  A closed piecewise linear curve is called integral if it is comprised of unit
intervals. Kenyon's problem asks whether for every integral curve $\gamma$ in
$\mathbb{R}^3$, there is a dome over $\gamma$, i.e. 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.
