ONE PIECE,
This is the dense shop of plane graphics, also called the mosaic of plane graphics.
2. Floor with the same regular polygon. Can a given regular polygon be put together into a plane figure without leaving any points?
Some space? Obviously, the key to the problem is to analyze the internal angle characteristics of regular polygons that can be used to completely pave the way.
When the internal angles of several polygons add up to form a fillet 360, a plane figure is paved. Actually, there are n sides.
Each internal angle of the shape is (n-2) 180, which requires that each internal angle of k regular n-polygons just covers the ground, so it is 360.
=k(n-2) 180/n, and k is a positive integer, so n can only be 3,4,6. So the floor is paved with the same regular polygon floor tiles.
Only regular triangle, regular quadrangle and regular hexagon floor tiles can be used. As we know, the sum of the internal angles of any quadrilateral is equal to 360.
Therefore, a group of irregular quadrilateral tiles with the same shape and size can also be paved into a seamless floor. Use any same triangle.
Can the shape cover the ground? Please spell it.
It is well known that two or more regular polygons are used to lay the floor. Some identical regular polygons can cover the ground, but
Some can't. In fact, we also see many plane patterns composed of more than two equilateral regular polygons, such as textbooks.
There are several situations listed in the table. Why can these regular polygon combinations be densely laid on the ground? This problem is essentially the intersection of related regular polygons.
The question of whether the sum of the angles at the intersection can be combined into a fillet.
We know that any congruent triangle and quadrilateral can be embedded in a plane (as shown in figure 1 and 2). And those greater than or equal to five sides.
Only special polygons can be embedded in a plane. The number of sides of a convex polygon that can be inlaid on a plane is less than 7. Over the years, looking for special
Plane mosaic of pentagons has become the dream of many mathematicians.
Let several angles add up to 360. Speaking of easy, let's come back and see why we are congruent with arbitrary triangles and quadrilaterals.
You can make a plane mosaic. The graph 1 is a planar mosaic composed of congruent arbitrary triangles. After careful observation, we found that this
The figure is translated by a parallelogram composed of triangles 1 and 2. We call it a characteristic polygon. Figure 2 congruence
Characteristic polygons embedded in arbitrary quadrilateral plane. It is found that the corresponding edges of these characteristic polygons are parallel. in other words
That is to say, if the characteristic polygons can be properly divided, polygons that can be embedded in the plane can be obtained.
As shown in Figure 3, a regular hexagon is a polygon that can be embedded in a plane. If it is divided into three parts as shown in Figure 3, it can be flattened.
A Pentagon inlaid with faces. As shown in Figure 4, it is a characteristic polygon that can be inlaid in a plane, and it can be obtained by dividing it into four parts as shown in Figure 4.
A pentagon with a plane mosaic. This is Marjory, a woman from San Diego? The meal was found on 1977.
If a set of figures with parallel edges is allowed, it will be too much for plane mosaic. The carpenter just pieced the wood together piece by piece.
Put together a big board.
Hope to adopt! thank you