How to make interesting mathematical manuscripts? The following are my carefully collected pictures of mathematical manuscripts, which are simple and beautiful. Let's see them together ~

The pictures of the first-grade math handwritten newspaper are simple and beautiful.

First-grade math handwritten newspaper pictures 1 interesting math stories? Four-color problem? certificate

? Four-color problem? It is a very famous proof problem in the history of world mathematics. It requires to prove that the colors between adjacent areas of any complex shape cannot be repeated as long as four colors are used on the plane map, that is to say, there can only be four bounded areas. In the past 150 years, many mathematicians have spent a long time and a lot of energy to prove this problem. A few days ago, it was reported in the newspaper that several college students spent more than ten hours assembling several electronic computers to prove this problem. I knew this more than twenty years ago? Four-color problem? , but has been unable to find a way to prove this. Now I'm exposed to it? Topology? , actual use? Topology? Principle one analysis, four-color problem? Just like Euler said. Seven bridges? Seven line segments that pass through four points and are not repeated? A stroke? Simple enough that even ordinary pupils can prove it.

According to? Topology? In principle, every area of any complex shape can be regarded as a point, and if there is a boundary between two areas, it can be regarded as a line between these two points. As long as it is proved that there are no more than four points with straight lines on a plane, it is proved? Four-color problem? .

Any point A on the plane can be connected with many points B, C, DX, Y and Z (as shown in figure 1), and point B can also be connected with other points, and points C, DX, Y and Z can also be connected with other points. But there is a principle: lines cannot cross each other, because once they cross, one line will cut off another line (as shown in Figure 2), and BC's line will cut off AD's line. But some people will say: there can be many lines between two points, and the AD line can bypass point B or point C (as shown by the dotted line in Figure 2), so there is no intersection. But such winding will produce a result: the points outside the closed figure become the points inside the closed figure. The following is to prove that there are no more than four interconnected points through the analysis of closed figures.

First-grade math handwritten newspaper picture 2

A point itself or a connecting line between two points can form one or more closed figures (as shown in Figure 3). From point A to point B, then to point C, and then back to point A (as shown in Figure 4), the three connected points will inevitably close the graph. If there are more than four points on the closed graph (as shown in Figure 5), the connection between each point and the first point A of the third point C will divide the whole closed graph into many small closed graphs. It is concluded that (1) any three connected points on the same plane must form at least one closed graph. In addition, we call it the three-point connection closure law.

Any fourth point on the plane can be inside or outside the closed figure formed by the three-point connection line (as shown in points D and D in Figure 6). Point), point D can be connected with points A, B and C respectively, D? Points can also be connected with points a, b and c respectively. The connecting line between point D and points A, B and C divides the closed figure ABC into three small closed figures, D? Of the three connecting lines between point A, point B and point C, one must be sandwiched between the other two lines. In figure 6, d? Line a is d? B-line sum

d？ Line c is caught in the middle and point a is closed. Figure BCD? Closed, which is the same as point D in the closed graph ABC. It is concluded that one of any four interconnected points on the same plane must be surrounded by a closed graph formed by the connection of the other three points. In addition, we call it the four-dotted line demarcation method.

First-grade math handwritten newspaper picture 3

So, is there a fifth point on the plane that can be divided into two parts? If it is related to the fourth point d, it must be in the closed graph ABC. Secondly, the fifth point cannot fall on every connection, otherwise the connection will be cut off. The fifth point can only be located in E 1, E2 and E3 (as shown in Figure 7), and the points in these three positions can only be connected with three points on the small closed graph surrounding it, but not with the fourth point. If there is a connection, all other connections will be cut off. It is concluded that only four points on the same plane can be connected with each other. If the fifth point is connected with these four points, the connection between two of them will be interrupted. In addition, we call it the five-point connection law. Is that what you need to prove? Four-color problem? .

The above are proved on the same plane? Four-color problem? . If the area map is distributed on the surface of a three-dimensional shape (such as a globe), we can regard the three-dimensional shape as a plane shape according to the basic principles of topology, and regard point D in Figure 6 as the front of the plane and point D' as the back of the plane. If there is a connection between these two points, there can be no connection unless they pass through a plane or through a space outside the surface of a three-dimensional shape. This three-dimensional shape can be any shape as long as there is no hole in the middle, because no matter how angular and uneven your surface is, it is the same as a sphere in topology, just like a balloon can be any shape before it is inflated, and it always approaches the sphere after it is inflated. But the situation with a hole in the middle of a three-dimensional shape is different. In the end, it will not become spherical, but will only become a tire-shaped ring of the wheel. The fourth point in front and the fifth point in the back can be connected through the middle hole. As mentioned above, when crossing from the space outside the solid surface, the crossing part actually forms a ring with the original solid shape, and finally it can also become the shape of a wheel inner tube. Therefore, it is concluded that only four points on a three-dimensional surface can be connected with each other without holes in the middle.