While waiting for the execution, he was still thinking about the universe and everything, including mathematics. One night, he saw a full moon shining into his cell through a square fence. His heart beat, and he thought: If the area of a circle is known, how can we make a square so that its area is exactly equal to the area of this circle? This question seems simple, but it stumbles Alasagora. In ancient Greece, painting tools were restricted, and only rulers and compasses were allowed.
Alaxagora has been thinking about this problem, even forgetting whether he is still a prisoner to be executed. Later, he was rescued by his good friend Perikles (an outstanding politician at that time) and released from prison. However, this problem was not solved by himself, nor by the mathematicians in ancient Greece, and became one of the three famous geometric problems in history. In the next two thousand years, there were countless mathematics to prove this point, but they never got the answer.
Second, cubic product. This problem is also one of the three difficult problems in geometry. According to legend, in ancient Greece, there was a plague on an island called Tyros one year. Residents of the island went to the temple to pray for Zeus and asked how to avoid disaster. Many days passed and the wizard finally conveyed the will of the gods. It turned out that Zeus thought people were not religious enough for him and his altar was too small. If you want to avoid the plague, you must make a new altar twice the size of this altar, and you can't change the shape of the cube. So people quickly measured the size and doubled the length, width and height of the altar. The next day, they dedicated it to Zeus. Unexpectedly, the plague has not stopped, but has become more popular. The people of Tiros panicked and prayed to Zeus again. The wizard once again conveyed Zeus' wishes. It turns out that the volume of the new altar is not twice that of the original altar, but eight times. Zeus felt that Tyrus was against his will, so he was even more angry. Of course, this is just a legend, but it is true that there is no answer to this question so far.
The problem is that just making a cube with a compass and a ruler without scales makes the volume of this cube twice that of the known original cube. Because no one has answered it so far, it has become the second biggest problem in geometry.
Third, the angle is divided into three parts. There is another legend about this problem. It is said that Alexandria in Egypt was a famous and prosperous capital in the 4th century BC. There is a round villa in the suburb of the city, where a princess lives. There is a river in the middle of the round villa, and the house where the princess lives is just built in the center of the circle. A door was opened on the north and south walls of the villa and a bridge was built over the river. The location of the bridge is just in a straight line with the north gate and the south gate. Every day, the items that the king gave the princess were sent in from the north gate, first put in the warehouse at the south gate, and then the princess sent someone to retrieve the room from the south gate. From the north gate to the princess mansion, and from the north gate to the bridge, the length of the two roads is exactly the same. The princess also has a younger sister, the little princess, and the king will build a villa for her. The little princess suggested that her villa should be built exactly like her sister's. The little princess's villa will be started soon. However, when the craftsmen decided on the location of the bridge and the north gate after the completion of the south gate, they found a problem: how can we make the distance from the north gate to the bedroom the same as the distance from the north gate to the bridge? In the end, craftsmen found that in order to have an equal distance, we must first solve the problem of equal division. As long as the problem can be solved, the position of the bridge and the north gate can be determined.
So the craftsmen tried to draw the position of the bridge with rulers and compasses, but after a long time, they still didn't solve it. In desperation, they had to consult Archimedes, the most famous mathematician at that time. Archimedes saw this problem and thought for a long time. He made a fixed mark on the ruler and solved the problem easily. Everyone admires him very much. But Archimedes said that this problem has not really been solved. Because once a ruler is marked, it is equal to marking, which is not allowed in ruler drawing. So this problem puzzled countless mathematicians for 2000 years, until more than 100 years ago, the German mathematician Klein made an unquestionable proof: it is impossible to solve these three problems only with rulers and compasses. In other words, this problem has not really been solved so far.