Timeline for Is it possible to partition $\mathbb R^3$ into unit circles?
Current License: CC BY-SA 2.5
18 events
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| Aug 22, 2023 at 21:50 | comment | added | Daniel Asimov | J.H. Conway and H.T. Croft proved that 3-space can be partitioned into congruent geometric circles, using the axiom of choice (Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 4 , October 1964). I have shown that for a continuous foliation of a connected open set U of 3-space by congruent geometric circles, there is a finite upper bound on the volume of such a U. (Forthcoming in a joint paper.) | |
| May 21, 2015 at 10:15 | history | edited | user9072 |
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| Jul 21, 2013 at 15:42 | answer | added | Wlodek Kuperberg | timeline score: 14 | |
| Jun 19, 2013 at 10:35 | comment | added | Dan Christensen | @Ryan I might be the Canadian you are remembering. The solution I know using round circles is the one described by Spencer below, and I learned it from Danny Arnon, who was a student of Mike Hopkins at the time. | |
| Jun 19, 2013 at 9:17 | answer | added | Dan Christensen | timeline score: 51 | |
| Nov 27, 2010 at 14:56 | answer | added | Spencer | timeline score: 32 | |
| Jun 19, 2010 at 8:03 | comment | added | Kevin H. Lin | What is the motivation of this question? | |
| Jun 18, 2010 at 21:12 | vote | accept | Zarathustra | ||
| Jun 18, 2010 at 19:29 | comment | added | Zarathustra | Ryan, it was given to me as an exercise. I've spent many hours trying to solve it. Now I understand I was supposed to use the axiom of choice, which I actually tried; presumably not hard enough. | |
| Jun 18, 2010 at 19:07 | answer | added | Joel David Hamkins | timeline score: 25 | |
| Jun 18, 2010 at 18:39 | comment | added | Joel David Hamkins | See related question: mathoverflow.net/questions/21327/… | |
| Jun 18, 2010 at 18:32 | answer | added | Péter Komjáth | timeline score: 69 | |
| Jun 18, 2010 at 18:16 | comment | added | Ryan Budney | A smooth circle is just a smooth compact connected 1-dimensional submanifold of $\mathbb R^3$, i.e. the image of a smooth non-constant periodic function $f: \mathbb R \to \mathbb R^3$. Round means having constant curvature and zero torsion, also there's the equivalent definition of O'Rourke's, in the comments to his answer below. | |
| Jun 18, 2010 at 18:13 | comment | added | Kiochi | What's the difference between a smooth circle and a round circle? | |
| Jun 18, 2010 at 18:01 | history | edited | Joseph O'Rourke |
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| Jun 18, 2010 at 17:55 | answer | added | Joseph O'Rourke | timeline score: 10 | |
| Jun 18, 2010 at 17:46 | comment | added | Ryan Budney | I know how to do it with smooth circles. Maybe 15 years ago some Hopkins student (I forget his name, he is Canadian) mentioned he knew how to do it with round circles. But I don't think he ever described the construction to me. I'm curious why you're interested in this question? | |
| Jun 18, 2010 at 17:39 | history | asked | Zarathustra | CC BY-SA 2.5 |