Timeline for Monotile that tiles when you apply a rubber band
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 14, 2020 at 16:14 | comment | added | Ville Salo | And I agree with you. | |
| Jun 14, 2020 at 15:15 | comment | added | Pietro Majer | What I'm saying is just that for the 9 squares tiling T with the diamond in the middle, there exists a path of tilings p(x) with p(0)=T and c(p(x)) strictly decreasing. | |
| Jun 14, 2020 at 15:06 | comment | added | Ville Salo | No, my question (after edit) contains a weak jam. | |
| Jun 14, 2020 at 14:47 | comment | added | Pietro Majer | I think it is not even a weak jam, for the length of the octagon is strictly decreasing in the process | |
| Jun 14, 2020 at 14:21 | comment | added | Ville Salo | So if I understand @PietroMajer correctly, this is not a strong jam. I give a weak jam in the question, and this is presumably a physical jam, so so far I suppose we believe the square is neither a physical nor a weak rubber band monotile, but do not yet know whether it is a weak rubber band monotile, i.e. it might not have a strong jam. To me strong jams seem like the most interesting of these, so would be nice to figure out whether they exist. (Sorry about the choice of terminology, it's maybe not the most convenient.) | |
| Jun 14, 2020 at 8:20 | comment | added | Pietro Majer | A small check also show that one can do the whole 90 degrees rotation of the diamond, till the tiling reaches the 3x3 minimal configuration (that is, in the process the 4 corner squares never overlap the central square) | |
| Jun 14, 2020 at 8:20 | comment | added | Pietro Majer | In the 9-square configuration, let's rotate the central diamond, and let the N,S,W,E squares move respectively S,N E,W, always touching the corresponding vertices of the diamond, and let the other square follow staying in touch. Then the length of the boundary of the convex hull (the rubber) decreases, because it is made by 4 unit segments plus 4 hypotenuses of 4 right-angled triangles with a unit cathetus, and a cathetus whose length decreases. | |
| Jun 14, 2020 at 2:15 | comment | added | Ville Salo | My guess is also that it's impossible, and your suggestion indeed should work for the square with the uniform rubber band. The comment of @GevaYashfe is also relevant for figuring out whether this also proves the square is not a weak rubber band monotile, i.e. whether this jam is strong. | |
| Jun 14, 2020 at 0:21 | comment | added | Geva Yashfe | This seems to work, and give arbitrarily large configurations with minor modification. However, it's not clear to me that it continues working if force is applied unevenly to the different sides (it looks like it might snap into a 3x3 square of squares if the symmetry of the configuration is broken?) Maybe the question should be extended to "typical" configurations in some sense, or to configurations in which some "external shell" is randomly perturbed. | |
| Jun 13, 2020 at 23:23 | history | answered | Gerhard Paseman | CC BY-SA 4.0 |