Thursday, December 28, 2006

Minimum length of thread needed

The typical procedure to weave a beaded spheroid is to start from a 5- or 6-bead group, then use the right-angle weave technique to extend outward by adding more 5- or 6-bead groups aroud the central (first)-bead group in a spiral fashion. Whenever a 5-bead group is introduced, negative surface curvature is created, and thus the surface would be curved inward. According the Euler theory, to close a spheroid, 12 pentagons are needed. Thus along the path of spiral, we can put any number of hexagon we want, but a total number of 12 pentagons are needed to get a closed spheroid. Of course, if the sequence is arbitray, many of the structures are going to have very high energy. When a spheroidal fullerene can be contructed in this way, the thread will go through each each exactly twice, the amount of thread needed will simply s*2*N*d, where s is the scaling parameter to take the part of thread outside the beads into account.

According to the work by the British chemist, Fowler, there exist a class of fullerene system which can not be described by this way. An example is tetrahedral C380 as shown in his book entitled "An Atlas of Fullerenes". For this kind of system, we probably have to use longer lenth of thread because some beads will be stitched more than two times.

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