Napoleon polygons

An n-gon is called Napoleon if the centers of the regular n- gons erected outwardly on its sides are vertices of a regular n-gon. In this note we obtain a new geometric characterization of Napoleon $n$- gons and give a new proof of the well-known theorem of Barlotti - Greber that an n-gon is Napoleon if and only if it is affine - regular. Moreover, we generalize this theorem by obtaining an analytic characterization of the n-gons leading to a regular n-gon after iterating the above construction k times.