Squaring the circle

This square and circle have the same area.

Squaring the circle is the problem proposed by ancient geometers of using a finite ruler-and-compass construction to make a square with the same area as a given circle. The Greek geometer Oenopides was among the first to lay down the restriction of the means permissible in constructions as the ruler and compass which became a canon of Greek geometry for all plane constructions."^[1] In 1882, it was proved to be impossible to square a circle using only a straightedge and compass. The term quadrature of the circle is synonymous.

1 Impossibility
2 Transcendence of π
3 "Squaring the circle" as a metaphor
4 See also
5 References
6 External links

Impossibility

Some apparent partial solutions gave false hope for a long time. In this figure, the area of the shaded figure is equal to the area of the triangle ABC (found by Hippocrates of Chios).

The problem dates back to the invention of geometry and has occupied mathematicians for millennia. It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility, though even the ancient geometers had a very good practical and intuitive grasp of its intractability. It should be noted that it is the limitation to just compass and straightedge that makes the problem difficult. If the straightedge is allowed to be a ruler or if other simple instruments, for example something which can draw an Archimedean spiral, are allowed, then it is not difficult to draw a square and circle of equal area.

Transcendence of π

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A solution of the problem of squaring the circle by straightedge and compass demands construction of the number $\sqrt{\pi}$ , and the impossibility of this undertaking follows from the fact that π (pi) is a transcendental number—that is, it is non-algebraic and therefore a non-constructible number. The transcendence of π was proved by Ferdinand von Lindemann in 1882. If you solve the problem of the quadrature of the circle, this means you have also found an algebraic value of π, which is impossible.

It is possible to construct a square with an area arbitrarily close to that of a given circle. If a rational number is used as an approximation of π, then squaring the circle becomes possible, depending on the values chosen. However, this is only an approximation and does not meet the constraints of the ancient rules for solving the problem. Several mathematicians have demonstrated workable procedures based on a variety of approximations.

Bending the rules by allowing an infinite number of ruler-and-compass constructions or by performing the operations on certain non-Euclidean spaces also makes squaring the circle possible. For example, although the circle cannot be squared in Euclidean space, it can in Gauss-Bolyai-Lobachevsky space.