## 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.