A closed piecewise linear curve is called integral if it is comprised of unit intervals. Kenyon's problem asks whether for every integral curve γ in ℝ3, there is a dome over γ, i.e. 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.
Alexey Glazyrin, Igor Pak, Domes over Curves, International Mathematics Research Notices, 2021;, rnab138, https://doi.org/10.1093/imrn/rnab138
International Mathematics Research Notices