Thinnest Rigid Packings of the Plane - Contd.

Here we elaborate a bit on this one year old post.

Let us recall the definition: 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. We are interested in thin and rigid packings of the plane, those that leave the largest possible fraction of the plane uncovered.

1. How does one form the thinnest rigid packing of the plane with unit squares? The layout may well be non-lattice! Note: As was noted in above linked page, even for unit circles, the thinnest rigid pack of the plane with them is not known for sure.

2. For a thin enough rectangle as unit, the configuration given in the above linked post

seems a good candidate for the thinnest rigid packing. Of course, as the rectangle unit get thinner, the coverage of the plane can be brought arbitrarily close to 0.

However, for fat rectangles (where length and breadth are close) tending to squares, this doesn't appear to be the way to thinnest packing.

3. For a general triangle, we can always have a rigid packing that leaves 1/2 of the plane uncovered:

But for thin enough triangles, it is better to form parallelograms with pairs of them and then to form analogs of above layout with rectangles - thus achieving arbitrarily low coverage of the plane.

4. Which is the convex shape for which the densest and thinnest rigid packs of the plane show least difference in coverage? Note: As noted above, for a thin rectangle or thin triangle as unit, the densest and thinnest rigid packs of the plane can show arbitrarily large difference in coverage.