On Nearly Optimal Paper Moebius Bands

TL;DR

This paper proves that nearly optimal paper Moebius bands are close to equilateral triangles, providing a sharp quantitative estimate on their shape when the aspect ratio is close to a specific value.

Contribution

It establishes a precise Hausdorff distance bound for nearly optimal paper Moebius bands, improving understanding of their geometric structure.

Findings

Nearly optimal paper Moebius bands are close to equilateral triangles.

Hausdorff distance between the band and the triangle is bounded by $18 \, ext{sqrt} \, \epsilon$.

Results are sharp and effective, refining previous theorems.

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.