Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.
It had been known for decades that the construction would be impossible if π were transcendental (not an algebraic irrational number or the root of any polynomial with rational coefficients), which was first proved in 1882 by Lindemann–Weierstrass theorem. The expression ‘squaring the circle’ is sometimes used as a metaphor for trying to do the impossible.
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The Daily Omnivore 