Timeline for Does every polyomino tile R^n for some n?
Current License: CC BY-SA 2.5
24 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 17, 2015 at 19:22 | vote | accept | Timothy Chow | ||
| May 15, 2015 at 7:49 | answer | added | Francisco Santos | timeline score: 38 | |
| Feb 19, 2015 at 20:17 | comment | added | domotorp | I heard it in Cambridge, it was Imre Leader and his student, whose name unfortunately I don't know. | |
| Feb 19, 2015 at 14:55 | comment | added | Timothy Chow | @domotorp: Very interesting! What else do you know? Where did you hear this? Who solved it? | |
| Feb 19, 2015 at 13:41 | comment | added | domotorp | I just heard that this problem has been solved recently, hopefully there will be a write-up soon. | |
| Feb 11, 2011 at 14:59 | comment | added | Timothy Chow | @fastforward: I don't know, but if your worry is that the hole can't be filled, it's enough to lift to $\mathbb{R}^4$. Suppose the first tile is in a horizontal plane, centered at the origin. In "our" 3D slice of $\mathbb{R}^4$, the second tile will be partially visible as a vertical stack of five cubes going through the hole. Shift over to the "next" 3D slice of $\mathbb{R}^4$ and the first tile will disappear but you'll see another vertical stack of five cubes. Shift over again and you'll see four cubes (the middle cube will be missing). The next two shifts will have five cubes again. | |
| Feb 11, 2011 at 12:37 | comment | added | fastforward | @all How can a 5x5 square with the middle square removed, tile any R^n? Which n ? | |
| Jan 16, 2011 at 10:37 | answer | added | domotorp | timeline score: 19 | |
| Jan 12, 2011 at 5:05 | answer | added | Adam Chalcraft | timeline score: 14 | |
| Jan 4, 2011 at 16:19 | answer | added | Erich Friedman | timeline score: 5 | |
| Dec 27, 2010 at 14:05 | comment | added | Zsbán Ambrus | What we do know is that all (possibly disconnected) polyminos made of at most four squares tile the plane. | |
| Dec 21, 2010 at 16:28 | comment | added | Timothy Chow | @Amit: I think the comments already address the connectedness issue just fine. I don't think it matters much what you assume about connectedness, so perhaps being vague about the issue is the best way to convey the fact that I don't think it's important. By the way, I'd argue that "glued together facet-to-facet" doesn't have to imply connected. | |
| Dec 21, 2010 at 2:30 | comment | added | Amit Kumar Gupta | @Tim, I was a bit confused, since the question as posed assumes polyominos have to be connected ("glued together facet-to-facet"), and you say you assumed Adam intended connectedness. But Adam himself has said that he doesn't want to assume connectedness (see his answer below), and people seem to have interpreted the question to not require connectedness anyways (Igor's initial comment, for instance). Perhaps you want to edit your question to reflect the fact that there are two related questions of interest, one where polyominos are assumed to be connected, and one where they're not. | |
| Dec 20, 2010 at 23:06 | answer | added | fedja | timeline score: 2 | |
| Dec 20, 2010 at 22:10 | answer | added | Adam Chalcraft | timeline score: 10 | |
| Dec 20, 2010 at 18:36 | comment | added | Igor Pak | @Tim & others: Sorry for the confusion. I am not claiming that this tile is a counterexample (though I did check 2- and 3-dim). All I am saying that since the problem is very much unclear in rather simple special cases, hoping that the answer is always yes is perhaps too optimistic. The connectivity is not mentioned in the question and doesn't seem terribly relevant; I agree with Tim on this. Anyhow, there is no need to trust my intuition - see for yourself if you can tile the space with this tile. | |
| Dec 20, 2010 at 16:24 | comment | added | Timothy Chow | @Denis and Tony: I think Adam assumed connected, but I admit I don't see why Igor's suggestion is a counterexample. Anyway, if it is, then I'd guess that there's a 2-dimensional connected counterexample obtained by connecting up the components of Igor's example in some way. @Moshe: Rotations are certainly allowed. As for reflections, if there's a counterexample when reflections are disallowed, that would still be interesting. | |
| Dec 20, 2010 at 15:40 | comment | added | Tony Huynh | @Igor & Timothy: I thought polyominos are inductively defined by gluing a unit hypercube to a previously constructed polyomino along a facet? Perhaps Timothy can clarify. Much thanks. | |
| Dec 20, 2010 at 8:54 | comment | added | Denis Serre | Don't you impose to polyominos to be connected ? | |
| Dec 20, 2010 at 8:46 | comment | added | Moshe Schwartz | Do you allow only tiling by translate of the polyomino, or are rotations and inversions allowed as well? | |
| Dec 20, 2010 at 4:39 | comment | added | Allen Knutson | The interval in $\mathbb R$ of length 25 minus three intervals of length 1, starting at 9, 12, and 16. | |
| Dec 20, 2010 at 4:35 | comment | added | Amit Kumar Gupta | Igor, what is "ooooooo..."? | |
| Dec 20, 2010 at 1:07 | comment | added | Igor Pak | It's a cute question, Tim, but I would guess the answer is no, perhaps already for 1-dim polyominos. Try ooooooooo oo ooo oooooooo and see how that works out | |
| Dec 19, 2010 at 23:29 | history | asked | Timothy Chow | CC BY-SA 2.5 |