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Squaring the circle

*** Shopping-Tip: Squaring the circle
Image:Squaring the circle.png right|thumb|This square and circle have the same area. '''Squaring the circle''' is the problem proposed by ancient geometers of using a finite compass and straightedge construction to make a square (geometry) square with the same area as a given circle. In 1882, the problem was proved to be impossible. The term ''numerical integration quadrature of the circle'' is synonymous.

Impossibility
Image:Hipocrat arcs.svg right|thumb|200px|Some apparent partial solutions gave false hope for a long time. In this figure, the area of the shaded figure is equal to the area of the triangle ABC (found by [[Hippocrates of Chios).]] The problem dates back to the invention of geometry and has occupied mathematicians for millennia. It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility, though even the ancient geometers had a very good practical and intuitive grasp of its intractability. It should be noted that it is the limitation to just Compass (drafting) compass and straightedge that makes the problem difficult. If the straightedge is allowed to be a ruler or if other simple instruments, for example something which can draw an Archimedean spiral, are allowed, then it is not difficult to draw a square and circle of equal area. The problem is to find an ''exact'' solution, in a finite number of steps. Methods to ''approximate'' a square with the area of a given circle were known already to Babylonian mathematics Babylonian mathematicians. Indian mathematics Indian mathematicians also found an approximate method, though less accurate, documented in the ''Sulba Sutras''. Indian mathematicians also gave a solution to the problem of circling the square.O'Connor, John J. and Robertson, Edmund F. (2000). [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html ''The Indian Sulbasutras''], MacTutor History of Mathematics archive, St Andrews University.

Transcendence of π
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 Pi π (pi) is a transcendental number—that is, it is algebraic number non-algebraic and therefore a non-constructible number. The transcendence of π was proved by Ferdinand von Lindemann in 1882. If you solve the problem of the quadrature of the circle, this means you have also found an algebraic value of π, which is impossible. 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.

Modern approximations
Though squaring the circle is an impossible problem, approximations to squaring the circle can be given by constructing lengths close to π. Among the correct approximate constructions to square the circle 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 equivalent to the 4 decimal places of π. 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...

which is equivalent to seven 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

(9^2 + \frac{19^2}{22})^{1/4} = 3.1415926525826461253...

which is equivalent to a remarkable 9 decimal places of π. R. Dixon in 1991 gave constructions for

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

and

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

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

"Squaring the circle" as a metaphor
The mathematical proof that the numerical integration quadrature of the circle is impossible has not proved to be a hindrance to the many people who have invested years in this problem anyway; having squared the circle is a famous crank (person) crank assertion. (''See also'' pseudomathematics.) The futility of undertaking exercises aimed at finding the quadrature of the circle has brought this term into use in totally unrelated contexts, where it is simply used to mean a hopeless, meaningless, or vain undertaking. Aleister Crowley used the metaphor in a different sense, to represent the goal of magick and mysticism. He implicitly associated his system of Thelema with pi. For more information, see Abrahadabra.

See also
* Tarski's circle-squaring problem * Doubling the cube

References

External links

- ''Squaring the circle'' at the MacTutor History of Mathematics archive
- ''Squaring the Circle'' at cut-the-knot
- ''Circle Squaring'' at MathWorld, includes information on procedures based on various approximations of π Category:Pi Category:Euclidean plane geometry cs:Kvadratura kruhu de:Quadratur des Kreises es:Cuadratura del círculo fa:تربیع دایره fr:Quadrature du cercle is:Ferningshringir it:Quadratura del cerchio lt:Apskritimo kvadratūra nl:Kwadratuur van de cirkel pl:Kwadratura koła pt:Quadratura do círculo ru:Квадратура круга sv:Cirkelns kvadratur zh:化圓為方

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