How many ways are there to quadruple a square? I have countless ways!
How many ways are there to divide a square into four equal pieces, each block of exactly equal size, area, and shape (as shown in Figure 1)?
Figure 1 Square
Please think in your head or take a stroke to draw, how many methods do you have, 2 kinds? 3 types? 4 kinds?
Most people can think of the following three methods, one is to divide the square into 4 small squares, one is to connect the diagonal into 4 triangles, and the other is to divide the square into 4 rectangles (as shown in Figure 2).
Figure 2 Three aliquot methods are commonly used
Here I add a qualification, that is, if the 1 aliquot of the two divisions can coincide with each other, then it belongs to the same division, such as dividing the square into 4 rectangles vertically and 4 rectangles horizontally, one of which can be completely coincident, which should be regarded as a method (as shown in Figure 3).
Figure 3 Different divisions have the same result
There are six main methods to solve the problem, namely splitting, analogy, association, tracing the origin, generalized momentum theorem, and systematic thinking (as shown in Figure 4).
Figure 4 General methodology
This time we use the lenovo method to solve the problem of the 4-aliquot square, and lenovo is a very effective way to solve the problem, which helps us expand our thinking infinitely.
In addition to the above three methods of 4-equal squares, how many other methods can you think of? What inspired each approach?
I said I have countless ways to divide the square in four equals, can you believe it?
Let me explain some of the major methods obtained by association.
1) Think of the dichotomy
4 equal squares, I first thought of the dichotomy, dichotomy is a splitting method, used to reduce the difficulty of the problem, the difficulty of the problem is reduced, the problem is easy to solve. We divide the square into two from the middle, just need to divide the rectangle on the left into exactly two identical shapes, and the right side solves it according to the method of the left, and we get the method of 4 equal squares.
After being divided into rectangles by vertical lines, we only need to consider how to divide the rectangle into two identical shapes, which shapes can form rectangles? We might think of rectangles, squares, triangles, so we get 3 methods, the most commonly used three divisions, which can also be obtained by dichotomy (as shown in Figure 5).
Figure 5 Dichotomy aliquot 1
Thinking about it a bit, we also think that 2 trapezoids can also form rectangles, and the hypotenuses of the trapezoid can be various angles, which makes for an infinite number of ways (as shown in Figure 6).
Fig. 6 Dichotomy is equally divided into trapezoids
We can also think of the L-shape of tetris can also form a rectangle, and the L-shape variation is also diverse, and can also derive countless methods (as shown in Figure 7).
Fig. 7 The dichotomy is divided into L-shapes
These methods are just appetizers, and we look at the methods below.
2) Think of puzzles
We may also think of puzzles, puzzles can be through the same small pieces, spliced into various shapes, such as 4 different colors of small pieces, spelled out into a shape similar to a square, we look a little will find that each color bulge filled into a depression, will become a square, and the 4 small pieces are exactly the same, that is to say, we can construct the puzzle block way 4 equal squares (as shown in Figure 8).
Figure 8 Puzzle
The bulge of each small piece can fill its own depression, and the raised small piece can be a circle, square, triangle, or a combination of square and triangle (as shown in Figure 9).
Figure 9 Puzzle four equal squares
The bulge of the puzzle can be any shape, so we have an infinite number of methods of quarter squares.
3) Think of the tenon structure
We can also think of the tenon structure, or ask how two square pieces of wood are spliced together, and we can also think of the tenon structure. Some of the tenon structures have 2 pieces of wood that are almost identical (as shown in Figure 10).
Fig. 10 Tenon and tenon structure
We only need to make the 2 pieces of wood of the tenon structure exactly the same, and we get another method of quadrangle squares, and the joint of the tenon structure can be rectangular, triangular or other graphics (as shown in Figure 11).
Fig. 11 Four-aliquot square of mortise and tenon structure method
The connection point of the tenon structure can be any rectangle, triangle, or other shape, so that we have an infinite number of methods of four equal squares.
4) Associated with tiles
Square stuff, we can easily think of tiles, almost every home toilet and kitchen will have tiles. We also often see a square tile with a symmetrical pattern of up and down left and right in the middle (as shown in Figure 12).
Figure 12 Tile with symmetrical graphic in the middle
We turn the small circle in the middle 45 degrees, and then each small semicircle and the next figure can be combined into a new shape, and the 4 figures are still exactly the same, so we get another method of quartering squares (as shown in Figure 13).
Fig. 13 Tile method four-part square 1
The circle in the middle can be turned at different angles, such as 60 degrees, so that it can be combined into countless new shapes. The middle shape can be a square or a regular octagon (as shown in Figure 14).
Fig. 14 Tile method four-equal square
The middle figure is as long as it is a symmetrical figure up and down, left and right, quadrilateral, octagonal, hexagonal or other symmetrical patterns, etc., so that we get countless methods of quadrangle squares.
5) Cross rotation
We found that the method of connecting the diagonal and dividing into 4 small squares contains a cross, but the cross rotates around the center at different angles, so if the cross turns to other angles, is it okay?
We rotate the cross at different angles and find that the four small shapes obtained are exactly the same, then the cross rotation can be a method of quadratical squares (as shown in Figure 15).
Fig. 15 Four-aliquot square by cross rotation method
The cross can be turned into an infinite number of angles, so we have an infinite number of methods of quadratical squares.
6) Associated with symmetry
A square is a perfectly symmetrical figure up and down, left and right, so can we use the center point of the square to solve the problem of the four equal squares?
We draw a curve to one side through the center of the square, and then this curve is rotated 90 degrees, 180 degrees, and 270 degrees, respectively, so that the square is divided into four parts, and the exact same four parts (as shown in Figure 16).
Fig. 16 Center symmetry method four aliquot square
We can draw a curve of any shape for a day from the center of the square to a side, and then rotate it as above, and we can divide a square in four equal parts, and since the curve is made of countless shapes, we also have countless methods of dividing squares.
7) Dichotomy + symmetry
We can also solve the problem of the quartile square by dichotomy and symmetry combination. The rectangle on the left is symmetrical up and down, left and right, with a center point, we draw a curve from the center point to the side of the rectangle, and then this curve rotates 180 degrees around the center point, so that the rectangle is divided into 2 parts, and the 2 parts are exactly the same (as shown in Figure 17).
Fig. 17 Dichotomy + symmetrical quartile square
We can draw countless curves to one side through the center point of a rectangle, so that we have an infinite number of methods of quadratical squares.
8) Think of mosaics
Squares are also easily associated with mosaics, which are graphics made up of countless small squares. Small squares can also be combined into various shapes (as shown in Figure 18).
Figure 18 Mosaic
If we divide the squares into 8× 8 = 64 small squares, then we will study how to combine them (as shown in Figure 19).
Figure 19 Analyze 64 small squares
A quarter of a large square contains 16 small squares that can be combined into various shapes, and another three equal squares can also be combined into the same shape (as shown in Figure 20).
Figure 20 Mosaic method four-aliquot square
Thus we have an infinite number of methods of quadrating squares. Because 1/4 of the graphs of the other methods above are a complete graph, if the title has this implicit requirement, then the mosaic combination is not possible.
With the exception of the controversial eighth mosaic combination, each of the other major methods has evolved an infinite number of methods of quartile squares.
We can use a mind map to summarize the above method of the quadratical square (as shown in Figure 20).
Fig. 21 Mind map of a four-equal square
What four-part square methods do you think of?