Every Jordan curve inscribes uncountably many rhombi

TL;DR

This paper proves that any simple closed curve in the plane, regardless of its shape or smoothness, contains uncountably many rhombi inscribed within it, extending classical geometric inscribing results.

Contribution

It establishes that all Jordan curves, with no regularity restrictions, inscribe uncountably many rhombi, a significant generalization of inscribed polygon theorems.

Findings

Every Jordan curve inscribes uncountably many rhombi.

No regularity conditions are needed on the Jordan curve.

The result applies to all simple closed curves in the plane.

Abstract

We prove that every Jordan curve in $\mathbb{R}^2$ inscribes uncountably many rhombi. No regularity condition is assumed on the Jordan curve.