The ancient Greeks were the first to consider the problem of constructions using compass and unmarked straightedge. Among many other things, they knew how to construct equilateral triangles, squares, regular pentagons, and regular hexagons using compass and straightedge. It wasn't until the late 18th century when further progress was made, and Gauss and Wantzel found an exact criterion describing the regular polygons that can be constructed using compass and straightedge; in particular, regular 17-gons, 257-gons, and 65537-gons can all be constructed with compass and straightedge, whereas a regular heptagon cannot.

One way of generalizing this result is to realize that constructing a regular n-gon is the same thing as dividing a circle into n arcs of equal length. It is then natural to ask when we can divide other closed curves into n pieces of equal length using a compass and straightedge. With my student Nitya Mani, who was a high-school student at the time, we solved this problem in the case of a family of curves called hypocycloids; these are essentially the shapes formed when playing with spirograph toys.