# Doubling the Cube

## Contents

## Classic Problem

**Doubling the cube** is the exercise to construct a cube whose volume is double that of a given cube.

## Solution

Let $C$ be the cube in question.

Let $x$ be the length of one of the edges of $C$.

Hippocrates of Chios demonstrated that the problem is equivalent to finding line segments of length $p$ and $q$ such that:

- $\dfrac x p = \dfrac p q = \dfrac q {2 x}$

from which:

- $2 x^3 = p^3$

and so:

- $\dfrac p x = \sqrt [3] 2$

Hence to find a cube whose volume is double that of $C$ is equivalent to finding the Cube Root of 2.

But Doubling the Cube by Compass and Straightedge Construction is Impossible.

Various techniques have been devised which use more elaborate apparatus than just the straightedge and compass:

### Archytas Curve

The Archytas curve can be used for Doubling the Cube.

### Intersection of Parabolas

The problem of Doubling the Cube can be solved by finding the intersection of two parabolas.

### Intersection of Parabola and Hyperbola

The problem of Doubling the Cube can be solved by finding the intersection of a parabola and a hyperbola.

### Conchoid of Nicomedes

The problem of Doubling the Cube can be solved by using a conchoid of Nicomedes.

### Cissoid of Diocles

The problem of Doubling the Cube can be solved by using a cissoid of Diocles.

## Fallacious Proofs

### Edw. J. Goodwin

*Doubling the dimensions of the cube octuples its contents, and doubling its contents increases its dimensions twenty-five plus one per cent.*

## Also see

## 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.

## Historical Note

The exercise to construct a cube double the volume of a given cube, using a compass and straightedge construction, was an exercise that the ancient Greeks failed to succeed in.

This was one of three such problems: the other two being Squaring the Circle and Trisecting the Angle.

There are several techniques available that use other tools, but these were considered unacceptably vulgar to the followers of Plato.

According to one version of the legend, the island of Delos was being ravaged by a plague. The people consulted the Oracle at Delphi, who instructed them that a new altar to Apollo had to be built which was twice the size of the existing one, which was cubic in shape.

Another version of the legend suggests that it was not a plague, but civil or political unrest causing the trouble.

The word *size* was misunderstood, and the new altar had each of the linear dimensions doubled, whereas apparently they should have doubled its volume instead. Hence, unfortunately, the gods were not appeased, and the desired abatement of the plague did not happen.

To quote Plato:

*It must be supposed, not that the god specially wished this problem solved, but that he would have the Greeks desist from war and wickedness and cultivate the Muses, so that their passions being assuaged by philosophy and mathematics, they might live in innocent and mutually helpful intercourse with one another.*

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $1 \cdotp 25992 \, 10498 \, 94873 \, 16476 \ldots$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Delian altar problem** - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1 \cdotp 25992 \, 10498 \, 94873 \, 16476 \ldots$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Delian problem** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Delian problem** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Delian (altar) problem** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**duplication of the cube**