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An improved bound on the optimal paper Moebius band

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A Correction to this article was published on 13 December 2023

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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.

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Correspondence to Richard Evan Schwartz.

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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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