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 
. Ramanujan (1913-1914),
Olds (1963), Gardner (1966, pp. 92-93), and (Bold 1982, p. 45) give geometric
constructions for 
. Dixon (1991) gives
constructions for 
and 
(Kochanski's approximation).
While the circle cannot be squared in Euclidean space, it can in Gauss-Bolyai-Lobachevsky Space (Gray 1989).
Bold, B. "The Problem of Squaring the Circle." Ch. 6 in Famous Problems of Geometry and How to Solve Them.New York: Dover, pp. 39-48, 1982.
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, pp. 91-102, 1966.
Gray, J. Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, 2nd ed. Oxford, England: Oxford University Press, 1989.
Hertel, E. "On the Set-Theoretical Circle-Squaring Problem." http://www.minet.uni-jena.de/Math-Net/reports/sources/2000/00-06report.ps.
Jesseph, D. M. Squaring the Circle: The War Between Hobbes and Wallis. Chicago: University of Chicago Press, 1999.
Klein, F. "Transcendental Numbers and the Quadrature of the Circle." Part II in "Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, pp. 49-80, 1980.
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 
." Quart.
J. Pure. Appl. Math. 45, 350-372, 1913-1914.
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 48, 1986.
Weisstein, Eric W. "Circle Squaring." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CircleSquaring.html
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circle squaring