The Impossible Dream: Doubling the Cube

The solution to this ancient geometry problem is that there is no solution

What does it mean to double the cube? And how can we know the task is impossible? This geometric puzzle — aka, the Delian Problem — besought the ancient Greeks. Pierre Wantzel presented the solution — that there is no solution — in a paper in 1837.

Using only a straightedge and a compass, we can construct a square. We can then construct a square of double area. Its side length is the diagonal of the first square.

Figure 1: BDFE is double ABCD — Euclid I:46

We can also construct a cube. But can we construct a cube of double volume? Such a cube would have an edge that is longer by a factor of the cube root of two.

We're talking about synthetic geometry. This is the geometry of constructions, such as one finds in Euclid’s Elements. Here are the rules:

• You can draw lines with an unmarked straightedge.
• You can draw circles with a compass.
• You can mark points, including the intersection points of the above constructions.