On Nearly Optimal Paper Moebius Bands

Abstract

Let $\epsilon <1/384$ and let $\Omega$ be a smooth embedded paper Moebius band of aspect ratio less than $\sqrt{3}+\epsilon$. We prove that $\Omega$ is within Hausdorff distance $18\sqrt{\epsilon }$ of an equilateral triangle of perimeter $2\sqrt{3}$. This is an effective and fairly sharp version of our recent theorems in [{\bf S0\/}] about the optimal paper Moebius band.

This is a sequel to my paper "The Optimal paper Moebius Band" (arXiv:2308.12641) but I tried to make it roughly self-contained