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?