Is it possible to construct a regular heptagon (a figure with seven sides) with just compass and straightedge? If so, could you please give me directions for how to do this?
No, it's not possible; in fact, the regular heptagon is the regular polygon with the least number of sides that is impossible to construct with compass and straightedge alone. It is, however, possible to construct it using a neusis ruler. A related question has been asked (and answered) here.