Angular trisection problem
That is, any angle is divided into three parts. 1837, the French mathematician Wang Zell first proved that the problem of angle bisection could not be drawn with rulers and compasses in ancient Greece. However, the problem can still be solved if the restrictions of drawing tools are relaxed. Archimedes' method is considered to be the simplest method. He skillfully used rulers and compasses to solve the problem with only a little mark. The in-depth study of the problem of diagonal bisection has led to the discovery of many drawing methods and the invention of drawing tools.
Double cube problem
That is, make a cube, so that its volume is twice that of the known cube. This problem originated from Greek myths and legends in 2000: one said that the plague hit the island of Telo (an island in the Aegean Sea), and a prophet claimed that he had received an Oracle from God, and he had to double the volume of the cube Apollo altar before the plague could stop; Another said Minos of King Crete built a tomb for his son, but he had to double the volume, but still kept the shape of a cube. Both legends show that the problem of cubic doubling originated from the need of architecture. 1938+0837, Guo Jie mathematician Wang Zell proved that it was impossible to draw the problem of tripling with a ruler in ancient Greece. The triple research promoted the establishment and development of cone theory.
Quadrature problem of circle
That is, make a square so that its area is equal to the area of a known circle. This is one of the most interesting questions in history. As early as the 5th century BC, many people studied it. There is even a special term in Greek, which means "working on the problem of turning a circle into a square". 1982, the German mathematician Lin Deman proved that it was impossible to turn a circle into a square in ancient Greece. In this way, the problem that has not been solved for more than two thousand years has been solved. There are many ways to solve this problem if the restrictions of drawing tools are relaxed. Among them, Leonardo da Vinci, a famous scholar in the Renaissance, designed a cylinder with a base equal to the known circle and a height half of the radius of the known circle to roll on the plane. The area of a rectangle is equal to the area of a known circle. The study of changing a rectangle into a square with equal area urges people to calculate the value of pi by scientific methods, which has a direct impact on the establishment of scientific methods such as exhaustive method.