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. We are interested in thin rigid packings of the plane, rigid packings that that leave the largest possible fraction of the plane uncovered.

https://mathworld.wolfram.com/RigidCirclePacking.html gives what probably is the thinnest rigid packing of the plane with unit circles.

1. What is the thinnest rigid packing of the plane with unit squares? Is it a non-lattice arrangement? I do not know any references.

2. Which is the convex shape for which the densest and thinnest rigid packings of the plane differ the least in terms of coverage of the plane? Is it the unit circle?

Note: For example, with thin rectangles or thin triangles, we observe (https://nandacumar.blogspot.com/2020/03/thinnest-rigid-packings-contd.html) that difference in coverage can be arbitrarily high: indeed, the densest packing of the plane is a perfect tiling and the thinnest rigid pack can leave an arbitrarily large fraction of the plane uncovered.

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. We are interested in thin rigid packings of the plane, rigid packings that that leave the largest possible fraction of the plane uncovered.

https://mathworld.wolfram.com/RigidCirclePacking.html shows a way to achieve the thinnest rigid packing of the plane with unit circles - with density zero.

1. What is the thinnest rigid packing of the plane with unit squares? Is it a non-lattice arrangement? I do not know any references.

2. Which is the convex shape for which the densest and thinnest rigid packings of the plane differ the least in terms of coverage of the plane?

Note: For example, with thin rectangles or thin triangles, we observe (https://nandacumar.blogspot.com/2020/03/thinnest-rigid-packings-contd.html) that difference in coverage can be arbitrarily high: indeed, the densest packing of the plane is a perfect tiling and the thinnest rigid pack can leave an arbitrarily large fraction of the plane uncovered.

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. We are interested in thin rigid packings of the plane, rigid packings that that leave the largest possible fraction of the plane uncovered.

https://mathworld.wolfram.com/RigidCirclePacking.html gives what probably is the thinnest rigid packing of the plane with unit circles.

1. What is the thinnest rigid packing of the plane with unit squares? Is it a non-lattice arrangement? I do not know any references.

2. Which is the convex shape for which the densest and thinnest rigid packings of the plane differ the least in terms of coverage of the plane? Is it the unit circle?

Note: For example, with thin rectangles or thin triangles, we observe (https://nandacumar.blogspot.com/2020/03/thinnest-rigid-packings-contd.html) that this difference can be arbitrarily high: indeed, the densest packing of the plane is a perfect tiling and the thinnest rigid pack can leave an arbitrarily large fraction of the plane uncovered.

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. We are interested in thin rigid packings of the plane, rigid packings that that leave the largest possible fraction of the plane uncovered.

https://mathworld.wolfram.com/RigidCirclePacking.html gives what probably is the thinnest rigid packing of the plane with unit circles.

1. What is the thinnest rigid packing of the plane with unit squares? Is it a non-lattice arrangement? I do not know any references.

2. Which is the convex shape for which the densest and thinnest rigid packings of the plane differ the least in terms of coverage of the plane? Is it the unit circle?

Note: For example, with thin rectangles or thin triangles, we observe (https://nandacumar.blogspot.com/2020/03/thinnest-rigid-packings-contd.html) that difference in coverage can be arbitrarily high: indeed, the densest packing of the plane is a perfect tiling and the thinnest rigid pack can leave an arbitrarily large fraction of the plane uncovered.