Squaring The Circle

Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing 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 an 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 the construction would be impossible if pi were transcendental, but pi was not proven transcendental until 1882. Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a finite number of steps, since there are rational numbers arbitrarily close to π.

The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.

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.

Read more about Squaring The Circle:  History, Impossibility, Modern Approximative Constructions, Squaring or Quadrature As Integration, "Squaring The Circle" As A Metaphor

Famous quotes containing the word circle:

    Perchance we may,
    Where now this night is day,
    And even through faith of still averted feet,
    Making full circle of our banishment,
    Amazed meet;
    Coventry Kersey Dighton Patmore (1823–1896)