Timeline for The Gömböc and monostatic objects
Current License: CC BY-SA 2.5
19 events
| when toggle format | what | by | license | comment | |
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| Jul 23, 2010 at 1:09 | comment | added | Robin Saunders | Since time has passed and some people may assume the ball's in my court, I should point out that I'm abysmal at calculus and am not going to be able to find an example (or proof of nonexistence thereof) myself. | |
| Jul 18, 2010 at 22:17 | comment | added | Robin Saunders | So the problem reduces to finding a function f from [-1,M] to the non-negative reals with the following properties: smooth on (-1,M) with non-positive second derivative; passes through 0 vertically at -1 and M; (x^2+f(x)^2) has positive derivative; and the integral of xf(x)^2 dx is 0. | |
| Jul 18, 2010 at 22:17 | comment | added | Robin Saunders | You're right, it doesn't follow directly. But we can put some constraints on what such an object could be like. Say we have a solid of revolution about the x axis, whose centre of mass is the origin. Take the cross section through the xy plane. Now the surface is a smooth, convex, plane curve, symmetric about the x axis; minima of the curve correspond to minima of the surface, so for there to be only one it must be one of the intersections with the axis. | |
| Jul 17, 2010 at 18:25 | comment | added | Peter Shor | @Robin: a three-dimensional surface of revolution isn't the same as a constant-mass plane curve, so it's not clear to me that this theorem directly applies. | |
| Jul 16, 2010 at 21:20 | comment | added | Joseph O'Rourke | @Robin: Ah! That makes sense. Nice application of the 4-vertex theorem! | |
| Jul 16, 2010 at 21:14 | comment | added | Robin Saunders | Well, it's pointed out on the creators' website (www.gomboc.eu) and also in the Wikipedia article that any plane curve must have at least two stable equilibria (as well as at least two unstable equilibria), as a consequence of the four-vertex theorem. So a surface of revolution wouldn't work even with my loosened criteria. | |
| Jul 16, 2010 at 20:54 | comment | added | Joseph O'Rourke | @Peter: My understanding is that Robin was loosening the criteria for Gömböc objects, so that surfaces of revolution do make sense: he just seeks a unique local min of $|X - o_X|$ with a big "saftey margin." | |
| Jul 16, 2010 at 18:19 | comment | added | Peter Shor | I expect that surfaces of revolution cannot be Gömböc objects. I don't know if there's a proof of this (if there isn't, it would be worth looking for one). | |
| Jul 16, 2010 at 11:43 | comment | added | Joseph O'Rourke | Perhaps restricting your question to surfaces of revolution would be a place to start. Do you know of the optimum 2D curve? Or good candidates? | |
| Jul 15, 2010 at 23:44 | history | edited | Robin Saunders | CC BY-SA 2.5 |
"My question then is" -> "So then I wondered"; hopefully this will clear it up for Igor and anyone else.
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| Jul 15, 2010 at 23:32 | comment | added | Robin Saunders | The precise question itself is stated further down; the "what if...?" was just used to introduce it. Sorry if that was a bit confusing. Ignoring the introductory background paragraph though, the rest of the question should be self-contained. Does it make sense, or is there something I could explain more clearly? | |
| Jul 15, 2010 at 23:05 | comment | added | Igor Pak | Can you perhaps rewrite the question to make is complete and precise? I am note sure I understand "What about...?" type of question. | |
| Jul 15, 2010 at 22:57 | history | edited | Robin Saunders | CC BY-SA 2.5 |
Pesky origins still causing problems (X and Y need not be concentric)! Hopefully this should fix it.
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| Jul 15, 2010 at 22:56 | comment | added | Joseph O'Rourke | @Robin: I see! Thanks for explaining. | |
| Jul 15, 2010 at 22:47 | comment | added | Robin Saunders | Technically you'll have a circle of "stable equilibria" near the apex of the cone. I know they're not truly stable, but informally they are "more like stable equilibria than anything else". More formally, they are still local minima of |X|. If placed on one of these points, the cone will roll around in circles, eventually coming to rest (due to friction) on a point other than the base. You'd also have to be careful rounding off edges in general, to avoid introducing extra local minima there. | |
| Jul 15, 2010 at 22:32 | history | edited | Robin Saunders | CC BY-SA 2.5 |
minor editing: removed superfluous "relative to the centre of mass" fluff
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| Jul 15, 2010 at 22:30 | comment | added | Joseph O'Rourke | How about a nearly flat cone with a flat base (rounded appropriately near edges to make it smooth)? Then the wide base will be a safe stable equilibrium point, and that seems to be the only stable position. Perhaps I am misinterpreting your question... | |
| Jul 15, 2010 at 22:24 | history | edited | Robin Saunders | CC BY-SA 2.5 |
considering |Y-X| needs them to be defined co-ordinate-wise, not in terms of radii
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| Jul 15, 2010 at 21:25 | history | asked | Robin Saunders | CC BY-SA 2.5 |