Doubling the Cube

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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 were devised which used 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.
-- Edw. J. Goodwin


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