On Nearly Optimal Paper Moebius Bands

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.