Thinnest Rigid Packings of the Plane

A packing 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.

Here is part of a rigid pack of the plane with copies of the same rectangle.



Questions: Is this the thinnest rigid pack (ie, one with the largest fraction of the plane uncovered) of the plane possible with this rectangle? If so, for any rectangle, is the thinnest rigid pack of the plane formed in this very way? What if the unit is a triangle? If the triangle is a right triangle, is it best to pair them into rectangles and form a rigid pack with these rectangles? And what about a general convex region (for circles, here is a conjectured thinnest rigid pack: http://mathworld.wolfram.com/RigidCirclePacking.html)?

Among all convex shapes of a specified area and perimeter, which gives the thinnest rigid pack of the plane?...
And as usual, higher dimensions...

Most, if not all of the above should be known. Updates should follow...