Doubling the cube

Problem

Given the edge of a cube, construct with straightedge and compass alone the edge of a cube with twice the volume.

Remarks

For an edge of length $1$, the new edge has length $\sqrt[3]{2}$. So the problem asks whether $\sqrt[3]{2}$ is a constructible number. The answer is no: doubling the cube is impossible. The problem has solutions with other tools, such as a marked ruler or conic sections.

History

The problem is also called the Delian problem, after a legend about an oracle on the island of Delos. Hippocrates of Chios showed that it is equivalent to finding two mean proportionals between a segment and a segment twice as long. Pierre Wantzel proved in 1837 that no straightedge and compass solution exists.

Explanations

L1 human

The problem asks to build a cube with twice the volume of a given cube, using only an unmarked ruler and a compass. It is one of the three classical construction problems of Greek geometry.