Is it possible to partition $\mathbb R^3$ into unit circles?
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3$\begingroup$ What's the difference between a smooth circle and a round circle? $\endgroup$– KiochiCommented Jun 18, 2010 at 18:13
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2$\begingroup$ 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. $\endgroup$– Ryan BudneyCommented Jun 18, 2010 at 18:16
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2$\begingroup$ See related question: mathoverflow.net/questions/21327/… $\endgroup$– Joel David HamkinsCommented Jun 18, 2010 at 18:39
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2$\begingroup$ 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. $\endgroup$– ZarathustraCommented Jun 18, 2010 at 19:29
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8$\begingroup$ @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. $\endgroup$– Dan ChristensenCommented Jun 19, 2013 at 10:35
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6 Answers
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The construction is based on a well ordering of $R^3$ into the least ordinal of cardinality continuum. Let $\phi$ be that ordinal and let $R^3=\{p_\alpha:\alpha<\phi\}$ be an enumeration of the points of space. We define a unit circle $C_\alpha$ containing $p_\alpha$ by transfinite recursion on $\alpha$, for some $\alpha$ we do nothing. Here is the recursion step. Assume we have reached step $\alpha$ and some circles $\{C_\beta:\beta<\alpha\}$ have been determined. If some of them contains (=covers) $p_\alpha$, we do nothing. Otherwise, we choose a unit circle containing $p_\alpha$ that misses all the earlier circles. For that, we first choose a plane through $p_\alpha$ that is distinct from the planes of the earlier circles. This is possible, as there are continuum many planes through $p_\alpha$ and less than continuum many planes which are the planes of those earlier circles. Let $K$ be the plane chosen. The earlier circles intersect $K$ in less than continuum many points, so it suffices to find, in $K$, a unit circle going through $p_\alpha$ which misses certain less than continuum many points. That is easy: there are continuum many unit circles in $K$ that pass through $p_\alpha$ and each of the bad points disqualifies only 2 of them.
answered Jun 18, 2010 at 18:32
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$\begingroup$ I understand your solution (provided I believe that there's such a well ordering). A little bit disappointed, but thank you! $\endgroup$ Commented Jun 18, 2010 at 21:12
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Even though this question is old, I'd like to give what I regard as a very beautiful solution. It is different from the others in that the circles used are not round (but they are unlinked). First observe that the circles $x^2 + y^2 = r^2$, $z = c$, for $r \geq 1$ and $c$ any real number, decompose all of $\Bbb{R}^3$ except an open cylinder into circles. At first glance, this seems to have accomplished nothing, since the open cylinder is homeomorphic to $\Bbb{R}^3$, so we have reduced the original problem to an equivalent problem. However, look at the left-hand figure of the included image, which shows an open cylinder embedded as a U shape, with the ends going to infinity in the same direction. Since this is just a deformation of the original embedding, we can decompose the complement into circles. To handle the interior, embed an open cylinder into it, as shown in the right-hand figure. We can decompose the complement of the smaller U-shaped cylinder into circles. We continue in this way, making sure that the embedded cylinders go off to infinity, so that every point of $\Bbb{R}^3$ is included at some finite stage.
It seems like we have never really solved the problem, but instead have just pushed it away so much that it vanishes into thin air!
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$\begingroup$ @Dan: welcome to mathoverflow! $\endgroup$ Commented Jun 19, 2013 at 10:34
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1$\begingroup$ This (generalized) proof appears in the short 1985 paper "Topological partitions of Euclidean space by spheres" (by Bankston & Fox) for partitioning $\mathbb R^{n+2}$ by $S^n$, apparently due to S. Kakutani when $n=1$. $\endgroup$ Commented Apr 1, 2021 at 18:03
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$\begingroup$ @ChrisGerig Thanks for the reference! I heard that argument by word of mouth in the 90's, and it's good to know the source. This link pulls up the paper for me: researchgate.net/publication/… $\endgroup$ Commented Apr 2, 2021 at 20:51
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Péter's proof is very clever and, while there is no real need to resurrect this thread, the following is quite straightforward in case one is not inclined to hunt for it in the literature on this subject:
Observe that you can cover a two-punctured sphere with circles. Now consider a family of circles lying in the $xy$ plane, radii 1, centred at the points $(4k+1,0,0)$ for $k \in \mathbb{Z}$. Each sphere about the origin intersects this family in exactly two places.
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5$\begingroup$ This is exactly the construction of Szulkin [MR0719756] (see my answer below). $\endgroup$ Commented Jul 22, 2013 at 16:15
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2$\begingroup$ But the question was about a partition into unit circles, and it's not obvious that an arbitrary two-punctured sphere can be covered with unit circles. $\endgroup$ Commented Nov 6, 2020 at 14:19
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In this article1, the authors prove that not only can you partition $R^3$ into congruent circles, but you can do so into unlinked congruent circles. They also prove a variety of other similar results: $R^3$ can be partitioned into isometric copies of any family of continuum many real analytic curves. And they consider the question in higher dimensions, and also the role of AC in the proofs: for example, in $R^3$ no AC is needed for circles, if different sizes are allowed.
1M. Jonsson and J. Wästlund: Partition of $R^3$ into curves, Mathematica Scandinavica 83 (1998) 192-204; JSTOR, author's website
answered Jun 18, 2010 at 19:07
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2$\begingroup$ Is it known if the axiom of choice is required for congruent circles to partition? $\endgroup$ Commented Jun 18, 2010 at 19:28
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1$\begingroup$ Ryan, I'm not sure. All the arguments I have heard use AC in this case, but I have never heard a proof that AC was required for such a thing (and it is difficult to see how such a proof would proceed). But perhas the geometers will come through with a positive construction... $\endgroup$ Commented Jun 18, 2010 at 19:40
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$\begingroup$ The article is: mscand.dk/article/view/13850/11850 $\endgroup$ Commented Apr 3, 2014 at 14:44
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1$\begingroup$ @JoelDavidHamkins : What paper is this? The links aren't working for me, and you didn't provide authors or a title or even a journal or year. $\endgroup$ Commented Nov 4, 2020 at 18:48
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1$\begingroup$ The paper is: PARTITIONS OF R^3 INTO CURVES, M. JONSSON and J. WAÎSTLUND, MATH. SCAND. 83 (1998), 192-204. The link works for me. $\endgroup$ Commented Nov 4, 2020 at 20:36
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A very nice, explicit, elementary partition (without the Axiom of Choice) of $\mathbb{R}^3$ into geometric circles of variable radii is given in: [MR0719756] Szulkin, Andrzej. $\mathbb{R}^3$ is the union of disjoint circles. Amer. Math. Monthly 90 (1983), no. 9, 640–641.
answered Jul 21, 2013 at 15:42
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Evelyn Sander says here, "Geometric circles of unit radius are called hoops. Using the Axiom of Choice, J.H. Conway and H.T. Croft showed that it is nevertheless possible to discontinuously fill three-space using disjoint hoops." The "nevertheless" was to contrast with filling continuously. This was a report on a talk by Daniel Asimov in 1994, who showed that it is not possible to fill continuously with hoops.
answered Jun 18, 2010 at 17:55
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1$\begingroup$ Does "geometric" mean round with a fixed radius? $\endgroup$ Commented Jun 18, 2010 at 17:57
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$\begingroup$ @Ryan: Yes. "A geometric circle in $R^3$ is the set of points in a fixed plane that lie a fixed positive distance from a center point that lies in the same plane." $\endgroup$ Commented Jun 18, 2010 at 17:58
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$\begingroup$ I mean, do all the circles have the same radius? Certainly their centres and axis can vary. $\endgroup$ Commented Jun 18, 2010 at 18:11
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$\begingroup$ Ah, sorry. Somehow I completely mis-read your answer. It makes sense now and my comments were off-track. $\endgroup$ Commented Jun 18, 2010 at 18:18
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$\begingroup$ I don't understand, didn't your original answer say something about "using algebraic topology it's possible to show..." ? $\endgroup$ Commented Jun 18, 2010 at 18:23