[math-fun] Calculus of variations problem: the one-wire cage.
Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull?
The seam on a baseball? On Aug 30, 2013, at 10:02 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Is the seam on a baseball well-defined? WFL On 8/30/13, Cris Moore <moore@santafe.edu> wrote:
The seam on a baseball?
On Aug 30, 2013, at 10:02 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I should confess that I have no idea what the answer is. A baseball-seam-type answer seems wrong to me. My intuition is that you're wasting the ends of the curve by putting them too close together. If you take a tiny section near the end of the curve and bend it to make it poke out at right angles, aren't you increasing the volume by adding a little tent? On Fri, Aug 30, 2013 at 12:15 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Is the seam on a baseball well-defined? WFL
On 8/30/13, Cris Moore <moore@santafe.edu> wrote:
The seam on a baseball?
On Aug 30, 2013, at 10:02 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Well, the first thing that comes to mind is a Hamiltonian path on the edges of a cube (or maybe a tetrahedron). But the corners would be curved, giving a baseball-seam-like thing. On Aug 30, 2013, at 10:36 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I should confess that I have no idea what the answer is. A baseball-seam-type answer seems wrong to me. My intuition is that you're wasting the ends of the curve by putting them too close together. If you take a tiny section near the end of the curve and bend it to make it poke out at right angles, aren't you increasing the volume by adding a little tent?
On Fri, Aug 30, 2013 at 12:15 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Is the seam on a baseball well-defined? WFL
On 8/30/13, Cris Moore <moore@santafe.edu> wrote:
The seam on a baseball?
On Aug 30, 2013, at 10:02 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Cristopher Moore Professor, Santa Fe Institute
An ordinary cube needs us to bend the wire into seven equal segments, giving an edge length of 1/7 and a volume of 1/343 > 0.00291. Can that be bettered easily? On Fri, Aug 30, 2013 at 12:39 PM, Cris Moore <moore@santafe.edu> wrote:
Well, the first thing that comes to mind is a Hamiltonian path on the edges of a cube (or maybe a tetrahedron). But the corners would be curved, giving a baseball-seam-like thing.
On Aug 30, 2013, at 10:36 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I should confess that I have no idea what the answer is. A baseball-seam-type answer seems wrong to me. My intuition is that you're wasting the ends of the curve by putting them too close together. If you take a tiny section near the end of the curve and bend it to make it poke out at right angles, aren't you increasing the volume by adding a little tent?
On Fri, Aug 30, 2013 at 12:15 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Is the seam on a baseball well-defined? WFL
On 8/30/13, Cris Moore <moore@santafe.edu> wrote:
The seam on a baseball?
On Aug 30, 2013, at 10:02 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Cristopher Moore Professor, Santa Fe Institute
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It occurs to me that I don't even know the answer to this question in 2 dimensions. My instinct says "semicircle". On Fri, Aug 30, 2013 at 12:52 PM, Allan Wechsler <acwacw@gmail.com> wrote:
An ordinary cube needs us to bend the wire into seven equal segments, giving an edge length of 1/7 and a volume of 1/343 > 0.00291. Can that be bettered easily?
On Fri, Aug 30, 2013 at 12:39 PM, Cris Moore <moore@santafe.edu> wrote:
Well, the first thing that comes to mind is a Hamiltonian path on the edges of a cube (or maybe a tetrahedron). But the corners would be curved, giving a baseball-seam-like thing.
On Aug 30, 2013, at 10:36 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I should confess that I have no idea what the answer is. A baseball-seam-type answer seems wrong to me. My intuition is that you're wasting the ends of the curve by putting them too close together. If you take a tiny section near the end of the curve and bend it to make it poke out at right angles, aren't you increasing the volume by adding a little tent?
On Fri, Aug 30, 2013 at 12:15 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Is the seam on a baseball well-defined? WFL
On 8/30/13, Cris Moore <moore@santafe.edu> wrote:
The seam on a baseball?
On Aug 30, 2013, at 10:02 AM, Allan Wechsler <acwacw@gmail.com>
wrote:
Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Cristopher Moore Professor, Santa Fe Institute
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For 2 dimensions it indeed is the semicircle. This is known as Dido's problem. -- Gene
________________________________ From: Allan Wechsler <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Friday, August 30, 2013 9:53 AM Subject: Re: [math-fun] Calculus of variations problem: the one-wire cage.
It occurs to me that I don't even know the answer to this question in 2 dimensions. My instinct says "semicircle".
On Fri, Aug 30, 2013 at 12:52 PM, Allan Wechsler <acwacw@gmail.com> wrote:
An ordinary cube needs us to bend the wire into seven equal segments, giving an edge length of 1/7 and a volume of 1/343 > 0.00291. Can that be bettered easily?
On Fri, Aug 30, 2013 at 12:39 PM, Cris Moore <moore@santafe.edu> wrote:
Well, the first thing that comes to mind is a Hamiltonian path on the edges of a cube (or maybe a tetrahedron). But the corners would be curved, giving a baseball-seam-like thing.
On Aug 30, 2013, at 10:36 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I should confess that I have no idea what the answer is. A baseball-seam-type answer seems wrong to me. My intuition is that you're wasting the ends of the curve by putting them too close together. If you take a tiny section near the end of the curve and bend it to make it poke out at right angles, aren't you increasing the volume by adding a little tent?
On Fri, Aug 30, 2013 at 12:15 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Is the seam on a baseball well-defined? WFL
On 8/30/13, Cris Moore <moore@santafe.edu> wrote:
The seam on a baseball?
On Aug 30, 2013, at 10:02 AM, Allan Wechsler <acwacw@gmail.com>
wrote:
> Which three-dimensional continuous differentiable curve of unit > arc-length > encloses the greatest volume in its convex hull?
a, I thought you mean a closed curve... On Aug 30, 2013, at 11:13 AM, Eugene Salamin <gene_salamin@yahoo.com> wrote:
For 2 dimensions it indeed is the semicircle. This is known as Dido's problem.
-- Gene
________________________________ From: Allan Wechsler <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Friday, August 30, 2013 9:53 AM Subject: Re: [math-fun] Calculus of variations problem: the one-wire cage.
It occurs to me that I don't even know the answer to this question in 2 dimensions. My instinct says "semicircle".
On Fri, Aug 30, 2013 at 12:52 PM, Allan Wechsler <acwacw@gmail.com> wrote:
An ordinary cube needs us to bend the wire into seven equal segments, giving an edge length of 1/7 and a volume of 1/343 > 0.00291. Can that be bettered easily?
On Fri, Aug 30, 2013 at 12:39 PM, Cris Moore <moore@santafe.edu> wrote:
Well, the first thing that comes to mind is a Hamiltonian path on the edges of a cube (or maybe a tetrahedron). But the corners would be curved, giving a baseball-seam-like thing.
On Aug 30, 2013, at 10:36 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I should confess that I have no idea what the answer is. A baseball-seam-type answer seems wrong to me. My intuition is that you're wasting the ends of the curve by putting them too close together. If you take a tiny section near the end of the curve and bend it to make it poke out at right angles, aren't you increasing the volume by adding a little tent?
On Fri, Aug 30, 2013 at 12:15 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Is the seam on a baseball well-defined? WFL
On 8/30/13, Cris Moore <moore@santafe.edu> wrote: > > The seam on a baseball? > > On Aug 30, 2013, at 10:02 AM, Allan Wechsler <acwacw@gmail.com> wrote: > >> Which three-dimensional continuous differentiable curve of unit >> arc-length >> encloses the greatest volume in its convex hull?
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Cristopher Moore Professor, Santa Fe Institute
I'm not sure if this curve is supposed to be a closed curve, or whether it's allowed to cross itself. But I don't think the answer would be any different in those 4 cases. What would change the answer is if the curve need not be the image of a closed interval, but instead can be the image of a half-closed or open interval. If the curve is the image of a closed interval, then you can get convex hulls arbitrarily close to the volume of the unit ball = 4pi/3, but necessarily less than that. But if the curve is instead the image of a half-closed or open interval, then you can reach 4pi/3 by arranging that the curve be dense in the unit sphere, which isn't hard to do, so I'll leave that as an exercise for the reader. --Dan On 2013-08-30, at 9:02 AM, Allan Wechsler wrote:
Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Apologies, but I misunderstood the problem as saying that the curve has to *lie* on the unit sphere. The problem Allan asks was a problem that George Bergman and Wolfgang Kahane of U.C. Berkeley worked on informally around 2003 and solved -- and it was pretty hard. But I don't think they published, and I don't know what the answer is. --Dan On 2013-08-30, at 9:02 AM, Allan Wechsler wrote:
Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I wonder if they knew of the work of P. A. P. Moran that Erich referred to. On Fri, Aug 30, 2013 at 1:34 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Apologies, but I misunderstood the problem as saying that the curve has to *lie* on the unit sphere.
The problem Allan asks was a problem that George Bergman and Wolfgang Kahane of U.C. Berkeley worked on informally around 2003 and solved -- and it was pretty hard. But I don't think they published, and I don't know what the answer is.
--Dan
On 2013-08-30, at 9:02 AM, Allan Wechsler wrote:
Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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What's the maximum volume if require the curve to be closed? In two dimensions, the answer is a circle; the circle encloses the most area, so it is the best convex solution. And if you have a nonconvex closed curve, you can replace it with the boundary of its convex hull, which is always a shorter curve, and scale up. But I have no idea what the best solution is in 3 dimensions; here maybe a baseball-seam shaped curve is correct. But I remember Conway claiming on this list that every extremal mathematical property he had ever seen claimed for the baseball-seam curve was false, so either there's something better, or Conway was unaware of this problem. Andy Andy On Fri, Aug 30, 2013 at 1:50 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I wonder if they knew of the work of P. A. P. Moran that Erich referred to.
On Fri, Aug 30, 2013 at 1:34 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Apologies, but I misunderstood the problem as saying that the curve has to *lie* on the unit sphere.
The problem Allan asks was a problem that George Bergman and Wolfgang Kahane of U.C. Berkeley worked on informally around 2003 and solved -- and it was pretty hard. But I don't think they published, and I don't know what the answer is.
--Dan
On 2013-08-30, at 9:02 AM, Allan Wechsler wrote:
Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Andy.Latto@pobox.com
I agree that the way the lobes of the baseball seam curve in towards each other isn't optimal. So another possibility is a series of semicircles drawn on 4 faces of a cube (you see what I mean, right?) Cris On Aug 30, 2013, at 12:11 PM, Andy Latto <andy.latto@pobox.com> wrote:
What's the maximum volume if require the curve to be closed?
In two dimensions, the answer is a circle; the circle encloses the most area, so it is the best convex solution. And if you have a nonconvex closed curve, you can replace it with the boundary of its convex hull, which is always a shorter curve, and scale up.
But I have no idea what the best solution is in 3 dimensions; here maybe a baseball-seam shaped curve is correct. But I remember Conway claiming on this list that every extremal mathematical property he had ever seen claimed for the baseball-seam curve was false, so either there's something better, or Conway was unaware of this problem.
Andy
Andy
On Fri, Aug 30, 2013 at 1:50 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I wonder if they knew of the work of P. A. P. Moran that Erich referred to.
On Fri, Aug 30, 2013 at 1:34 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Apologies, but I misunderstood the problem as saying that the curve has to *lie* on the unit sphere.
The problem Allan asks was a problem that George Bergman and Wolfgang Kahane of U.C. Berkeley worked on informally around 2003 and solved -- and it was pretty hard. But I don't think they published, and I don't know what the answer is.
--Dan
On 2013-08-30, at 9:02 AM, Allan Wechsler wrote:
Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I should mention here that the problem generalizes to higher finite dimensional spaces. On Fri, Aug 30, 2013 at 2:57 PM, Cris Moore <moore@santafe.edu> wrote:
I agree that the way the lobes of the baseball seam curve in towards each other isn't optimal. So another possibility is a series of semicircles drawn on 4 faces of a cube (you see what I mean, right?)
Cris
On Aug 30, 2013, at 12:11 PM, Andy Latto <andy.latto@pobox.com> wrote:
What's the maximum volume if require the curve to be closed?
In two dimensions, the answer is a circle; the circle encloses the most area, so it is the best convex solution. And if you have a nonconvex closed curve, you can replace it with the boundary of its convex hull, which is always a shorter curve, and scale up.
But I have no idea what the best solution is in 3 dimensions; here maybe a baseball-seam shaped curve is correct. But I remember Conway claiming on this list that every extremal mathematical property he had ever seen claimed for the baseball-seam curve was false, so either there's something better, or Conway was unaware of this problem.
Andy
Andy
On Fri, Aug 30, 2013 at 1:50 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I wonder if they knew of the work of P. A. P. Moran that Erich referred to.
On Fri, Aug 30, 2013 at 1:34 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Apologies, but I misunderstood the problem as saying that the curve has to *lie* on the unit sphere.
The problem Allan asks was a problem that George Bergman and Wolfgang Kahane of U.C. Berkeley worked on informally around 2003 and solved -- and it was pretty hard. But I don't think they published, and I don't know what the answer is.
--Dan
On 2013-08-30, at 9:02 AM, Allan Wechsler wrote:
Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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And just to add to the profusion of problems, what if the curve -- of length L -- is constrained to lie on the unit sphere? Let the volume of the maximum convex hull of such curves be denoted by f(L). Then what is the function f ??? And of course the same thing could be asked for *closed* curves of length L constrained to lie on the unit sphere. That maximum could be denoted by g(L). What's g ??? We are *assuming* that the sup of all the volumes in either case is actually attained, since intuitively that seems likely, and probably not too hard to prove. But just in case, we should probably redefine f(L) (and g(L)) to be the sup of volumes of convex hulls over all curves (resp. closed curves) of length L on S^2. --Dan On 2013-08-30, at 12:36 PM, Dan Asimov wrote:
Not to mention infinite-dimensional Hilbert space.
--Dan
On 2013-08-30, at 12:23 PM, Allan Wechsler wrote:
I should mention here that the problem generalizes to higher finite dimensional spaces.
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Yes, intuitively, the attainment of the sup seems like a typical end-of-first-year topology exercise. On Fri, Aug 30, 2013 at 3:58 PM, Dan Asimov <dasimov@earthlink.net> wrote:
And just to add to the profusion of problems, what if the curve -- of length L -- is constrained to lie on the unit sphere? Let the volume of the maximum convex hull of such curves be denoted by f(L). Then what is the function f ???
And of course the same thing could be asked for *closed* curves of length L constrained to lie on the unit sphere. That maximum could be denoted by g(L). What's g ???
We are *assuming* that the sup of all the volumes in either case is actually attained, since intuitively that seems likely, and probably not too hard to prove.
But just in case, we should probably redefine f(L) (and g(L)) to be the sup of volumes of convex hulls over all curves (resp. closed curves) of length L on S^2.
--Dan
On 2013-08-30, at 12:36 PM, Dan Asimov wrote:
Not to mention infinite-dimensional Hilbert space.
--Dan
On 2013-08-30, at 12:23 PM, Allan Wechsler wrote:
I should mention here that the problem generalizes to higher finite dimensional spaces.
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participants (6)
-
Allan Wechsler -
Andy Latto -
Cris Moore -
Dan Asimov -
Eugene Salamin -
Fred Lunnon