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 Kochanski's
approximation ).
While the circle cannot be squared in Euclidean space , it can in Gauss-Bolyai-Lobachevsky
Space (Gray 1989).
See also Banach-Tarski Paradox ,
Geometric Construction ,
Kochanski's
Approximation ,
Quadrature ,
Squaring ,
Wallace-Bolyai-Gerwien Theorem
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References Bold, B. "The Problem of Squaring the Circle." Ch. 6 in Famous
Problems of Geometry and How to Solve Them. Conway, J. H. and Guy, R. K. The
Book of Numbers. Dixon,
R. Mathographics. Dunham, W. "Hippocrates'
Quadrature of the Lune." Ch. 1 in Journey
through Genius: The Great Theorems of Mathematics. Gardner, M. "The Transcendental Number Pi." Ch. 8
in Martin
Gardner's New Mathematical Diversions from Scientific American. Gray, J. Ideas
of Space: Euclidean, Non-Euclidean, and Relativistic, 2nd ed. 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. 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. 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. 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.
Cite this as:
Weisstein, Eric W. "Circle Squaring."
From MathWorld https://mathworld.wolfram.com/CircleSquaring.html
Subject classifications
.
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).
." Quart. J. Pure. Appl. Math. 45, 350-372,
1913-1914.Wells, D.