# 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.