# Duplication of the cube

2010 Mathematics Subject Classification: Primary: 51M04 [MSN][ZBL]

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)