A closed polygonal curve is called integral if it is comprised of unit intervals. Kenyon's problem asks whether for every integral curve $γ$ in $\mathbb{R}^3$, 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. I will survey both positive and negative results on the subject.
Joint work with Alexey Glazyrin. The talk is aimed at a general audience.