Circle squaring is the problem of constructing 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.
The Indiana pi bill of 1897 was a legislative attempt to endorse an erroneous circle-squaring method; among its incompatible consequences
were the values
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 ,
Indiana
Pi Bill ,
Kochanski's Approximation ,
Quadrature ,
Squaring ,
Tarski's Circle-Squaring Problem ,
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. Edington, W. E. "House Bill No. 246, Indiana State
Legislature, 1897." Proc. Indiana Acad. Sci. 45 , 206-210, 1935. 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. Numberphile.
"Squaring the Circle." Featuring J. Grime. Mar. 25, 2013. https://www.youtube.com/watch?v=CMP9a2J4Bqw . 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. Referenced on Wolfram|Alpha Circle Squaring
Cite this as:
Weisstein, Eric W. "Circle Squaring."
From MathWorld https://mathworld.wolfram.com/CircleSquaring.html
Subject classifications
and 
(Edington 1935).
. 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.