**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 - iow, 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?