Domes over Curves

A closed piecewise linear curve is called integral if it is composed of unit intervals. Kenyon’s problem asks whether for every integral curve $\gamma $ in ${\mathbb{R}}^3$, there is a dome over $\gamma $, that is, whether $\gamma $ is a boundary of a polyhedral surface whose faces are equilateral t...

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Bibliographic Details
Published in:International mathematics research notices 2022-09, Vol.2022 (18), p.14067-14104
Main Authors: Glazyrin, Alexey, Pak, Igor
Format: Article
Language:English
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Summary:A closed piecewise linear curve is called integral if it is composed of unit intervals. Kenyon’s problem asks whether for every integral curve $\gamma $ in ${\mathbb{R}}^3$, there is a dome over $\gamma $, that is, whether $\gamma $ is a boundary of a polyhedral surface whose faces are equilateral triangles with unit edge lengths. First, we give an algebraic necessary condition when $\gamma $ is a quadrilateral, thus giving a negative solution to Kenyon’s problem in full generality. We then prove that domes exist over a dense set of integral curves. Finally, we give an explicit construction of domes over all regular $n$-gons.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnab138