Regular hexagons can be densely laid, because each internal angle is 120, and each splicing point can just accommodate three internal angles; Regular pentagons cannot be densely laid, because each internal angle is 108 degrees, and 360 degrees is not an integer multiple of 108, so there is no gap and overlap in the internal angles of each splicing point. Except for regular triangles, regular quadrangles and regular hexagons, other regular polygons are not allowed to closely arrange planes.
As we all know, we should cover the floor when paving it, so that there can be a gap between the floor tiles and the tiles. If the floor tile used is square, and each corner of it is a right angle, then put four squares together, and the four corners of the vertex of an ordinary square are just put together to form a 360-degree fillet.
Each angle of a hexagon is 120 degrees. When three regular hexagons are put together, the sum of the three angles on the vertex of the common hexagon is exactly 360 degrees. Besides squares and rectangles, regular triangles can also be densely laid on the ground. Because every internal angle of a regular triangle is 60 degrees, when six regular triangles are put together, the sum of the degrees of the six angles at the vertices of the common triangle is exactly 360 degrees.
It is precisely because the sum of several angles on the vertices of a square and a regular hexagon is exactly 360 degrees that the ground can be paved densely and beautifully.
In fact, the understanding of crystal structure can not be separated from the problem of geometric density. For a single regular polygon, only regular triangles, squares and regular hexagons can be used, and the symmetry axes involved are only 1, 2, 3, 4 and 6. However, if various polygons are used for dense paving, there may be 5 or 7 or more symmetry axes.
This problem was put on the table by China mathematician Wang Hao in 196 1 year. In 1976, mathematician Penrose constructed the most classic dense pattern with two different diamonds (36/144,72/108).