There is a problem in [Richard Kenyon's list](https://gauss.math.yale.edu/%7Erwk25/openprobs/index.html) ([Wayback Machine](https://web.archive.org/web/20191209170902/https://gauss.math.yale.edu/%7Erwk25/openprobs/index.html)) which I would like to post here, because although I have thought about it from time to time, I have not been able to make the slightest progress on it:



> 
> **Question:** "Given a closed polygonal path $p$ in $R^3$ composed of unit segments, is there an immersed polygonal surface
> whose faces are equilateral triangles of edge length $1$, spanning $p$?"
> 
> 
> 


The condition that the configuration is an "immersed" surface means that the whole configuration may be parametrized by a continuous locally one-to-one mapping from a compact connected 2-dimensional manifold with boundary into $R^3$. In other words at most two triangles may share an edge, and triangles sharing a vertex form an embedded topological disk, but two triangles which do not share an edge or a vertex may intersect.


I cannot answer the question even for the case where $p$ is a quadrilateral. Has there been any partial results for this problem?



Equilaterally triangulated surfaces with prescribed boundary
We resolve Kenyon's problem in [this paper](https://arxiv.org/abs/2005.02555 "Alexey Glazyrin, Igor Pak: Domes over curves"). We discuss in Section 5 a number of further conjectures and open problems.


