The problem that rulers and rulers can't draw is the problem that they can't draw with rulers and rulers. Among them, the most famous is the classic problem known as the three difficult problems of geometry: ■ Angle trisection problem: bisecting any angle; ■ Cube problem: Make a cube, so that its volume is twice that of the known cube; ■ The problem of turning a circle into a square: make a square so that its area is equal to the area of a known circle. The above three problems were put forward in ancient Greece 2400 years ago, but they could not be solved under the restriction of Euclidean geometry. It was not until 1837 that the French mathematician Mancer proved for the first time that "angle bisection" and "bicubic" were not problems in drawing ruler. Then in 1882, German mathematician Lin Deman proved that π is a transcendental number, and "turning a circle into a square" was also proved to be a ruler drawing work.

Drawing with a ruler is not a problem. There are two other famous problems: regular polygon method. Use only rulers and compasses to make regular pentagons. Use only rulers and compasses to make a regular hexagon. Making a regular heptagon with a ruler and compasses is a seemingly simple problem, but many famous mathematicians are at a loss because a regular heptagon can't be made with a ruler. Using only rulers and compasses to make a correct nonagon is impossible, because using only rulers and compasses is not enough to divide an angle into three equal parts. Thinking of solving problems: Gauss got the regular drawing method of regular heptagon in his sophomore year, and gave the conditions for drawing regular polygons: the number of sides of the regular drawing polygon must be the product of the non-negative integer power of 2 and different Fermat prime numbers, which solved the unsolved problem for two thousand years. ■ Only compasses are allowed to divide the circumference of a known center into four parts. This question was refuted by Napoléon Bonaparte, which is a challenge to all French mathematicians.

Edit related extensions of this paragraph.

Draw with rusty compasses (that is, compasses with fixed radius) ■ Draw regular pentagons with only rulers and rusty compasses ■ Draw with rusty compasses, knowing two points A and B, and find a point C as a ruler of AB = BC = CA.

. ■ Given two points A and B, just use a compass with a fixed radius to find C, so that C is the midpoint of line AB. ■ Ruler drawing is based on the ancient Greek idea of "as simple as possible". Can you express it more concisely? Follow this line of thought, and you will have a more concise expression. /kloc-in the 20th century, a mathematician proposed to draw with a ruler and a compass with a fixed radius. 1672 proves that if "doing a straight line" is interpreted as "doing two points in a straight line", then what a ruler can do can be done only with a compass! Several cases of making a new point from a known point: two arcs intersect, a straight line intersects with an arc, and two straight lines intersect. If there is already a circle, then whatever the ruler can do, you can do it with the ruler alone! . A nonagon can be drawn with a ruler.

Edit this historical development?

"Gauge" is a compass and a tool for drawing circles. China ancient Oracle Bone Inscriptions has the word "gauge". "Moment" is like a square ruler used by carpenters now, which is formed by the intersection of two feet at right angles. The two are connected with wooden poles to make them firm. A short ruler is called a hook, and a long ruler is called a stock. The use of torque is an ancient invention of China. There is "inside the stone statue of Wuliangsi in Licheng, Shandong Province" which can be measured with scale, and can also replace compasses. There are also rectangular characters in Oracle Bone Inscriptions, which can be traced back to before Dayu's flood control (2000 BC). The second volume of Historical Records records that Dayu's water control is based on "the left is the right". Zhao Shuang's "Zhou Kuai" notes that "Yu governs water, ... looking at the shape of mountains and rivers, he decides to take part in the competition, ... it is necessary to apply the Pythagorean principle. This also shows that the moment originated in distant ancient China. There are also many works about rules in the Spring and Autumn Period. Volume 7 of Mozi says, "A lathe worker (a craftsman who makes cars) abides by his rules and makes the world a Fiona Fang. In the fourth volume of Mencius, it is said that "Li Lou (a legendary man with strong eyesight) lost his light (that is, the legendary Lu Ban). Rulers have been widely used in painting and making tools. Because China's ancient moments are calibrated, they are widely used and have great practicability. The ancient Greeks paid more attention to the role of rules and moments in training thinking and intelligence in mathematics, but ignored the practical value of rules. Therefore, the use of rules and moments is limited when drawing, and the problem of ruler drawing is put forward. It is only the limited use of scale-free rulers and compasses in painting. Anasargeras of ancient Greece first proposed that painting should be limited in size. He was imprisoned and sentenced to death because of political entanglements. In prison, he thought about turning a circle into a square and other related issues to pass the distressing idle life. He can't have a standard drawing tool, so he can only draw a circle with a rope and use a broken stick as a ruler. Of course, there can't be scales on these rulers. In addition, for him, time is running out, so he naturally thought of using a ruler to solve the problem in a limited time. Later, Euclid's Elements of Geometry specifically defined this law in theoretical form. Due to the great influence of geometric elements, the ruler painting advocated by the Greeks has always been observed and passed down. Due to the limitation of ruler drawing, some seemingly simple geometric drawing problems cannot be solved. The most famous are three classical drawing problems called three geometric problems in ancient Greece: cubic product problem, bisecting arbitrary angle problem and turning a circle into a square problem. At that time, many famous Greek mathematicians devoted themselves to studying these three problems. Although it can be solved with the help of other tools or curves, it has not been solved because of the limitation of ruler drawing. In the next two thousand years, countless mathematicians racked their brains for this and all ended in failure. It was not until 1637 that Descartes founded analytic geometry that there was a criterion for the possibility of drawing with a ruler. 1837, wanzil first proved that the cubic product problem and the problem of bisecting any angle belong to the impossible problem of drawing with a ruler. 1882, Lin Deman proved that π is an irrational number and it is impossible to change a circle into a square with a ruler.

Edit the progress promoted by this paragraph.

From the above, it can be seen that the three major problems of geometry can be easily solved if drawing tools are not restricted. Many mathematical achievements in history are by-products obtained in order to solve three major problems, especially the study of drawing with conic curve and ruler.

, found some famous curves, and so on. Not only that, the three major problems are also related to the branches of modern algebra such as equation theory and group theory. Geometric Ruler Drawing Problem "Geometric Ruler Drawing Problem" means that drawing is limited to the use of rulers and compasses. The ruler here refers to a ruler that can only draw straight lines without scale. The problem of geometric ruler drawing includes the following four problems: 1. Turn a circle into a square-find a square so that its area is equal to the known circle; 2. Divide any corner into three equal parts; 3. Double Cube-Find a cube, and make its volume twice that of the known cube. 4. Make a regular heptagon. The above four problems have puzzled mathematicians for more than two thousand years, but in fact, the first three problems have been proved impossible to be solved with a ruler and compass in a limited number of steps. The fourth problem was solved by Gauss in algebra. He also regarded it as a masterpiece of his life and told him to carve the regular heptagon on his tombstone. But later, his tombstone was not engraved with a heptagon, but with a 17 star, because the sculptor in charge of carving the monument thought that the heptagon was too similar to the circle, so everyone would be confused.