[math-fun] Correction Re: Villarceau circles
Thanks to Fred for pointing out that I couldn't have meant 2-spheres in R^5 (that case is already excluded, as mentioned). What I meant to write is 2-spheres in R^6. (In fact, even just topologically: I don't know if topological 2-spheres can foliate a non-empty open set of R^6.) --Dan I wrote: << On 3/28/09, Dan Asimov <dasimov@earthlink.net> wrote:
P.S. For completeness I'll add that the above readily implies that there do exist non-empty open sets in R^(2k+1+p), for any p >= 0, that are foliated by congruent round k-spheres for k = 0,1,3,7. And conversely, for any k not in {0,1,3,7}, there does not exist a non-empty open set in R^(2k+1-p), for p >= 0 and 2k+1-p >= 0, that is foliated by congruent round k-spheres.
But this leaves unresolved infinitely many cases of k-spheres in n-space. First open case: Can congruent round 2-spheres foliate a non-empty open set in R^5 ??? ^^^
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
I think I must be missing something. Take a family of parallel, concentric 2-spheres whose radii span the open interval (1,2). Say they are in the abc-hyperplane. Their union is a 3-dimensional set. Now cross that set with an open 3-ball in the def-hyperplane. The resulting set is open, and 6-dimensional. I don't think any of the 2-spheres intersect any other. Doesn't that answer your question constructively, and in the affirmative? Maybe I'm missing one of the requirements for foliation. On Fri, Mar 27, 2009 at 10:51 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Thanks to Fred for pointing out that I couldn't have meant 2-spheres in R^5 (that case is already excluded, as mentioned).
What I meant to write is 2-spheres in R^6. (In fact, even just topologically: I don't know if topological 2-spheres can foliate a non-empty open set of R^6.)
--Dan
I wrote:
<< On 3/28/09, Dan Asimov <dasimov@earthlink.net> wrote:
P.S. For completeness I'll add that the above readily implies that there do exist non-empty open sets in R^(2k+1+p), for any p >= 0, that are foliated by congruent round k-spheres for k = 0,1,3,7. And conversely, for any k not in {0,1,3,7}, there does not exist a non-empty open set in R^(2k+1-p), for p >= 0 and 2k+1-p = 0, that is foliated by congruent round k-spheres.
But this leaves unresolved infinitely many cases of k-spheres in n-space. First open case: Can congruent round 2-spheres foliate a non-empty open set in R^5 ??? ^^^
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
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Your original question was about _congruent_ spheres, which would preclude my concentric trick. I think you threw the baby out with the bath when you relaxed your constraints all the way to mere topological 2-spheres. On Sat, Mar 28, 2009 at 1:17 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I think I must be missing something. Take a family of parallel, concentric 2-spheres whose radii span the open interval (1,2). Say they are in the abc-hyperplane. Their union is a 3-dimensional set. Now cross that set with an open 3-ball in the def-hyperplane. The resulting set is open, and 6-dimensional. I don't think any of the 2-spheres intersect any other. Doesn't that answer your question constructively, and in the affirmative? Maybe I'm missing one of the requirements for foliation.
On Fri, Mar 27, 2009 at 10:51 PM, Dan Asimov <dasimov@earthlink.net>wrote:
Thanks to Fred for pointing out that I couldn't have meant 2-spheres in R^5 (that case is already excluded, as mentioned).
What I meant to write is 2-spheres in R^6. (In fact, even just topologically: I don't know if topological 2-spheres can foliate a non-empty open set of R^6.)
--Dan
I wrote:
<< On 3/28/09, Dan Asimov <dasimov@earthlink.net> wrote:
P.S. For completeness I'll add that the above readily implies that there do exist non-empty open sets in R^(2k+1+p), for any p >= 0, that are foliated by congruent round k-spheres for k = 0,1,3,7. And conversely, for any k not in {0,1,3,7}, there does not exist a non-empty open set in R^(2k+1-p), for p >= 0 and 2k+1-p >= 0, that is foliated by congruent round k-spheres.
But this leaves unresolved infinitely many cases of k-spheres in n-space. First open case: Can congruent round 2-spheres foliate a non-empty open set in R^5 ??? ^^^
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
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Allan Wechsler -
Dan Asimov