Duplication of the Cube

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Duplication of the Cube

Along with squaring the circle and trisecting an angle, duplication of the cube, also called cube duplication and the Delian problem, is considered one of the three unsolvable problems of mathematical antiquity. First asked by ancient Greek mathematicians, duplication of the cube asks: If given a length of an edge of a cube, construct a second cube having double the volume of the first—that is, with only the use of an unmarked straightedge and compass.

According to tradition, the problem of duplication of the cube arose when the Greeks of Athens sought the assistance of the oracle at Delos in order to gain relief from a devastating epidemic. The oracle told them that to do so they must double the size of the altar of Apollo which was in the shape of a cube.

Their first attempt at doing this was a misunderstanding of the problem: They doubled the length of the sides of the cube. This, however, gave them eight times the original volume since (2x)³ =8x³.

In modern notation, in order to fulfill the instructions of the oracle, mathematicians must go from a cube of side x units to one of y units where y³ =2x³, so that y = 2^1/3x.

Thus, essentially, given a unit length, they needed to construct a line segment of length 2^1/3 units. Now, there are ways of doing this but not by using only a compass and an unmarked straight edge—which were the only tools allowed in classical Greek geometry.

Thus, there is no solution to the Delian problem that the Greeks would accept and, presumably, the epidemic continued until it ran its accustomed course.

The first proof for the duplication of the cube was performed by French mathematician and philosopher Rene´ Descartes (1596–1650) in 1637. Today, mathematicians can solve the problem with a geometric construction called a Neusis construction, also called a verging construction.