Squaring the circle
From The Art and Popular Culture Encyclopedia
|
Related e |
|
Wikipedia
Featured:  Illustration: Portrait fantaisiste du marquis de Sade (1866) by H. Biberstein |
Squaring the circle is a problem proposed by ancient geometers. It is the challenge to construct a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. More abstractly and more precisely, it may be taken to ask whether specified axioms of Euclidean geometry concerning the existence of lines and circles entail the existence of such a square.
In 1882, the task was proven to be impossible, as a consequence of the Lindemann-Weierstrass theorem which proves that pi (π) is a transcendental, rather than algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients. It had been known for some decades before then that if π were transcendental then the construction would be impossible, but that π is transcendental was not proven until 1882. Approximate squaring to any given non-perfect accuracy, on the other hand, is possible in a finite number of steps, as a consequence of the fact that there are rational numbers arbitrarily close to π.
The term quadrature of the circle is sometimes used synonymously, or may refer to approximate or numerical methods for finding the area of a circle.
"Squaring the circle" as a metaphor
The futility of exercises aimed at finding the quadrature of the circle has lent itself to metaphors describing a hopeless, meaningless, or vain undertaking.
For example, in Spanish, the expression "descubriste la cuadratura del círculo" ("you discovered the quadrature of the circle") is often used derisively to dismiss claims that someone has found a simple solution to a particularly hard or intractable problem.
