Timeline for Bicoloring of $\mathbb{N}^2$, avoiding set of patterns, is the maximal limit density rational?
Current License: CC BY-SA 3.0
16 events
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| Sep 11, 2015 at 7:05 | comment | added | Anthony Quas | So it's likely the case that there are Wang tilings where the frequency of one tile is irrational. The Kari-Culik tiles have the property that they only tile the plane aperiodically. K-C exhibit some tilings of the plane. Work by my student, Jason Siefken, can be used to show that for the class of tilings that K-C exhibit, some tiles show up with irrational frequencies. Unfortunately it was recently shown that there are other legal tilings than these. Still, one might hope that by adding further local restrictions, one can get back to the original set of tilings, solving Q2b | |
| Sep 10, 2015 at 21:29 | comment | added | Ilya Bogdanov | EDIT: It seems that Per is right, and the question is equivalent to the question of irrationality of occurrence density of some (translational) Wang tile. I've typed it quite briefly; sorry if this needs many explanations. | |
| Sep 10, 2015 at 21:28 | history | edited | Ilya Bogdanov | CC BY-SA 3.0 |
A better construction is provided.
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| Sep 10, 2015 at 17:12 | comment | added | Ilya Bogdanov | @Per: I'm not sure that it is that easy to achieve. This way, you need to assign to tiles the squares with different numbers of black cells. But then some arrangement with incomplete squares may happen to be better! | |
| Sep 10, 2015 at 17:09 | comment | added | Per Alexandersson | So, if there is a non-periodic Wang tiling, where the limit percentage of the presence of some tile is irrational, then Q2b has a positive answer. | |
| Sep 10, 2015 at 17:04 | comment | added | Ilya Bogdanov | That's what I meant in a comment for the original post... | |
| Sep 10, 2015 at 17:00 | comment | added | Per Alexandersson | So, (I need to read the details a few times more), this says Q2 is YES, but Q1 is then NO, (there is non-periodicity). This opens up the possibility to have an irrational limit! | |
| Sep 10, 2015 at 16:52 | comment | added | Anthony Quas | I see - so you're only imposing restrictions on regions that have maximal density. | |
| Sep 10, 2015 at 16:50 | comment | added | Ilya Bogdanov | I've added an explanation for this to part 4. | |
| Sep 10, 2015 at 16:49 | history | edited | Ilya Bogdanov | CC BY-SA 3.0 |
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| Sep 10, 2015 at 16:46 | comment | added | Anthony Quas | How can you "impose restrictions on the neighbouring tilings" in the original class of tilings? | |
| Sep 10, 2015 at 16:45 | history | edited | Ilya Bogdanov | CC BY-SA 3.0 |
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| Sep 10, 2015 at 16:44 | comment | added | Ilya Bogdanov | I do not claim that there are no periodic colorings at all; I only claim that the arrangement of maximal possible density cannot be periodic, since it should correspond to a periodic tiling. | |
| Sep 10, 2015 at 16:43 | comment | added | Anthony Quas | This doesn't sound right. The conditions are monotonic: given any legal configuration, any "smaller" configuration is legal. So in particular, unlike Wang tilings, any one of these systems contain periodic points (e.g. the all 0 point). | |
| Sep 10, 2015 at 16:38 | history | edited | Ilya Bogdanov | CC BY-SA 3.0 |
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| Sep 10, 2015 at 16:33 | history | answered | Ilya Bogdanov | CC BY-SA 3.0 |