Domes over Curves. (2nd June 2021) - Glazyrin, Alexey; Pak, Igor
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 $γ $ in ${\mathbb{R}}^3$, there is a dome over $γ $, that is, whether $γ $ is a boundary of a polyhedral surface whose faces are equilateral triangles with unit edge lengths. First, we give an algebraic necessary condition when $γ $ 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.