Timeline for Blocking visibility with cylinders
Current License: CC BY-SA 3.0
23 events
| when toggle format | what | by | license | comment | |
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| Sep 19, 2017 at 11:49 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Image links broken; now fixed.
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| May 11, 2011 at 7:51 | comment | added | Yaakov Baruch | Awesome graphics! Yaakov "There Is No Light At The End Of The Tunnel" Baruch. | |
| May 11, 2011 at 0:55 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
added 21 characters in body; added 12 characters in body; Post Made Community Wiki
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| May 11, 2011 at 0:51 | comment | added | Joseph O'Rourke | @Louigi: Indeed the problem generalizes to $\mathbb{R}^d$. @Gerhard and Aaron: I can't keep up with all the suggestions! :-) | |
| May 11, 2011 at 0:48 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Added image of forest of blocking cylinders.
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| May 10, 2011 at 23:28 | comment | added | Louigi Addario-Berry | What about in four dimensions? | |
| May 10, 2011 at 20:31 | comment | added | Gerhard Paseman | Alternatively, cover the unit sphere with six lunes(?) representing shadows of the cylinders on a sphere (looking from infinity into the central point). There are eight spherical triangles remaining to be covered. I think eight lunes suffice and they will be not much larger than the existing lunes. Tell me how much larger (I don't have my spherical trig slide rule handy) the lunes need to be, and I'll try to tell you how many more cylinders are needed and where to put them. Gerhard "The Light Is This Way" Paseman, 2011.05.10 | |
| May 10, 2011 at 20:23 | comment | added | Gerhard Paseman | For construction purposes, take 4 large (and 4 larger) cylinders at a sufficient distances and place their axes parallel to 4 of the faces of a central large (and a central larger) octahedron. This may be a helpful visualizing step to take before reading Aaron's comments above. Gerhard "Ask Me About System Design" Paseman, 2011.05.10 | |
| May 10, 2011 at 20:16 | comment | added | Aaron Meyerowitz | It would be nice to find a small collection of disjoint unit spheres which block the view (as in a recent planar problem) then embed each one in a cylinder keeping the cylinders disjoint (possibly using pairs of spheres). Gerhard's idea (with some touching spheres) probably fits this construction with a sphere centered on each of the 14 rays through a vertex or face center of a cube. Equivalently, through the rays on the face centers of he 14 faces of a cuboctehedron. Other semi-regular polyhedra might work as well. | |
| May 10, 2011 at 20:15 | comment | added | Aaron Meyerowitz | I agree that Gerhard's idea should work but I can't easily pin down the details. There are 8 small apertures close to the origin whose view must be blocked and 8 appropriately tilted flat disks of radius 2 (or 4 or 6) not too far out should do that if allowed. Instead use a cyclinder (or a group of 2 or 3 parallel and touching cylinders). With the freedom to rotate each (group) one should be able to avoid any interior intersections. | |
| May 10, 2011 at 12:54 | answer | added | Yaakov Baruch | timeline score: 5 | |
| May 10, 2011 at 11:52 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Mention half-baked idea on irregular weavings.
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| May 10, 2011 at 11:30 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
added 33 characters in body
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| May 10, 2011 at 11:12 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Image of Gerhard's construction.
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| May 9, 2011 at 23:08 | comment | added | Joseph O'Rourke | @Gerhard: Your idea is plausible but remains a little vague to me. I'll mull it over. Thanks! | |
| May 9, 2011 at 18:24 | comment | added | Gerhard Paseman | Perhaps my visualization skills could use some assistance. What about 6 cylinders tangent (or near tangent) to a unit sphere (say a certain subset of your red, blue, and green in your picture) and then 8 small groups of 2 or 3 cylinders close but not touching to block the remaining lines of sight? Gerhard "Ask Me About System Design" Paseman, 2011.05.09 | |
| May 9, 2011 at 17:34 | comment | added | Gerhard Paseman | You can mimic larger cylinders by groups of small cylinders. Should you not then be able to place large groups far enough away to patch any holes that a close and densely packed arrangement leaves? Gerhard "The Forest Is The Trees" Paseman, 2011.05.09 | |
| May 9, 2011 at 15:17 | comment | added | Mark Bennet | I posted a bad comment and deleted it. I think the question is interesting for dounbly infinite cylinders. If cylinders are allowed to be singly infinite take a hexagon of seven doubly infinite cylinders eg vertically and cut the middle one to allow space for p. (previous comment was six in a triangle - but until you get ten in a triangle there is no central one, and when you have ten it is only the central hexagon which counts) | |
| May 9, 2011 at 15:11 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
added 57 characters in body
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| May 9, 2011 at 15:05 | comment | added | Joseph O'Rourke | @Olivier: But I intended that all cylinders must be infinite in length. Likely my figure misled in that regard. Edited to make clearer. | |
| May 9, 2011 at 15:03 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Tried to clarify that infinite in length
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| May 9, 2011 at 14:47 | comment | added | Olivier Bégassat | can you arrange $3$ cylinders vertically with their centers forming a equilateral triangle around $p$, and then seal off the top and bottom holes with (I'd guess) $4$ other cylinders for each hole? | |
| May 9, 2011 at 14:31 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |