A symmetry property of some harmonic algebraic curves
  The aim of this note is to give a surprising symmetry property of some harmonic algebraic curves: when all the roots $z_i$ of a complex polynomial $P$ lie on the unit circle $\U$, the points of $\U$ different from the $z_i$, and such that $\Arg(P(z))=\theta$, form a regular $n$-gon, where $n$ is the degree of $P$. 
Jean-Christophe Aval (LaBRI), Jean-Fran\c{c}ois Marckert (LaBRI)