An Elementary Proof of Inscribed Squares for Smooth Curves via the H-Shape Construction

Abstract

We give a new, elementary proof that every C² Jordan curve in the plane inscribes at least one square. The existence of inscribed squares on smooth curves has been known since the work of Schnirelman (1929, 1944), and Stromquist (1989) extended the result to C¹ curves. Our contribution is a different proof method: the H-shape construction, a seven-point geometric configuration that encodes inscribed rectangles and reveals the role of the Z/4Z rotational symmetry. The proof shows that the 90° antisymmetry of the parameter space forces inscribed squares into orbits of size 4 that can only be created or destroyed in pairs at fold bifurcations, preserving a parity invariant. The argument uses only the implicit function theorem, compactness, and parity counting.Subjects (MSC 2020):** 51M04 (Elementary problems in Euclidean geometry); 55M25 (Degree, winding number); 57R35 (Differentiable mappings; transversality)