Squaring the circle  

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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.

In literature

Literature

The problem of squaring the circle has been mentioned by poets such as Dante and Alexander Pope.

The character Meton of Athens in the play The Birds by Aristophanes (first performed in 414 BCE) mentions squaring the circle.

Dante's Paradise canto XXXIII lines 133–135 contain the verses:

As the geometer his mind applies

To square the circle, nor for all his wit

Finds the right formula, howe'er he tries

Pope, in his 1743 poem Dunciad, wrote:

Mad Mathesis alone was unconfined,

Too mad for mere material chains to bind,

Now to pure space lifts her ecstatic stare,

Now, running round the circle, finds it square.

The Gilbert and Sullivan comic opera Princess Ida features a song which satirically lists the impossible goals of the women's university run by the title character, such as finding perpetual motion. One of these goals is "And the circle – they will square it/Some fine day."

See also

• The two other classical problems of antiquity were doubling the cube and trisecting the angle, described in the compass and straightedge article. Unlike squaring the circle, these two problems can be solved by the slightly more powerful construction method of origami, as described at mathematics of paper folding.
• For a more modern related problem, see Tarski's circle-squaring problem.
• The Indiana Pi Bill, an 1897 attempt in the Indiana state legislature to dictate a solution to the problem by legislative fiat.
• Squircle, a mathematical shape with properties between those of a square and those of a circle.
• Squared circle, a professional wrestling ring.