In the history of mathematics, three problems have been firmly established for more than 2000 years with amazing power. Elementary geometry has been studied for at least 3000 years. During this period, scholars who strive to develop elementary geometry have encountered many problems, but these three problems have been puzzling scholars. The problems are "cubic product", "turning a circle into a square" and "angle trisection". Because these three issues stand firm, they are now collectively referred to as "three major issues."

Cubic double product

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 a new altar, making each side twice as long as the old altar. However, the plague has not stopped, but has become more rampant, which surprised and frightened them. 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.

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 you can make a right-angled triangle, and the lengths of the two sides between the right angles are the perimeter 2πr and radius r of the known circle, then the area of the triangle is

( 1/2)(2πr)？ = the square of π r

Equal to the area of a 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 is not a problem that Archimedes can solve.

The third division of an angle

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.