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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.

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    $\begingroup$ See this question and answers for packings with zero density. $\endgroup$
    – TonyK
    Commented Mar 13, 2020 at 18:52
  • $\begingroup$ Why do you write "probably" the thinnest? The packing by Böröczky cited on that page and in the linked question has density zero. Are you suggesting there may be ones with negative density? $\endgroup$
    – Yoav Kallus
    Commented Mar 13, 2020 at 18:54
  • $\begingroup$ thanks... made a corrective edit. $\endgroup$
    – Nandakumar R
    Commented Mar 13, 2020 at 19:50
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    $\begingroup$ The question now is: is there a shape which does not admit density zero rigid packings? $\endgroup$
    – Wojowu
    Commented Mar 13, 2020 at 20:06
  • $\begingroup$ For unit squares I believe you should be able to use the structure in the mathworld illustration, replacing the circles with alternating coordinate-aligned squares and 45-degree-rotated squares so that each square of one orientation is wedged between three squares of the other orientation. $\endgroup$
    – Yoav Kallus
    Commented Mar 13, 2020 at 20:50

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