Squaring the circle

Squaring the circle is the problem proposed by ancient geometers of using a finite compass and straightedge construction to make a square with the same area as a given circle. In 1882, the problem was proven to be impossible. The term quadrature of the circle is synonymous.

History

Methods to approximate the area of a given circle with a square were known already to Babylonian mathematicians. The Rhind papyrus in 1800BC gives the area of a circle as $64/81 d^2$ , where $d$ is the diameter of the circle. Indian mathematicians also found an approximate method, though less accurate, documented in the Sulba Sutras. Indian mathematicians also gave an approximate solution to the problem of circling the square.O'Connor, John J. and Robertson, Edmund F. (2000). The Indian Sulbasutras, MacTutor History of Mathematics archive, St Andrews University.

The first person to be associated with the problem in Greece was Anaxagoras, who worked on it while in prison. Hippocrates of Chios squared certain lunes, thought to be in the hope it will lead to a solution. Antiphon the Sophist believed that inscribing regular polygons within a circle and doubling the number of sides will eventually fill up the area of the circle, and since a polygon can be squared, it means the circle can be squared. Even then there were skeptics - Eudemus argued that magnitudes cannot be divided up without limit, so the area of the circle will never be used up. The problem was even mentioned in Aristophenes's play Birds.

It is believed that Oenopides was the first person who required a plane solution (that is, using only a compass and straightedge). James Gregory attempted a proof of its impossibility in Vera Circuli et Hyperbolae Quadratura (The True Squaring of the Circle and of the Hyperbola) in 1667. Although his proof was incorrect, it was the first paper to attempt to solve the problem using algebraic properties of π. It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility.

Impossibility

A solution of the problem of squaring the circle by compass and straightedge demands construction of the number

\sqrt{\pi}

, and the impossibility of this undertaking follows from the fact that π is a transcendental number—that is, it is non-algebraic and therefore a non-constructible number. If one solves the problem of the quadrature of the circle, this means one has also found an algebraic value of π, which is impossible. Johann Heinrich Lambert conjectured that π was transcendental in 1768 in the same paper he proved its irrationality, even before the existence of transcendental numbers was proved. It wasn't until 1882 before Ferdinand von Lindemann proved its transcendence.

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 compass and straightedge operations 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.

Note that the transcendence of π implies the impossibility of exactly "circling" the square, as well as of squaring the circle.

Modern approximative constructions

Though squaring the circle is an impossible problem, approximations to squaring the circle can be given by constructing lengths close to π. It takes only minimal knowledge of elementary geometry to convert any given rational approximation of π into a corresponding compass-and-straightedge construction, but constructions made in this way tend to be very long-winded in comparison to the accuracy they achieve. After the exact problem was proved unsolvable, some mathematicians have applied their ingenuity to finding elegant approximations to squaring the circle, defined roughly (and informally) as constructions that are particularly simple among other imaginable constructions that give similar precision.

Among the modern approximate constructions was one by E. W. Hobson in 1913. This was a fairly accurate construction which was based on constructing the approximate value of 3.14164079..., which is accurate to 4 decimals.

Indian mathematician Srinivasa Ramanujan in 1913, C. D. Olds in 1963, Martin Gardner in 1966, and B. Bold in 1982 all gave geometric constructions for

\frac{355}{113} = 3.1415929203539823008\dots

which is accurate to 6 decimal places of π.

Srinivasa Ramanujan in 1914 gave a ruler and compass construction which was equivalent to taking the approximate value for π to be

\left(9^2 + \frac{19^2}{22}\right)^{1/4} = \sqrt

^*{\frac{2143}{22}} = 3.1415926525826461253\dots

giving a remarkable 8 decimal places of π.

In 1991, Robert Dixon gave constructions for

\frac{6}{5} (1 + \varphi)

and

\sqrt{{40 \over 3} - 2 \sqrt{3}\  }

(Kochanski's approximation), though these were only accurate to 4 decimal places of π.