Squaring the circle

Impossibility[edit]

The solution of the problem of squaring the circle by compass and straightedge requires the construction of the number ${\displaystyle {\sqrt {\pi }}}$, the length of the side of a square whose area equals that of a unit circle. If ${\displaystyle {\sqrt {\pi }}}$ were a constructible number, it would follow from standard compass and straightedge constructions that ${\displaystyle \pi }$ would also be constructible. In 1837, Pierre Wantzel showed that lengths that could be constructed with compass and straightedge had to be solutions of certain polynomial equations with rational coefficients.^[23][24] Thus, constructible lengths must be algebraic numbers. If the circle could be squared using only compass and straightedge, then ${\displaystyle \pi }$ would have to be an algebraic number. It was not until 1882 that Ferdinand von Lindemann proved the transcendence of ${\displaystyle \pi }$ and so showed the impossibility of this construction. Lindemann's idea was to combine the proof of transcendence of Euler's number ${\displaystyle e}$, shown by Charles Hermite in 1873, with Euler's identity $${\displaystyle e^{i\pi }=-1.}$$ This identity immediately shows that ${\displaystyle \pi }$ is an irrational number, because a rational power of a transcendental number remains transcendental. Lindemann was able to extend this argument, through the Lindemann–Weierstrass theorem on linear independence of algebraic powers of ${\displaystyle e}$, to show that ${\displaystyle \pi }$ is transcendental and therefore that squaring the circle is impossible.^[16][17]


Bending the rules by introducing a supplemental tool, allowing an infinite number of compass-and-straightedge operations or by performing the operations in certain non-Euclidean geometries makes squaring the circle possible in some sense. For example, Dinostratus' theorem uses the quadratrix of Hippias to square the circle, meaning that if this curve is somehow already given, then a square and circle of equal areas can be constructed from it. The Archimedean spiral can be used for another similar construction.^[25] Although the circle cannot be squared in Euclidean space, it sometimes can be in hyperbolic geometry under suitable interpretations of the terms. The hyperbolic plane does not contain squares (quadrilaterals with four right angles and four equal sides), but instead it contains regular quadrilaterals, shapes with four equal sides and four equal angles sharper than right angles. There exist in the hyperbolic plane (countably) infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area, which, however, are constructed simultaneously. There is no method for starting with an arbitrary regular quadrilateral and constructing the circle of equal area. Symmetrically, there is no method for starting with an arbitrary circle and constructing a regular quadrilateral of equal area, and for sufficiently large circles no such quadrilateral exists.^[26][27]