Timeline for On rigid packings of the plane with a constraint
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 5, 2022 at 4:45 | comment | added | Nandakumar R | An observation on the above pictured 'half-lattice' pack with triangles. It appears for rigidity against both translation and rotation, every unit triangle needs to be held in place by at 3 points at least. In this arrangement, each unit is being held down at points (their vertices) as far apart from one another as possible. That should give low density for the pack but on the downside, each unit touches 6 other units and not the minimum 3 required. So, the density of pack might be reducible. | |
| Apr 30, 2022 at 12:53 | comment | added | Nandakumar R | Thanks for pointing this out. Added that bit to the question. | |
| Apr 30, 2022 at 10:30 | comment | added | Joseph O'Rourke | Note you specified translation: "no unit can be translated without disturbing others." It is a different problem if the arrangement must also be rigid w.r.t. rotations. | |
| Apr 30, 2022 at 6:39 | comment | added | Nandakumar R | Considering the above pictured arrangement with equilateral triangle units, it appears: with the centroid of one unit as center, we can rotate that unit a little bit without disturbing other units - and that unit comes loose from the arrangement. So, it seems this way cannot achieve a rigid pack with 1/2 as packing fraction - at least with equilateral triangles. | |
| Apr 29, 2022 at 17:39 | comment | added | Joseph O'Rourke | Yes. At least it is a target to beat. | |
| Apr 29, 2022 at 17:07 | comment | added | Nandakumar R | Yes. But I am not sure if that is the lowest density constrained pack. Thanks. | |
| Apr 29, 2022 at 14:51 | history | answered | Joseph O'Rourke | CC BY-SA 4.0 |