Duplication of the cube

The problem of constructing a cube having twice the volume of a given cube; it is one of the classical problems of Antiquity, to find an exact construction with ruler and compass. If an edge of the given cube has length 1, the length $x$ of an edge of the desired cube is equal to $2^{1/3}$ and is determined by the cubic equation $x^3-2=0$. However, an exact construction of the segment $2^{1/3}$ by means of ruler and compass is impossible, in view of the unsolvability of the cubic equation by square roots. The first rigorous proof of the unsolvability of the problem of the duplication of the cube by ruler and compass was given in 1837 by P. Wantzel.

References

[1]	, Encyclopaedia of elementary mathematics , 4. Geometry , Moscow-Leningrad (1963) pp. 205–227 (In Russian)

Comments

Classical Greek mathematicians realized (but did not prove) that the problem cannot be solved with ruler and compass (i.e. the cube cannot be constructed this way). Since Antiquity numerous solutions by other means are known.

Like the other famous problems of quadrature of the circle and trisection of an angle, the problem of duplication of the cube belongs to the branch of geometric constructions, and is, in algebraic formulation, treated in Galois theory.

The problem of duplication of the cube is also known as the Delian problem or the problem of doubling the cube. Cf. [a2], pp. 154-158, for some details such as the origin of the name "Delian problem" and Menaechmus' "solution" by intersecting a parabola and a hyperbola.

References

[a1]	I. Stewart, "Galois theory" , Chapman & Hall (1973) pp. Chapt. 5
[a2]	E.E. Kramer, "The nature and growth of modern mathematics" , Princeton Univ. Press (1982)