Re: [math-fun] just rubber bands
Since no one else has jumped in to reply to Jim Propp's post, I'll take a stab at it. Maybe one reason nobody responded to the question is that it isn't really a mathematical question, or even a single question; it's an amorphous stew of several questions, mixing math, physics, and perhaps esthetics. On the subject of esthetics, let me say that I don't find the rubber-band cube in the video at all appealing; it's a far cry from the sort of elegance one finds in modular origami (let me mention here the work of Jeannine Mosely, who takes common objects like business cards or egg cartons and unleashes their latent potential as modules). On the subject of physics, let's notice that the frictional properties of rubber bands play a role in the rubber-band ball. Once one of the outmost rubber-bands start to deviate from a great circle, it will experience even more contractive force, causing it to shrink further, unless there's friction. I'm not sure what specific question (if any) Jim has in mind, but let me mention that, just as one can nearly cover the surface of a cube of side-length L by rubber-bands of length 4L, one can nearly cover the surface of a tetrahedron of side-length L by rubber-bands of length 2L. To see where the rubber-bands go, draw a line connecting the midpoints of two opposite edges of the tetrahedron, and now look at how planes perpendicular to this line cut the tetrahedron. The cross-sections are rectangles of varying shape with common perimeter 2L. (To see this, check that the perimeter varies linearly as one moves the cutting plane, and that it starts and ends with value 2L.) James Propp
It's a long shot, but ... There might just be a connection with the construction of polyhedra from interlaced strips of paper. I've never taken an interest in the matter, so can't supply a reference --- maybe Cundy & Rollett, or somewhere in Gardner? WFL On 2/6/11, James Propp <jamespropp@gmail.com> wrote:
Since no one else has jumped in to reply to Jim Propp's post, I'll take a stab at it.
Maybe one reason nobody responded to the question is that it isn't really a mathematical question, or even a single question; it's an amorphous stew of several questions, mixing math, physics, and perhaps esthetics.
On the subject of esthetics, let me say that I don't find the rubber-band cube in the video at all appealing; it's a far cry from the sort of elegance one finds in modular origami (let me mention here the work of Jeannine Mosely, who takes common objects like business cards or egg cartons and unleashes their latent potential as modules).
On the subject of physics, let's notice that the frictional properties of rubber bands play a role in the rubber-band ball. Once one of the outmost rubber-bands start to deviate from a great circle, it will experience even more contractive force, causing it to shrink further, unless there's friction.
I'm not sure what specific question (if any) Jim has in mind, but let me mention that, just as one can nearly cover the surface of a cube of side-length L by rubber-bands of length 4L, one can nearly cover the surface of a tetrahedron of side-length L by rubber-bands of length 2L. To see where the rubber-bands go, draw a line connecting the midpoints of two opposite edges of the tetrahedron, and now look at how planes perpendicular to this line cut the tetrahedron. The cross-sections are rectangles of varying shape with common perimeter 2L. (To see this, check that the perimeter varies linearly as one moves the cutting plane, and that it starts and ends with value 2L.)
James Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Sun, 6 Feb 2011, Fred lunnon wrote:
It's a long shot, but ...
There might just be a connection with the construction of polyhedra from interlaced strips of paper.
I've never taken an interest in the matter, so can't supply a reference (*) --- maybe Cundy & Rollett, or somewhere in Gardner?
WFL
(*) Peter Hilton & Jean Pedersen, Build your own polyhedra.
On 2/6/11, James Propp <jamespropp@gmail.com> wrote:
Since no one else has jumped in to reply to Jim Propp's post, I'll take a stab at it.
Maybe one reason nobody responded to the question is that it isn't really a mathematical question, or even a single question; it's an amorphous stew of several questions, mixing math, physics, and perhaps esthetics.
On the subject of esthetics, let me say that I don't find the rubber-band cube in the video at all appealing; it's a far cry from the sort of elegance one finds in modular origami (let me mention here the work of Jeannine Mosely, who takes common objects like business cards or egg cartons and unleashes their latent potential as modules).
On the subject of physics, let's notice that the frictional properties of rubber bands play a role in the rubber-band ball. Once one of the outmost rubber-bands start to deviate from a great circle, it will experience even more contractive force, causing it to shrink further, unless there's friction.
I'm not sure what specific question (if any) Jim has in mind, but let me mention that, just as one can nearly cover the surface of a cube of side-length L by rubber-bands of length 4L, one can nearly cover the surface of a tetrahedron of side-length L by rubber-bands of length 2L. To see where the rubber-bands go, draw a line connecting the midpoints of two opposite edges of the tetrahedron, and now look at how planes perpendicular to this line cut the tetrahedron. The cross-sections are rectangles of varying shape with common perimeter 2L. (To see this, check that the perimeter varies linearly as one moves the cutting plane, and that it starts and ends with value 2L.)
James Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The rubber band shapes reminded me of a problem that I have been wondering about since last summer. If I have a smooth rock, what is the minimum length of string that I need to tie the string around the rock so that the rock can't slip out of the knot? How many knots are needed? Hilarie
Depends on the shape of the rock --- if it has a "neck" somewhere, the lower bound is the circumference of the neck. The worst case must be a sphere --- the edges of a blown-up regular tetrahedron probably gives a lower bound on the total length in that case, though it doesn't allow a single piece of string --- a blown-up regular octahedron does, giving you an upper bound. WFL On 2/8/11, Hilarie Orman <ho@alum.mit.edu> wrote:
The rubber band shapes reminded me of a problem that I have been wondering about since last summer. If I have a smooth rock, what is the minimum length of string that I need to tie the string around the rock so that the rock can't slip out of the knot? How many knots are needed?
Hilarie
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If you use this problem with a class, you should probably include "Everybody wants a rock to wind a piece of string around" ( http://tmbw.net/wiki/Lyrics:We_Want_A_Rock ) --Joshua Zucker
participants (5)
-
Fred lunnon -
Hilarie Orman -
James Propp -
Joshua Zucker -
Richard Guy