Timeline for Tetrahedron insphere iteration
Current License: CC BY-SA 3.0
25 events
| when toggle format | what | by | license | comment | |
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| S Oct 20, 2013 at 22:28 | history | bounty ended | Gil Kalai | ||
| S Oct 20, 2013 at 22:28 | history | notice removed | Gil Kalai | ||
| S Oct 18, 2013 at 18:05 | history | bounty started | Gil Kalai | ||
| S Oct 18, 2013 at 18:05 | history | notice added | Gil Kalai | Reward existing answer | |
| Oct 16, 2013 at 0:03 | comment | added | The Masked Avenger | Sorry for the delay @Wlodek. From a point external to a sphere choose three tangents through that point. Consider two triangles with vertices on these tangents, one inside the sphere, the other lying on a tangent plane "opposite" from the external point. The idea (now refuted) was that the internal triangle would be more regular than the external triangle. Perhaps this perspective might help in studying the triangles. | |
| Oct 12, 2013 at 15:01 | answer | added | Joseph O'Rourke | timeline score: 11 | |
| Oct 12, 2013 at 12:42 | vote | accept | Joseph O'Rourke | ||
| Oct 12, 2013 at 2:00 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
added 85 characters in body
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| Oct 11, 2013 at 22:47 | comment | added | j.c. | I'm reposting my answer as a comment here, as it was never really an answer to begin with, and fedja's nice answer deserves to be at the top. The proof of Theorem 2 in this paper homepages.warwick.ac.uk/~masdbl/inscribed.pdf by Mark Pollicott mentions that tetrahedra do not converge. | |
| Oct 11, 2013 at 19:42 | answer | added | Joseph O'Rourke | timeline score: 4 | |
| Oct 11, 2013 at 15:08 | answer | added | fedja | timeline score: 24 | |
| Oct 11, 2013 at 14:58 | comment | added | Wlodek Kuperberg | ... and then what, Mr. Avenger...? What do these cross sections have to do with the iterated process in n+1 dimensions? | |
| Oct 11, 2013 at 13:53 | comment | added | Suvrit | Your question also reminded me of a related question, but on doing averaging operations over random polygons, until one attains an ellipse! See e.g.,: cs.cornell.edu/cv/ResearchPDF/EllipsePoly.pdf | |
| Oct 11, 2013 at 2:46 | comment | added | The Masked Avenger | In n+1 dimensions, the n-sections that contain the n-faces of the induced inscribed simplex. | |
| Oct 11, 2013 at 1:08 | comment | added | Wlodek Kuperberg | Which cross sections specifically do you suggest, Mr. Avenger? | |
| Oct 10, 2013 at 23:53 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Better phrasing.
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| Oct 10, 2013 at 17:52 | comment | added | The Masked Avenger | Using cross sections, this suggests Joseph's result for arbitrary finite dimensions. | |
| Oct 10, 2013 at 17:43 | comment | added | Gerardo Arizmendi | Its not hard to see that if $a,b,c$ are the original angles then at the next step the angles are are $(a+b)/2$, $(a+c)/2$, $(b+c)/2$, and this implies the claim for the triangles. | |
| Oct 10, 2013 at 17:42 | comment | added | Joseph O'Rourke | @SashaKolpakov: Scale or not as you wish. The limit is an equiangular triangle. I am focusing on the shape, not the size, so if it makes it more transparent, imagine rescaling at each step. | |
| Oct 10, 2013 at 17:38 | comment | added | SashaKolpakov | Do you mean you rescale the triangle on each iteration, so that its circum-circle/in-circle has radius 1? | |
| Oct 10, 2013 at 17:36 | comment | added | Igor Rivin | You are a very trusting man :) | |
| Oct 10, 2013 at 17:34 | comment | added | Joseph O'Rourke | @IgorRivin: I know just from the word of a colleague. I do not have a reference. The angles change by a linear map... | |
| Oct 10, 2013 at 17:31 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
[Edit removed during grace period]
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| Oct 10, 2013 at 17:25 | comment | added | Igor Rivin | How do you know this? Is there a reference? Inquiring minds are puzzled... | |
| Oct 10, 2013 at 17:15 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |