Squaring a Circle
Given a circle with radius , we would like to know whether we can obtain a square whose area is the same as the area of the circle by Euclidean construction. It is easy to see that the length of each side of such square is . In other words, the problem reduce to determining whether is a constructible number. To simplify the problem further, since we can take square and square root by Euclidean construction, then it is equivalent to see whether is constructible or not.
The problem of squaring a circle has been of great interest to mathematicians in the history of mathematics. One of them is Hippocrates, who had a very interesting strategy to tackle the problem. However, his method was doomed to failure because in 1880, a German mathematician called Ferdinard von Lindemann, completely settled the problem by showing that is not a constructible number!
