problem is that the form is very simple, but in fact it has profound connotation. It is required that only compasses and scaleless rulers can be used for drawing, and only rulers and compasses can be used for a limited number of times. But the basic figures that rulers and compasses can make are: draw a straight line, draw a circle, draw the intersection of two straight lines, draw the intersection of two circles, and draw the intersection of a straight line and a circle. A certain figure is operable, that is, from several points, the limited number of the above basic figures is compounded. The idea of modern algebra is implied in this process. After more than 2000 years of hard exploration, mathematicians finally figured out that these three classic problems are "drawing problems that can't be completed with a ruler". Realizing that some things are really impossible is a big leap in mathematical thinking. However, once the conditions of drawing are changed, the problem becomes different. For example, if there are scales on the ruler, it is possible to fold the cube in half and divide it equally at any angle. Mathematicians have deduced many stories on these issues. Until recently, a mathematician from China and an ambitious middle school student successively solved two drawing problems about "rusty compasses" (compasses with fixed radius) put forward by the famous American geometer Pedro, which added a wonderful stroke to ruler drawing. Or describe it as follows: these are three drawing problems. Only use compasses and rulers to solve the following problems. Until 1 9th century, it proved impossible:1. Cubic product, that is, find the edge of a cube so that the volume of the cube is twice that of a given cube. 2. To turn a circle into a square is to make a square so that it is equal to a given circle area. 3. Angle trisection, that is, dividing a given arbitrary angle into three equal parts.

Edit the cubic product of this paragraph.

There is a myth about triple multiplication: when the plague was prevalent in Delos, Greece, the residents were afraid and prayed to Apollo, the patron saint of the island. The prophetic nuns in the temple told them God's instructions: "Double the cube altar in front of the temple and the plague can stop." This shows that this great god likes mathematics very much. The residents were very happy after receiving this instruction, and immediately began to build new altars, making each side twice as long as the old altar, but the plague did not stop, but it became more rampant.

Let them all be surprised and scared. As a result, a scholar pointed out the mistake: "When the edge is doubled, the volume becomes eight times. God wants twice, not eight times." Everyone thought this statement was correct, so they changed it to a god and put two altars with the same shape and size as the old ones, but the plague was still not eliminated. People are puzzled and ask God again. This time, God replied, "The altar you made is indeed twice the size of the original one, but its shape is not a cube. I want it to be twice as big and cubic in shape. The residents suddenly realized that they went to Plato, a great scholar at that time, for advice. It was enthusiastically studied by Plato and his disciples, but it was never solved, which consumed the brains of many mathematicians in later generations. Because of this legend, the cubic product problem is also called Tiros problem.

Edit this paragraph and turn a circle into a square.

The problem of Fiona Fang is contemporary with that of Tyrus studied by the Greeks. The famous Archimedes transformed this problem into the following form: it is known that the radius of a circle is r, the circumference is 2πr and the area is πr2. Therefore, if we can make a right-angled triangle, and the lengths of the two sides between the right angles are the perimeter 2πr and the radius r of the known circle respectively, then the area of the triangle is (1/2)(2πr)(r)=πr2, which is equal to the area of the known circle. It is not difficult to make a square with the same area from this right triangle. But how do you make the sides of this right triangle? That is, how to make a line segment equal to the circumference of a known circle, Archimedes could not solve it.

Edit this paragraph and divide it into three parts.

The problem of bisecting any angle may appear earlier than those two problems, and there is no relevant record in history. But there is no doubt that its appearance is natural, and even we can think of it ourselves now. Greek mathematicians had already thought of the method of bisecting any angle as early as 500-600 BC, just as we learned in geometry textbooks or geometric paintings: take the vertex of a known angle as the center and an appropriate radius as both sides of the arc intersection angle to get two intersection points, and then draw an arc with an appropriate length as the center and the intersection points of the two arcs are connected with the vertex to divide the known angle into two halves. Since it is so easy to bisect a known angle, it is natural to change the question slightly: How about bisection? In this way, this problem naturally arises.

The editing results of three geometric problems in this paragraph and their significance

How to prove the results of three difficult problems in ancient Greek geometric drawing, that is, turning a circle into a square, cubic product and angle trisection? Let's discuss it with questions. (1) As we all know, turning a circle into a square was put forward by the famous ancient Greek scholar Anaquel Sa Gollers, but Anaquel Sa Gollers failed to solve his own problems all his life. In fact, the side length of the square in this problem of turning a circle into a square is the arithmetic square root of the circle area. Let's assume that the radius of a circle is 1, then the side length of a square is the root sign π. Until 1882, the problem of changing a circle into a square finally got a reasonable answer. German mathematician Lindemann (1852 ~ 1939) successfully proved in this year that pi = 3. 14 15926 ... is a transcendental number, so it is impossible to make a transcendental number by drawing with a ruler, so the problem of turning a circle into a square by drawing with a ruler is solved. German mathematician Lin Deman

(2) The result of the problem of cubic product and angle bisection. Until 1830, the French mathematician Galois, 18 years old, created a theory that was later named "Galois Theory". This theory can prove that cubic product and angle trisection are problems that ruler drawing can't do. 1837, the French mathematician Vanzer (18 14 ~ 1848) finally proved that it is impossible to draw the angle trisection and cubic product with a ruler.

(3) The Significance of Three Geometric Drawing Problems Although all three geometric drawing problems have been proved impossible to be done by drawing with a ruler, mathematicians have made wave after wave of explorations to solve these problems, and finally got many new achievements and found many new methods. At the same time, it reflects that mathematics, as a science, is a vast and deep ocean, and there are still many unknown mysteries waiting for us to discover.