Timeline for Dissecting a square
Current License: CC BY-SA 3.0
33 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 30, 2011 at 8:10 | vote | accept | Colin D Wright | ||
| May 29, 2011 at 11:31 | vote | accept | Colin D Wright | ||
| May 29, 2011 at 11:31 | |||||
| May 29, 2011 at 11:30 | vote | accept | Colin D Wright | ||
| May 29, 2011 at 11:31 | |||||
| May 26, 2011 at 21:44 | comment | added | Colin D Wright | Have done already, as soon as I saw your posting there. I don't have much time at the moment, but I'm certainly interested. Currently trying to work out how to DM you there, but I have an early start to a long day tomorrow, so I might have to try another time. Thanks again. | |
| May 26, 2011 at 21:40 | comment | added | fedja | OK. By the way, join AoPS. We like such questions there and, if you really stand by what you say on your webpage, we need such people there :). | |
| May 26, 2011 at 20:47 | comment | added | Colin D Wright | @fedja That's wonderful. No, it's not a solution I had, so that partly answers my intended question. Thank you. I have much to think about, and much to work on. And now my brain hurts. | |
| May 26, 2011 at 20:26 | comment | added | fedja | Too many or's. Choose something. Meanwhile, I'll make my choice. I do not want to turn it into a topology problem, so I'll not assume the pieces connected. I do not want it to become some amenability question either, so I suggest to assume that each set is a closure of its interior with boundary of zero measure. Under these assumptions, look at artofproblemsolving.com/Forum/viewtopic.php?f=48&t=408538 Is it one of the solutions you knew? | |
| May 26, 2011 at 19:33 | history | edited | Colin D Wright | CC BY-SA 3.0 |
Correcting typos
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| May 26, 2011 at 17:57 | answer | added | Juris Steprans | timeline score: 2 | |
| May 26, 2011 at 17:53 | comment | added | Colin D Wright | @fedja Nice exploration of the pathological - thank you. Perhaps we should say that the sets are connected, or have non-empty interior, or that the interiors of their closures are disjoint, or something similar. Something to better capture the "intuitive" concept of piece. But I like the pathological, and will think on it further. | |
| May 26, 2011 at 17:45 | comment | added | fedja | Equal=obtained from each other by rotation and translation; piece=a measurable set; disjoint=intersecting by a set of measure 0; cut=undefined. touching at the center=containing the center Then you can have as many as you want though you'll call it shameless cheating (just take small squares and add a few isolated points to each). Anyway, to talk about pieces when distinguishing solutions is a bit awkward. It makes much more sense to talk about the corresponding set of rigid motions. Then so far the solutions are finitely many. | |
| May 26, 2011 at 17:40 | comment | added | Colin D Wright | Congruent - reflections allowed. It's interesting to separate out those dissections that use reflection from those that don't. Again, I believe they can be classified. | |
| May 26, 2011 at 17:10 | comment | added | Mark Bennet | What is meant by 'identical' - is this congruent, or directly congruent (are reflections allowed)? | |
| May 26, 2011 at 15:03 | comment | added | Colin D Wright | @fedja I'd be interested to see the assumptions that would let you get arbitrarily many pieces. | |
| May 26, 2011 at 14:38 | comment | added | fedja | The short answer to that is "Yes". I guess the first question to ask here is if we can get arbitrarily many pieces but that would require the precise definitions. | |
| May 26, 2011 at 14:30 | comment | added | Colin D Wright | Do I need to re-write the question again to make it clear what I mean by "cut" and "piece"? People are producing incomplete sets of dissections, but no one is talking about proofs of completeness. Is that just because it's hard? | |
| May 26, 2011 at 14:20 | answer | added | Aaron Meyerowitz | timeline score: 13 | |
| May 26, 2011 at 14:01 | comment | added | Colin D Wright | @Tapio we should probably add "connected," and perhaps strengthen that to "pathwise connected" | |
| May 26, 2011 at 13:54 | comment | added | Tapio Rajala | @unknowngoogle: I understood the question as writing the square as a disjoint union of sets which are isometric to one another and which have distance $0$ to the center and of a set of Lebesgue measure zero (the part which gets cut out). | |
| May 26, 2011 at 13:49 | comment | added | Qfwfq | What do you mean by "pieces"? Does it mean "any subset" without any regularity or connectedness assumption? | |
| May 26, 2011 at 13:45 | comment | added | Colin D Wright | @Qiaochu OK, understood. I had both recommended to me, and I had a spare 30 minutes. Apologies, and thank you. | |
| May 26, 2011 at 13:43 | comment | added | Qiaochu Yuan | @Colin: also, cross-posting on math.SE (math.stackexchange.com/questions/41499/…) is discouraged. Generally we want people to decide on one site, and if it turns out not to be appropriate (or doesn't get any answers on math.SE) then ask on the other site. | |
| May 26, 2011 at 13:34 | history | edited | Colin D Wright | CC BY-SA 3.0 |
Clarified and re-phrased the question
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| May 26, 2011 at 13:30 | comment | added | Colin D Wright | @Qiaochu I will edit my question @Beni Yes, I've replied to it. | |
| May 26, 2011 at 13:27 | comment | added | Beni Bogosel | @Colin: Do you see the answer below? | |
| May 26, 2011 at 13:25 | comment | added | Colin D Wright | OK, I guess I'm still trying to work out how this site works, and what people want. I was hoping that people would answer the question of how many ways there are to dissect the square, and how they would prove they got them all. I was hoping to pattern much my answer to theirs. Hmm. | |
| May 26, 2011 at 13:21 | comment | added | Qiaochu Yuan | @Colin: that question may be clear, but that is not the question you asked. The question you asked is "did I miss any solutions, and how can I prove that my answer is complete," and neither of these questions is possible to answer without knowing what your answer is. | |
| May 26, 2011 at 13:20 | comment | added | Colin D Wright | Of course there are infinitely many solutions. The question is, what families of solutions exist, and are there any sporadics. Sorry, I'm new here, but I thought that the question: How many ways are there to dissect a square into congruent pieces such that all of them touch the centre? was pretty clear. If there are infinitely many solutions, characterise them. | |
| May 26, 2011 at 13:20 | comment | added | Beni Bogosel | No, there are not infinitely many. The pieces must be identical, and must touch the center point. | |
| May 26, 2011 at 13:19 | answer | added | Beni Bogosel | timeline score: 2 | |
| May 26, 2011 at 13:17 | comment | added | Steven Gubkin | It seems to me that there are infinitely many solutions. Voting to close. | |
| May 26, 2011 at 13:15 | comment | added | Abel Stolz | What exactly is your question? How can we help you to show that your answer is "complete", when we don't know your answer, let alone your proof? | |
| May 26, 2011 at 13:10 | history | asked | Colin D Wright | CC BY-SA 3.0 |