Circle Squaring

Construct a Square equal in Area to a Circle using only a Straightedge and Compass. This was one of the three Geometric Problems of Antiquity, and was perhaps first attempted by Anaxagoras. It was finally proved to be an impossible problem when Pi was proven to be Transcendental by Lindemann in 1882.

However, approximations to circle squaring are given by constructing lengths close to $\pi=3.1415926\ldots$. Ramanujan (1913-14), Olds (1963), and Gardner (1966, pp. 92-93) give geometric constructions for $355/113=3.1415929\ldots$. Dixon (1991) gives constructions for $6/5(1+\phi)=3.141640\ldots$ and $\sqrt{4+[3-\tan(30^\circ)]^2}=3.141533\ldots$.

While the circle cannot be squared in Euclidean Space, it can in Gauss-Bolyai-Lobachevsky Space (Gray 1989).

See also Geometric Construction, Quadrature, Squaring

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 190-191, 1996.

Dixon, R. Mathographics. New York: Dover, pp. 44-49 and 52-53, 1991.

Dunham, W. ``Hippocrates' Quadrature of the Lune.'' Ch. 1 in Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 20-26, 1990.

Gardner, M. ``The Transcendental Number Pi.'' Ch. 8 in Martin Gardner's New Mathematical Diversions from Scientific American. New York: Simon and Schuster, 1966.

Gray, J. Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, 2nd ed. Oxford, England: Oxford University Press, 1989.

Meyers, L. F. ``Update on William Wernick's `Triangle Constructions with Three Located Points.''' Math. Mag. 69, 46-49, 1996.

Olds, C. D. Continued Fractions. New York: Random House, pp. 59-60, 1963.

Ramanujan, S. ``Modular Equations and Approximations to $\pi$.'' Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.