THE ACRIS TWIN BOUNDARY LAWS A Dual Geometric System Linking Complementary Pathways to the Circular Limit

This paper introduces the Acris Twin Boundary Laws, a pair of geometric invariants describing complementary boundary behaviours between regular polygons and circles. The first law, the Universal Polygon Boundary Law (UPBL), previously published, establishes a monotonic contraction of polygonal boundaries toward the circular boundary under equal-area constraints. The second law, introduced here, describes the complementary behaviour of regular polygons relative to an inscribed circle, revealing a monotonic reduction in boundary mismatch as polygonal complexity increases. Although derived from opposite geometric constructions, both laws converge toward the same circular limit. Under equal-area conditions, polygonal boundaries approach the circle through contraction; under inscribed-circle conditions, polygonal boundaries approach the circle through progressive boundary equivalence. Together, these laws establish a unified geometric duality linking boundary efficiency, impedance minimisation, and convergence for all regular polygons with n≥5. The Acris Twin Boundary Laws demonstrate that, regardless of whether a circle bounds a polygon externally through area equivalence or internally through inscription, increasing polygonal complexity causes the polygonal boundary to converge toward the unique efficiency of the circle. This twin-law structure has not previously been formalised as a unified geometric framework and provides a new perspective for analysing polygon-circle relationships.