This post continues Thinnest rigid packings of the plane

A packing of the plane with copies of any shape is called rigid (or "stable") if every unit is fixed by its neighbors, i.e., no unit can be translated without disturbing others in the packing. Let us consider 'constrained rigid packs' - a rigid pack of identical units with the additional constraint that any 2 given units can touch each other at most at one point (we allow both corner to corner or corner to edge contacts).

• Given a general triangle T of unit area, how does one find a constrained rigid packing with copies of T such that packing density is maximized (minimized) - ie. the highest(lowest) fraction of the plane is covered? Specifically, for any T, can one approach a perfect packing arbitrarily closely with such a constrained rigid pack?

• Which specific triangle gives maximum(minimum) density for constrained rigid pack?

One can ask same questions with triangle replaced by other convex polygonal shapes.