A Generalized Mathematical Framework for the Scaling and Convergence of Iteratively Inscribed Regular Polygons

While classical Euclidean geometry establishes the isolated area bisection of midpoint-inscribed squares, the general case of arbitrary boundary partition ratios t in (0, 1) across general regular k-gons warrants a unified formal treatment. To the author's knowledge, this work develops an explicit closed-form framework for the specified iterative construction, extending the midpoint case to arbitrary partition ratios t in (0, 1) and governing regularity preservation, edge contraction, area scaling, and infinite series convergence. For any regular polygon of k vertices and initial area A_0 partitioned at continuous parameter t in (0, 1), the edge length contracts geometrically according to L_n = L_0 [Lambda_k(t)]^(n/2), while the polygonal area scales as A_n = A_0 [Lambda_k(t)]^n, where Lambda_k(t) = 1 - 4t(1 - t) sin^2(pi / k). We prove the edge-length and area scaling laws by mathematical induction. Using the resulting geometric progression, we further establish the absolute convergence of the cumulative area series to the closed-form limit S_inf = A_0 / [ 4t(1 - t) sin^2(pi / k) ]. We establish the tight universal bound 1/4 <= Lambda_k(t) < 1 for all k >= 3 and t in (0, 1), with the lower bound attained uniquely at k = 3 and t = 1/2.