Let \(v_k\) and \(v_{k+1}\) be the polar angles of two vertices of the regular polygon \(S_n\): \[v_k=\frac{\pi(2k-1)}{n}, \quad v_{k+1}=\frac{\pi(2k+1)}{n}.\] For a regular polygon centered at the origin, the outward normal angle of the connecting edge is the average of these vertex angles: \[\text{Normal Angle}=\frac{v_k+v_{k+1}}{2} = \frac{2\pi k}{n}.\] Since a full circle is \(2\pi\), the direction of the \(k\)-th edge is uniquely identified by the fraction \(k/n\).
By the properties of the Minkowski sum for convex polygons, the boundary of the resulting shape is formed by the union of the original edges. Edges sharing the exact same normal direction merge into a single extended edge, while edges with distinct directions form separate sides. Therefore, the total number of edges in the resulting Minkowski sum is exactly the number of strictly unique fractions \(k/n\) across all given polygons.