## The equilateral small octagon of maximal width

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Christian Bingane and Charles Audet
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## Abstract:

A small polygon is a polygon of unit diameter. The maximal width of an equilateral small polygon with $n=2^s$ vertices is not known when $s \ge 3$. This paper solves the first open case and finds the optimal equilateral small octagon. Its width is approximately $3.24%$ larger than the width of the regular octagon: $\cos (\pi /8)$. In addition, the paper proposes a family of equilateral small $n$-gons, for $n=2^s$ with $s\ge 4$, whose widths are within $O(1/n^4)$ of the maximal width.## References

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## Additional Information

**Christian Bingane**- Affiliation: Département de mathématiques et de génie industriel, Polytechnique Montréal, Montreal, Quebec H3C 3A7, Canada
- ORCID: 0000-0002-1980-5146
- Email: christian.bingane@polymtl.ca
**Charles Audet**- Affiliation: Département de mathématiques et de génie industriel, Polytechnique Montréal, Montreal, Quebec H3C 3A7, Canada
- MR Author ID: 619525
- ORCID: 0000-0002-3043-5393
- Email: charles.audet@polymtl.ca
- Received by editor(s): August 12, 2021
- Received by editor(s) in revised form: January 5, 2022
- Published electronically: March 30, 2022
- Additional Notes: This work was financed by the IVADO Fundamental Research Projects Grant PRF-2019-8079623546
- © Copyright 2022 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 2027-2040 - MSC (2020): Primary 52A40, 52A10, 52B55
- DOI: https://doi.org/10.1090/mcom/3733
- MathSciNet review: 4435955