I can rule out any such polygons with an odd prime number of vertices. Note that if the vertices of a polygon are given by complex numbers: {z_0, z_1, ..., z_(n-1)} then the algebraic area is equal to: Im[z_0 z_1* + z_1 z_2* + ... + z_(n-1) z_0*] / 2 where * indicates complex conjugation and Im is the imaginary part. So the area of a re-entrant polygon with vertices: {w^(a_0), w^(a_1), ..., w^(a_(n-1))} is proportional to: Im[w^(a_0 - a_1)] + Im[w^(a_1 - a_2)] + ... + Im[w^(a_(n-1) - a_0)] Note that none of these terms are zero. When n is prime, the *only* integer linear relationships between the imaginary parts of roots of unity are generated by relationships of the form: Im[w^k] + Im[w^-k] = 0 so must have an even number of terms. Our sum, however, has an odd number of terms, so cannot be equal to zero. Best wishes, Adam P. Goucher