I've been reading Wolfram's recent articles about graph/mesh/grid structures as an analogy for physical space, and it seems to me that there will be a problem getting the notion of distance to work out, since the natural metric on eg a square grid or triangle grid is L1, even in the limit of a very dense grid becoming a patch of the plane.

Is there any simple rule for generating a mesh-like graph that, with the length of all edges considered to the be the same, in the limit of very fine scale, closely approximates Euclidean distance in the plane?

Looking around, I was able to find this paper (https://projecteuclid.org/download/pdf_1/euclid.cmp/1104286245) which shows that if you allow the length of each edge in the graph to match its actual length in the plane, there is a solution with a simple production rule, but is it possible with all graph edges considered to be the same length?