This is the dense shop of plane graphics, also called the mosaic of plane graphics.
2. Floor with the same regular polygon. For a given regular polygon, can it be spliced into a plane figure without leaving gaps? Obviously, the key to the problem is to analyze the characteristics of the internal angles of regular polygons that can be used to completely pave the way. When the internal angles of several polygons spliced together around a point are added to form a fillet 360, a plane figure is paved. In fact, each internal angle of a regular N-polygon is (n-2) 180, which requires that each of the k regular N-polygons has an internal angle spliced at a point, just covering this point. So 360 = k (n-2) 180/n, and k is a positive integer, so n can only be 3,4,6. So we can only use regular triangle, regular quadrangle and regular hexagon tiles to pave the floor. As we know, the sum of the internal angles of any quadrilateral is equal to 360. So we use one. Please spell it.
We know that two or more regular polygons are used to lay floor tiles. Some identical regular polygons can cover the floor, while others can't. In fact, we also see many plane patterns composed of more than two equilateral regular polygons, such as those listed in textbooks. Why can these regular polygon combinations cover the floor densely? This problem is essentially whether the sum of intersection angles of related regular polygons can be combined into rounded corners.
We know that any congruent triangle and quadrilateral can be embedded in a plane (as shown in figure 1 and 2). Only special polygons with five or more sides can be plane mosaic. The number of sides of a convex polygon that can be inlaid on a plane is less than 7. For many years, it has become the dream of many mathematicians to find a special pentagon for plane mosaic.
Let several angles add up to 360. Speaking of easy, let's come back and see why any congruent triangle or quadrilateral can be inlaid in a plane. The graph 1 is a planar mosaic composed of congruent arbitrary triangles. After careful observation, we find that this figure is translated by a parallelogram composed of triangles 1 and 2. We call it a characteristic polygon. Fig. 2 is a characteristic polygon of plane mosaic of congruent arbitrary quadrilateral. It is found that the corresponding edges of these characteristic polygons are parallel. In other words, if the characteristic polygon can be properly divided, a polygon that can be embedded in the plane can be obtained.
As shown in fig. 3, a regular hexagon is a characteristic polygon that can be inlaid on a plane. If it is divided into three parts as shown in Figure 3, you can get a Pentagon that can be embedded in a plane. As shown in Figure 4, it is a characteristic polygon that can be inlaid in a plane. Divide it into four parts as shown in Figure 4, and get a pentagon that can be embedded in the plane. This is Marjory, a woman from San Diego? 6? 1 rice 1977 found it.
If a set of figures with parallel edges is allowed, it will be too much for plane mosaic. It was the carpenter who put this wood together into a big board.