Part 5/7:

By conceptualizing the dynamics of the closed loop and its mappings, mathematicians can confidently infer the existence of self-intersections—conditions crucial for establishing that inscribed rectangles must exist within any arbitrary closed loop.

A Shift Towards Squares

While the rectangle proof provides a fascinating glimpse into the world of topology, the inscribed square problem remains an unsolved enigma. To accomplish the latter requires not only mapping midpoints and distances but also tracking angles, leading to notions concerning embeddings in higher dimensions.