TITLE: Equilaterally triangulated surfaces with prescribed boundary
QUESTION [16 upvotes]: There is a problem in Richard Kenyon's list 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?

REPLY [7 votes]: We resolve Kenyon's problem in this paper.  We discuss in Section 5 a number of further conjectures and open problems.