squaring the circle
/ pi -> py /
Given a circle, can you construct a square with exactly the same area, using only compass and straightedge? The phrase 'squaring the circle' has even entered everyday language as a byword for an impossible task — and that reputation is mathematically earned. People chased it from antiquity until 1882, when it was finally laid to rest, not by geometry but by a deep fact about the number pi.
A circle of radius 1 has area pi. A square of the same area must have side sqrt(pi), so squaring the circle amounts to constructing a length of sqrt(pi) — and that is possible if and only if pi itself is constructible. Now every constructible number is algebraic: it satisfies some polynomial equation with rational coefficients (indeed one of degree a power of 2). But in 1882 Ferdinand von Lindemann proved that pi is transcendental — it satisfies NO polynomial equation with rational coefficients at all. A transcendental number cannot possibly be constructible, so sqrt(pi) cannot be constructed, and the circle cannot be squared.
This is the deepest of the three classical impossibilities. Trisection and doubling fail because of a degree-3 algebraic obstruction; squaring the circle fails because pi is not algebraic in the first place — a far stronger statement that needed nineteenth-century analysis to prove. The impossibility is, once more, specific to the classical tools: with other curves or instruments one can produce lengths equal to pi. And approximations are easy — you can get a square as close to the circle's area as you like — but an EXACT classical construction is provably out of reach.
A unit circle has area pi; an equal square has side sqrt(pi). Lindemann (1882) proved pi is transcendental, satisfying no rational-coefficient polynomial. Since every constructible number is algebraic, pi and sqrt(pi) are not constructible — the circle cannot be squared.
Because pi is transcendental, no exact compass-and-straightedge square matches a circle's area.
Approximate squarings exist and can be made as accurate as you wish; only an EXACT construction is impossible. The proof rests on pi being transcendental (Lindemann, 1882) — a stronger fact than the degree-3 obstructions that kill trisection and cube-doubling.