 Regular polygon edge rotation criterion

For $n\ge 3$ cyclically ordered planar vertices, let $e_j=v_{j+1}-v_j$, with indices modulo $n$. They describe a convex regular polygon exactly when $e_0\ne 0$ and there is a single choice of sign for which every successive edge vector is obtained by a rotation through $2\pi /n$ in that orientation. Necessity follows from equal sides and equal exterior angles. Conversely, the edge vectors have the form $R^j{e}_0$. Their sum is zero because the corresponding geometric series runs through the $n$ distinct roots of unity. Summing its partial series shows that the vertices are a translated and scaled copy of those roots, so they form a convex regular polygon. A larger turn such as $4\pi /5$ can instead trace a star, which is why merely checking a constant turning angle is insufficient.