# Doubling the Cube by Compass and Straightedge Construction is Impossible

## 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

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $2$: The Logic of Shape: Problems for the Greeks - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**duplication of the cube**