Doubling the Cube by Compass and Straightedge Construction is Impossible

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Theorem

There is no compass and straightedge construction to allow a cube to be constructed whose volume is double that of a given cube.


Proof

Suppose it is possible.

Then from a cube of edge length $L$ we can construct a new cube with edge length $\sqrt [3] 2 L$.

$\sqrt [3] 2$ is algebraic of degree $3$.

This contradicts Constructible Length with Compass and Straightedge.

$\blacksquare$


Also known as

The problem of Doubling the Cube is known as the Delian problem, after the location (Delos) of the altar whose dimensions were under the question.


Some sources refer to the problem of Doubling the Cube as duplicating the cube.

However, the position taken by $\mathsf{Pr} \infty \mathsf{fWiki}$ is that duplication can also mean making an exact copy of, which could cause misunderstanding.


Sources

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