The problem of how to turn a circle into a square has perplexed mathematicians for thousands of years.

The reason why this problem is difficult to answer is that it is not only a geometry problem, but also an algebra problem. In the problem of drawing with rulers and compasses, given a number of angles or length of line segments, the essence is to give some real numbers, while the rule of drawing with non-graduated rulers and compasses ensures the length of the angles or segments made, which is the combination of sum, difference, product, quotient and square root of a given real number. As a result, each ruler drawing problem actually corresponds to an algebraic problem.

Back to the problem of turning a circle into a square. Suppose that the radius of a given circle is unit length 1, the area of the known circle is π, and the side length of a square whose area is equal to it should be π. When asked whether the square with side length can be made with a ruler, the corresponding algebraic problem is: 1, can we get square π by finite addition, subtraction, multiplication and division and square operation?

Weight the circle into a square

Ruler drawing cannot turn a circle into a square, but if you put aside the limitations of ruler mapping, in 1925, Polish-American mathematician Alfred Tarski (Alfred Tarski) proposed another version of the circle into square problem: can a given circle be divided into a finite number of pieces, and then these pieces can be reassembled into a square piece with no surplus, and the spliced square cannot have a gap? The area of the circle and the square should be equal.

Tarski's question was answered in 1990. The Hungarian mathematician Mikl ó s Laczkovich proved that circles can be divided and reorganized into squares. But this kind of segmentation method can not be cut out by the usual scissors, and the segmented fragments have extremely complex irregular shapes. Ratskovich transformed this geometric problem into a graph theory problem. He used two different vertices sets to draw a graph (graph)-- one vertex set corresponds to a circle, the other corresponds to a square-- and then established an one-to-one correspondence between the two vertex sets. In the end, he proved that by dividing a circle into up to 1050 pieces, they could be spliced into squares without even having to rotate the pieces.

However, Ratskovich's proof did not end the new problem of turning a circle into a square. In fact, he only proved the existence, that is, proved that it is feasible to turn a circle into a square, but failed to give a specific mode of operation, nor the shape of each fragment. Through his proof, we don't know how the circle is divided. In order to intuitively understand the process of turning a circle into a square, mathematicians still need to go back to geometry and give a clear description of the shape of each fragment.

Visualization attempt

"almost all the parts of the square have been spliced out, and you can't even draw the missing part, because it looks like it has no shape." Andrew Marks, a mathematician at the University of California, Los Angeles, commented that although this circle-squaring method still requires additional fragments, it is still a dramatic step forward.

A year later, Max and Spencer Unger, now at the University of Toronto in Canada, made improvements and came up with the first really effective way to square a circle. They split the circle into as many as 10200 pieces and reassembled it into a square without leaving a gap in the zero measure. The shapes of these fragments are still very complex, and although they are mathematically clearly described, they are difficult to visualize.

"I bet you can reassemble the circle into a square in less than 20 pieces," Max said, "but I won't bet too much."