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.)
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