Timeline for Polygons uniquely inducing arrangements
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| May 21, 2017 at 1:56 | comment | added | user44143 | Might there be a unique smallest or unique convex inducing polygon? | |
| May 21, 2017 at 0:07 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Image links broken; now fixed.
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| May 4, 2012 at 13:42 | comment | added | Zsbán Ambrus | There are no simple polyhedrons with exactly 5 sides, so any simple arrangement of 5 planes will be a counterexample for Q2. | |
| Apr 29, 2012 at 13:46 | vote | accept | Joseph O'Rourke | ||
| Apr 24, 2012 at 1:49 | answer | added | Gjergji Zaimi | timeline score: 4 | |
| Feb 10, 2012 at 16:57 | comment | added | Gerhard Paseman | Here's an idea which might contribute: imagine the hyperplanes as made of cardboard or styrofoam with serrations at thethe intersections so you can break off pieces or reattach them. Take an arrangement and remove the unbounded bits of planes and discard. The goal is to find a simple polytope among the bounded remainder. There are generally many pointy bits, so start removing those, but you may need to reapply some so don't throw them away yet. Keep going until you are down to something which has few pieces belonging to each plane. Gerhard "And Then A Miracle Occurs" Paseman, 2012.02.10 | |
| Feb 10, 2012 at 15:22 | comment | added | Joseph O'Rourke | @Gerhard: Cool idea! I'll think about it. Meanwhile, here's that image: cs.smith.edu/~orourke/cantmate-small.gif | |
| Feb 10, 2012 at 15:05 | comment | added | Gerhard Paseman | You had a picture in your gallery of something I call a "Klingon triangle"; it was by Jeff Erickson KHi Jeff!)Gerhard and was a counterexample to some result about one polygon that could not be transformed to another using certain motions that would create interesting looking prisms. That might induce a unique arrangement, but the pentagon does not because you have four lines that can "flex" around a middle vertex. Gerhard "Ask Me About System Design" Paseman, 2012.02.10 | |
| Feb 10, 2012 at 13:54 | comment | added | Joseph O'Rourke | @Gjergji: Very nice idea! Can you expand on "smallest"? It seems if your idea works, it settles the question in any dimension. | |
| Feb 10, 2012 at 13:12 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Illustration of Gjergji's idea.
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| Feb 10, 2012 at 12:45 | comment | added | Roland Bacher | The proof of the paper by Ackerman, Pinchasi, Scharf and Scherfenberg shows also that there exists a homologically non-trivial Hamiltonian cycle for simple arrangements of the projective plane. | |
| Feb 10, 2012 at 3:06 | history | edited | Joseph O'Rourke |
edited tags
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| Feb 10, 2012 at 2:58 | comment | added | Gjergji Zaimi | For Q2, if I'm not mistaken, you can take the line arrangement induced on one of the planes, pick a simple inducing polygon there and then find the smallest polyhedron attached to this polygon. | |
| Feb 10, 2012 at 2:47 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |
