Ruler and compass construction of a pentagon

Which geometric figures are constructible with a straightedge and a compass?

A geometric figure is constructible with the only use of ruler and compass if its segments are constructible. A segment of length \(a\) is constructible if the number \(a\) is constructible. It can be proved that the set of constructible numbers contains the rational numbers and, if it contains \(a\) and \(b\), then it also contains \(a \pm b\), \(a b\), \(a/b\) and \(\sqrt{a b}\).

Why the classical problem of doubling the cube has no solution?

The problem is the following: given a cube, construct another cube of twice the given cubic volume. If the first cube has side \(a\), and volume \(a^3\), then the second cube must have volume \(2a^3\) and a side of length \(\sqrt[3]{2}a\). Thus the problem can be solved if \(\sqrt[3]{2}a\) can be constructed, i.e. if it belongs to the set of constructible numbers. It is possible to prove that this is not the case, for a general proof have a look at Mathematics and Logic by M.Kac and S.M. Ulam.