This open problem taught me what topology is

Summary by Krater

This video explores the inscribed square problem and its topological proof, examining how continuous mappings of loop points onto three-dimensional surfaces help demonstrate that any smooth closed curve contains inscribed rectangles of all possible aspect ratios, and discussing non-orientable surfaces like the Möbius strip and the Klein bottle.

From the video

Answers: Do all closed continuous curves have an inscribed square?

• Inscribed square problem
• Continuous curves
• Topology
• Möbius strip
• Klein bottle
• Torus
• Borsuk-Ulam theorem

What it concludes

• Every smooth closed curve contains inscribed rectangles of all possible aspect ratios.
• A Möbius strip cannot be embedded in 3D with its edge on the xy-plane and its interior entirely above the xy-plane without intersecting itself.
• Any closed non-orientable surface cannot be mapped into 3D without self-intersection.