= Regular polygon edge rotation criterion
{title2=$e_{j+1}=R_{\pm2\pi/n}e_j$}
For $n\geq3$ cyclically ordered planar <vertices>, let $e_j=v_{j+1}-v_j$, with indices modulo $n$. They describe a convex <regular polygon> exactly when $e_0\ne0$ 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 $R^je_0$. 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.
\Image[/regular-polygon-edge-rotation.png]
{title=Equal sides, equal turns and the fundamental turn distinguish an ordinary <regular polygon>}
{description=Original constructions: a square satisfies the edge-rotation criterion; a nonsquare rhombus has unequal turns; a five-point star uses the wrong turn, $144^\circ$ instead of $72^\circ$. Arrows indicate traversal direction.}
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