I consider a graph $G$ (possibly infinite, but locally finite) embedded in the Euclidean plane $\mathbb{E}^2 \cup \{\infty\}$ such that each local perturbation of the embedding "increases the total length". That is, for any sufficiently small neighborhood $U \subset \mathbb{E}^2$, the total length of $G \cap U$, if not empty, is minimized subject to fixed boundary points $G \cap \partial U$.
So all edges must be straight. If a vertex has degree 3, then it must be the Fermat point of its neighbors. If a vertex has degree 4, the adjacent edges must form two collinear pairs. In general, the outgoing unit vectors along the edges around a vertex must sum up to 0. I consider such embeddings as an analogy to minimal surfaces. Examples include, of course, infinite Steiner trees --- or finite Steiner trees if I use the point at infinity wisely.
I failed to find reasonable literature about such embeddings, hence would like to ask for references. In particular, I wonder if such an embedding may appear as (a nice supgraph of) the graph of some quasi-crystallographic tiling.
