Comment Conveyor belt problem... (Score 1) 60
Interesting article, but I don't understand why the conveyor belt problem (as described) is unsolved. Start with one pulley. Obviously a band around it works. Assume a solution exists for some finite number of pulleys, N. Since the support of the pulley locations is compact, one can always and uniquely determine the exterior of the spanning belt. Place an additional pulley exterior to this belt. There are only three topologically relevant cases -- (an pair of in the case of more than two of) the "nearest neighbor" exterior pulleys carry a belt that is "convex" (outside both), "concave" (inside both), or "mixed" (inside one, outside the other). In all three cases it can be shown that one can add the pulley and still satisfy the conditions of the problem. Hence one has 1, N and N+1, a proof by topological induction. The only additional bit of work on the proof is to note that one can avoid problems with pathological interior loopings (if necessary -- I don't really think that it is) or adding the N+1 pulley INSIDE the belt by simply reordering the inductive process for any given pattern to maintain the belt in a maximally convex state as one proceeds, that is starting with any belt and then adding the pulleys ordered by their distance from the original pulley. Not only is there "a" spanning belt, but there will be in most cases an enormous permutation of spanning belts. As in, all of the permutations one can construct by adding pulleys in circular distance order from any pulley treated as the original pulley until they are all entrained.