The roads on both sides of the street are often paved with bricks with geometric patterns. The shape of floor tiles is usually square and rectangular. We have also seen regular hexagonal floor tiles. Whether it is a square, rectangular or hexagonal floor tile, the middle of a floor can be paved without gaps or overlaps. This is called dense paving.

As we all know, when paving the floor, the floor should be covered, and there should be no gaps between the floor tiles. If the floor tile used is square, and each corner of it is a right angle, then four squares are put together, and the four corners of the common vertex just spell a 36-degree fillet. Every corner of a regular hexagon is 120 degrees. When three regular hexagons are put together, the sum of the three angles on the vertex of an ordinary hexagon is exactly 36 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 36 degrees.

It is precisely because the sum of several angles on the vertices of a square and a regular hexagon is exactly 36 degrees that the ground can be paved densely and beautifully.

1, regular triangles (equilateral triangles) and squares can be closely arranged, and each vertex of it has three regular triangles (equilateral triangles) and two squares.

2. Regular triangles (equilateral triangles) and regular hexagons can also be closely arranged, and each vertex has two regular triangles and two regular hexagons.

3. Square and regular octagon can also be densely laid, 1 square, with 2 regular octagons at each vertex.