Given a cube, find another cube which is twice as large, with nothing more than a compass and a straight-edge. One of the three geometric problems of antiquity.

This classical mathematical problem is sometimes called the Delian problem, as it was originally set by the oracle of Delios in the 5th Century BC. The Delians were instructed that, in order to rid themselves of a plague, they must double the altar of their god -- its shape being a perfect cube.
The problem exercised many mathematicians over the next two and a half millennia, but only in the 19th Century was it proved to be impossible, using the newly-developed mathematical techniques of group theory.

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