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July 16th, 2003

quiz time for all you math smartypantses @ 12:51 am

Current Mood: thoughtful thoughtful
Current Music: Soul Coughing, Soundtrack to Mary

Take as long as you like. Use a #2 pencil. Do not pee on your monitor.

Given a two-dimensional sheet of rubber, what shape would create the smallest net amount of internal tension when the three pieces are glued edge-to-edge to create a hemisphere?

or stated another way: what's the optimal piece shape to assemble half a sphere out of three pieces of (flat) rubber?
 
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From:metallikat
Date:July 16th, 2003 01:59 am (UTC)
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Hah. I just spent 25 minutes on it and got stuck, so i'm not posting my workings here - but it'd be 3 rounded-off triangles, wouldn't it?
From:metallikat
Date:July 16th, 2003 02:09 am (UTC)
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Found this:
http://www.neubert.net/DESCarte.html

God, I'm not surprised I had problems if Descartes was wrangling with it.
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From:absintheminded
Date:July 16th, 2003 10:23 am (UTC)
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Unless I'm misunderstanding what you mean by 'internal tension', this depends on factors such as the stiffness and thickness of the rubber, since the edges will have to be beveled in order to get the same curvature inside the pieces as at the boundaries. Otherwise you'll have a polyhedron which is topologically the same as a hemisphere; but a hemisphere is topologically the same as a flat disk with no curvature.

Intuition inclines me to agree with zeocin -- well, almost. Just three triangles meeting in a point will do, no rounding-off is required; moreover, I'd venture that the triangles ought to be equilateral. Proving that this arrangement has the minimal internal tension would require more information, however.
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From:zunger
Date:July 16th, 2003 12:24 pm (UTC)
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I'm kinda confused by the question. What sort of pieces do we have, and how are we allowed to attach them? Are we working with support struts here? (Since otherwise, three pieces of rubber will just flop around)
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From:moof
Date:July 16th, 2003 12:36 pm (UTC)
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the pieces are individual pieces of rubber, shaped however you like. they're infinitely stretchable. you attach them edge to edge. flopping around is fine.

(I'm not sure if in the physical version of this problem I'm gonna bother with supports or not.)
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From:absintheminded
Date:July 16th, 2003 02:26 pm (UTC)
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ok, i think i see what you mean. the edges will need to be curved so that the top angle for each triangle is 360/3=120 degrees, and the bottom edge will have two right angles for each.
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From:bolteroo
Date:July 16th, 2003 02:46 pm (UTC)
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this is a burning man question, isn't it?
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From:moof
Date:July 16th, 2003 02:52 pm (UTC)
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Heh. Actually, it isn't. It's a clothey thing. :)
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From:merovingian
Date:July 17th, 2003 02:30 am (UTC)
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My immediate guess is that a series of three tree-like structures, with arbitrarily thin and dense "cilia," would flex into the hemisphere most easily.

But wouldn't be practical for any use.

For practical use, I would guess it'd be three vaguely lenticular "football" shapes like volleyballs use.
From:hattifattener
Date:July 19th, 2003 12:48 am (UTC)
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You're sick clever.

Okay, going along the same lines, what about cutting the hemisphere in a spiral shape (plus a small disk at the apex) and then unwinding it? By making the spiral infinitesimally thin, the curvature of the strip should go to zero. Then just snip it at two arbitrary points to make it into three pieces as requested.
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From:moof
Date:July 19th, 2003 01:19 am (UTC)
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The only problem with this is that it sucks if you're actually trying to make the thing out of real, live rubber sheeting. :)
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From:merovingian
Date:July 20th, 2003 08:39 pm (UTC)

Re:

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I think the genral trick of "approach arbitrary thinness" does the job, since then it can fold instead of stretching.
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From:allida
Date:July 17th, 2003 09:32 pm (UTC)
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i am going to venture to say, if you have to use three, and not two, that something along the lines of a baseball, but you would have to figure it out a little more than that

a baseball is made out of two pieces that are ALL curves, and that is what makes it more stable than a round ball, which usually is done with between three and six lozenges with chopped off tips sewn/glued to two round things at top and bottom. With three pieces, I think it gets more complicated. The easiest way to get a sphere is to do four triangles, with curved sides, but the points where they meet will be funky. So i am going to guess something like a baseball, two "Y" shaped pieces, and one round piece. Or maybe three "Y" shaped pieces, but i am not sure how to do it exactly. The other thing to try would be to make something like this (__) and two small spheres for top and bottom, glueing together the ends of the parentheses, and glueing the underscore to the spheres.

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