Doubling the cube in the context of "Compass and straightedge"

Pandrosion of Alexandria (Ancient Greek: Πανδροσίων) was a mathematician in fourth-century-AD Alexandria, discussed in the Mathematical Collection of Pappus of Alexandria and known for having possibly developed an approximate method for doubling the cube. She is likely the earliest known female mathematician.

Pierre Laurent Wantzel (5 June 1814 – 21 May 1848) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge.

In a paper from 1837, Wantzel proved that the problems of doubling the cube and trisecting the angle are impossible to solve if one uses only a compass and straightedge. In the same paper he also solved the problem of determining which regular polygons are constructible: a regular polygon is constructible if and only if the number of its sides is the product of a power of two and any number of distinct Fermat primes (i.e., the sufficient conditions given by Carl Friedrich Gauss are also necessary).

In mathematics, an impossibility theorem is a theorem that demonstrates a problem or general set of problems cannot be solved. These are also known as proofs of impossibility, negative proofs, or negative results. Impossibility theorems often resolve decades or centuries of work spent looking for a solution by proving there is no solution. Proving that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a proof that works in general, rather than to just show a particular example. Impossibility theorems are usually expressible as negative existential propositions or universal propositions in logic.

The irrationality of the square root of 2 is one of the oldest proofs of impossibility. It shows that it is impossible to express the square root of 2 as a ratio of two integers. Another consequential proof of impossibility was Ferdinand von Lindemann's proof in 1882, which showed that the problem of squaring the circle cannot be solved because the number π is transcendental (i.e., non-algebraic), and that only a subset of the algebraic numbers can be constructed by compass and straightedge. Two other classical problems—trisecting the general angle and doubling the cube—were also proved impossible in the 19th century, and all of these problems gave rise to research into more complicated mathematical structures.