Thursday, December 28, 2006

Minimum length of thread needed to weave a fullerene

What is the minimum length of thread you need to build a stable beaded fullerene. By stable I mean the beaded fullerene should satisfy VSEPR principle everywhere. Probably not many people working on beadings worry about this kind of problem. I am not sure how hard the problem is in general. But the lower bound for the minimum length is quite easy. Since every bead has to be stitched at least twice to get a stable structure, therefore the minimum length must be greater than 2*N*d, where N is the number of beads and d the the diameter of bead. Better estimation can be found if the ratio of the size of hole and diameter of bead is taken into account. Furthermore, if the part of thread outside the beads is included, slight improvement can be obtained. This is just simple exercise in geometry, which can be worked in principle. We can also use empirical approach by performing a real experiment to find out the actual length used, then divide this number by the theoretical minimum 2*N*d, the scaling factor.

More complicated issue is that the lower bound with the scaled factor included is in general not the minimum length. It is possible to finish the beaded molecule, some beads may always be stitched through more than twice! This problem is not easy to explain. I will talk about this later.

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