# An improved bound on the optimal paper Moebius band

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## Abstract

We show that a smooth embedded paper Moebius band must have aspect ratio at least

\begin{aligned}\lambda _1= \frac{2 \sqrt{4-2 \sqrt{3}}+4}{\root 4 \of {3} \sqrt{2}+2 \sqrt{2 \sqrt{3}-3}} =1.69497\ldots \end{aligned}

This bound comes more than 3/4 of the way from the old known bound of $$\pi /2=1.5708\ldots$$ to the conjectured bound of $$\sqrt{3} = 1.732\ldots$$

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## Notes

1. The smoothness requirement (or some suitable variant) is necessary in order to have a nontrivial problem. Given any $$\epsilon >0$$, one can start with the strip $$[0,1] \times [0,\epsilon ]$$ and first fold it (across vertical folds) so that it becomes, say, an $$(\epsilon /100) \times \epsilon$$ “accordion”. One can then easily twist this “accordion” once around in space so that it makes a Moebius band. The corresponding map from $$M_{\epsilon }$$ is an isometry but it cannot be approximated by smooth isometric embeddings.

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## Funding

Funding was provided by National Science Foundation (Grant No. DMS-1807320), Simons Foundation, Institute for Advanced Study.

## Author information

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Correspondence to Richard Evan Schwartz.

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Supported by N.S.F. Grant DMS-1807320.

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Schwartz, R.E. An improved bound on the optimal paper Moebius band. Geom Dedicata 215, 255–267 (2021). https://doi.org/10.1007/s10711-021-00648-5

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• DOI: https://doi.org/10.1007/s10711-021-00648-5