A Necklace of Equilaterals

Let

P

be an

n

-gon (a polygon with

n

sides). Construct equilateral triangles on the sides of

P

. The third vertices of the triangles form another

n

-gon,

E(P)

. Given an

n

-gon

Q

, is it possible to find an

n

-gon

P

such that

Q=E(P)

? This Demonstration constructs

P

(with blue interior) for the case

n=7

. In general, for odd

n

the problem has a unique solution whereas for even

n

the problem either has no solution or an infinite number of solutions.