A simple closed loop will always have four points that form a perfect isosceles trapezoid no matter how weirdly you draw it.
Every smooth Jordan curve contains a hidden geometric structure that cannot be escaped. This mathematical proof shows that you can always find the vertices of any possible isosceles trapezoid inscribed within the loop. This property holds true regardless of how much you stretch or wiggle the shape. It reveals a universal rule of geometry that applies to everything from a perfect circle to a squiggly doodle. This finding deepens our understanding of how fundamental shapes are embedded in the topology of the world.
Inscriptions of Isosceles Trapezoids in Jordan Curves
arXiv · 2604.27717
We construct a Lagrangian Floer homology whose chain complex is generically generated by the inscriptions of isosceles trapezoids in a smooth Jordan curve. This is an extension of Greene and Lobb's Jordan Floer homology (arXiv:2404.05179), which we also call Jordan Floer homology. Its non-triviality re-establishes that every smooth Jordan curve inscribes every isosceles trapezoid. By consideration of the spectral invariants associated with the real filtration known as the action filtration, we e