Doubling the cube

As with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible to construct by using only a compass and straightedge, but even in ancient times solutions were known that employed other methods.

[2][3][4] The nonexistence of a compass-and-straightedge solution was finally proven by Pierre Wantzel in 1837.

In algebraic terms, doubling a unit cube requires the construction of a line segment of length x, where x3 = 2; in other words, x =

This is a consequence of the fact that the coordinates of a new point constructed by a compass and straightedge are roots of polynomials over the field generated by the coordinates of previous points, of no greater degree than a quadratic.

This implies that the degree of the field extension generated by a constructible point must be a power of 2.

We are required to construct a line segment defined by two points separated by a distance of

Respectively, the tools of a compass and straightedge allow us to create circles centred on one previously defined point and passing through another, and to create lines passing through two previously defined points.

An exercise of elementary analytic geometry shows that in all three cases, both the x- and y-coordinates of the newly defined point satisfy a polynomial of degree no higher than a quadratic, with coefficients that are additions, subtractions, multiplications, and divisions involving the coordinates of the previously defined points (and rational numbers).

Restated in more abstract terminology, the new x- and y-coordinates have minimal polynomials of degree at most 2 over the subfield of

But this is not a power of 2, so by the above: The problem owes its name to a story concerning the citizens of Delos, who consulted the oracle at Delphi in order to learn how to defeat a plague sent by Apollo.

The oracle responded that they must double the size of the altar to Apollo, which was a regular cube.

The answer seemed strange to the Delians, and they consulted Plato, who was able to interpret the oracle as the mathematical problem of doubling the volume of a given cube, thus explaining the oracle as the advice of Apollo for the citizens of Delos to occupy themselves with the study of geometry and mathematics in order to calm down their passions.

[8] This may be why the problem is referred to in the 350s BC by the author of the pseudo-Platonic Sisyphus (388e) as still unsolved.

[9] However another version of the story (attributed to Eratosthenes by Eutocius of Ascalon) says that all three found solutions but they were too abstract to be of practical value.

[11] In modern notation, this means that given segments of lengths a and 2a, the duplication of the cube is equivalent to finding segments of lengths r and s so that In turn, this means that But Pierre Wantzel proved in 1837 that the cube root of 2 is not constructible; that is, it cannot be constructed with straightedge and compass.

[12] Menaechmus' original solution involves the intersection of two conic curves.

Other more complicated methods of doubling the cube involve neusis, the cissoid of Diocles, the conchoid of Nicomedes, or the Philo line.

Pandrosion, a probably female mathematician of ancient Greece, found a numerically accurate approximate solution using planes in three dimensions, but was heavily criticized by Pappus of Alexandria for not providing a proper mathematical proof.

[13] Archytas solved the problem in the 4th century BC using geometric construction in three dimensions, determining a certain point as the intersection of three surfaces of revolution.

False claims of doubling the cube with compass and straightedge abound in mathematical crank literature (pseudomathematics).

In music theory, a natural analogue of doubling is the octave (a musical interval caused by doubling the frequency of a tone), and a natural analogue of a cube is dividing the octave into three parts, each the same interval.

In this sense, the problem of doubling the cube is solved by the major third in equal temperament.