Domes over Curves

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.