When the ancient Greeks said rulers, they meant rulers without scales. They feel from a lot of drawing experience that they can draw all kinds of geometric figures that meet the requirements with only two drawing tools: ruler and compass. Therefore, the ancient Greeks stipulated that rulers and compasses can only be used for a limited number of drawings, which is called ruler-compass drawing method. In the long-term drawing practice, people have made a large number of drawings that meet the given conditions according to the requirements of ruler drawing, and even some complex drawing problems can be made through limited steps. From the 6th century BC to the 4th century BC, the ancient Greeks encountered three painting problems that troubled them. Angle trisection problem: divide any given angle into three equal parts. Cubic product problem: find the side length of a cube, so that the volume of this cube is twice that of the known cube. Turn a circle into a square: find a square and make its area equal to that of a known circle. These are three famous problems in ancient geometric drawing, which were put forward before the publication of Geometry Elements. With the spread of geometric knowledge, they were later widely spread around the world. On the surface, these three problems are simple, and their drawing seems to be possible. Therefore, there are many people who have been engaged in the study of the three difficult problems of geometry for more than 2000 years. People also put forward various solutions, such as Archimedes, Pappus and others all found a good method of angle bisection and Boluo method to solve the cubic product problem. But all these methods, either do not conform to the ruler drawing method or approximate solution, can not be regarded as the solution of the problem. During this period, mathematicians also reformed the problem and found many problems closely related to the three major problems, such as finding the line segment to be equal to the circle, dividing the circle equally, inscribed the circle with a regular polygon and so on. But no one can think of a solution to the problem. In this way, the three painting problems have racked many people's brains, and countless people have tried countless times without success. Later, some people realized that the positive results were hopeless, and turned to the opposite side to wonder if these three problems were impossible for the rulers to do. Mathematicians began to think about which figures can be drawn with rulers and which can't. What is the standard? Where is the boundary? But this is still a very difficult problem. The discovery of Gauss turned the wheel of history to17th century. Descartes, a French mathematician, founded analytic geometry, which provided an algebraic research method for judging the possibility of drawing ruler and ruler, and made a new turning point for solving three major problems. The first breakthrough was the German mathematician Gauss. He was born in a poor family in Brunswick on April 30th, 1777. His grandfather is a farmer, his father is a day laborer and his mother is a mason's daughter. Neither of them has a school education. Because of his poor family, in winter nights, in order to save fuel and lamp oil, his father always lets his children go to bed after dinner. Gauss climbed into the attic, secretly lit the homemade radish oil lamp and read in the dim light. His childhood cleverness won the love of a duke. /kloc-was sent to Caroline College by the Duke when he was 0/5 years old, and 1795 came to study at the University of G? ttingen. Because of Gauss's diligence, in the second year after he entered school, he made a polygon of positive 17 according to the ruler drawing method. Then Gauss proved an important theorem of drawing with a ruler: If an odd prime P is fermat number, then a positive P- polygon can be drawn with a ruler, otherwise it cannot be drawn. It can be concluded that polygons with 3 sides, 5 sides and 17 sides can be made, but polygons with 7 sides, 1 1 sides and 13 sides cannot be made. Gauss not only made many outstanding achievements in mathematics, but also made important contributions in physics and astronomy. He is known as the "Prince of Mathematics". After Gauss died, according to his wishes, people carved a positive 17 polygon on his tombstone to commemorate his outstanding mathematical discovery when he was young. After the birth of analytic geometry, people know that straight lines and circles are the trajectories of linear equations and quadratic equations respectively. In algebra, the problem of finding the intersection points of straight lines, straight lines and circles, and circles is only a problem of solving linear equations or quadratic equations. Through finite addition, subtraction, multiplication, division and square root, the final solution can be obtained from the coefficients (known quantities) of the equation. Therefore, the question whether a geometric quantity can be made with a ruler compass is equivalent to whether a known quantity can be obtained by adding, subtracting, multiplying, dividing and squaring. In this way, on the basis of analytic geometry, combined with the experience of Gauss and others, people have a deeper understanding of the possibility of drawing with a ruler, and come to the conclusion that the line segments or points that can be drawn with a ruler can only be drawn through finite addition, subtraction, multiplication, division and square root (square correction and positive value). Three major problems: 1: bisecting any angle (0