Background: Given a convex region C. One can define a graph corresponding to any planar arrangement of copies of C - each unit C is a node and an edge connects it to another node iff the two C units share some finite length of boundary. Such a graph is necessarily planar. As is well known, the average degree of a planar graph can at most be 6.

Now, the intuitive (parallelogram based) tilings of the plane with any triangle seem to correspond to graphs (defined above) with average degree = 3 or 4. In other words, there appears a gap between these values and the maximum average degree of a planar graph (6).

Question: Can one have any plane-filling arrangement (not necessarily a tiling) with any triangle where the average degree of the corresponding graph is between 4 and 6?

Note: With squares, it is easy to form a tiling with the corresponding graph having degree 6 at every node.

Generalization: Given any convex 2D shape C, not necessarily one that tiles. Assume the planar arrangement(s) with infinite copies of C that maximizes the average degree of the corresponding graph is known. How good are these arrangements at achieving max packing density?