Domes over Curves. (2nd June 2021)

• Record Type: Journal Article
• Authors: Glazyrin, Alexey; Pak, Igor
• Is Part Of: International mathematics research notices. Volume 2022:Number 18(2022)
• Page Start: 14067
• Page End: 14104
• Publication Date: 2021-06-02
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab138
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: 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.