About this video

A talk by Professor Igor Pak (UCLA), Title: Domes over curves Abstract: A closed PL-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.