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Photo Source: Pixabay intuition tells us that given a circle, there must be a square with an area equal to it. But how to draw this square? This problem of "turning a circle into a square" has perplexed mathematicians for thousands of years. They first proved that it is impossible to turn a circle into a square only by drawing with rulers, and then they thought: can you divide the circle into a limited number of pieces and then splice these pieces into squares? Now there are really mathematicians who have realized this.

The difficult problem of drawing rulers and compasses around 450 BC, the ancient Greek mathematician Anaxagoras raised an interesting geometric question: can a square equal to the area of a given circle be made only with rulers and compasses? This seemingly simple problem of "turning a circle into a square" has become a classic problem in the field of ruler mapping. In the following 2000 years, many mathematicians tried to solve it, but failed.

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 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. Can we use a ruler to make this square with side length? the corresponding algebraic problem is: given 1, can √ π be obtained by finite addition, subtraction, multiplication and division and square operation?

This question was not finally answered until 1882. The German mathematician Ferdinand von Linderman (Ferdinand von Lindemann) proved that π (and its square root) is a transcendental number, that is, it is not the root of any rational coefficient polynomial equation and cannot be obtained by finite algebraic operations. Therefore, there is no solution to the equivalent algebra problem, so there is no solution to the problem of turning a circle into a square under the drawing of a ruler.

Mathematicians in the 16th and 17th centuries tried to figure the circle into a square by solving ruler gauges. (photo Source: Middle Temple Library / Science Source) A square ruler cannot turn a circle into a square, but what if you put aside the limitation of the ruler? In 1925, the Polish-American mathematician Alfred Tarski proposed another version of the circle-to-square problem: can a given circle be divided into a finite number of pieces and then reassembled into a square? There can be no surplus in the fragments, there can be no gap in the spliced square, and the area of the circle and square should be equal.

This version of the problem is less limited by ruler mapping, but requires not only that the area of a square be equal to that of a given circle, but that each part of it comes from a given circle-which is closer to the literal meaning of "turning a circle into a square". And in both versions of the problem, the number of operations allowed is limited, so the new problem of turning a circle into a square is still a difficult problem.

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.

In 2016, Lukash Grabowski of the University of Lancaster in the UK and Andr á s M á th é and Oleg Pikhurko of the University of Warwick published a paper showing how to divide a circle. In their proof, most of the fragments obtained by dividing the circle have a clear shape. They also found that these pieces cannot be spliced together to form a complete square, and that there are still some small "gaps" that need to be filled with extra pieces. But these fragments are so small that they have no area, which mathematicians call a "set of zero measures".

"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.

Divide the circle and square of the same area into a pile of the same pieces so that the two can be transformed into each other.

This gives mathematicians room to continue to improve the method of turning circles into squares. In a previously published preprint paper, Matai, Pikurco and Jonathan Noel of the University of Victoria in Canada split the circle into about 10200 pieces that were simpler and easier to visualize and reorganized them into squares. Mathematicians still want to further simplify these fragments, especially to reduce the number of fragments. Max has done some computer experiments that show that the process of turning a circle into a square can be done with only 22 pieces, and he even thinks that the number can be reduced, although there is no proof.

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

Reference link:

Https://www.quantamagazine.org/an-ancient-geometry-problem-falls-to-new-mathematical-techniques-20220208/

This article comes from the official account of Wechat: global Science (ID:huanqiukexue), written by Bai Defan, revision: Erqi

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