A Necklace of Equilaterals

Let be an -gon (a polygon with sides). Construct equilateral triangles on the sides of . The third vertices of the triangles form another -gon, . Given an -gon , is it possible to find an -gon such that ? This Demonstration constructs (with blue interior) for the case . In general, for odd the problem has a unique solution whereas for even the problem either has no solution or an infinite number of solutions.