球中柱

Prof. Mou said the other day that his group discovered an interesting structure which looks like a sphere with a hole puncturing through the north and south poles(球中柱). Topologically, this structure is equivalent to the toroidal carbon nanotube, since both of these types of structures have genus one (one hole). The general classification scheme of toroidal carbon nanotube that Chuang, Fan, and I developed can generate this kind of structure easily. Basically, what we need to do is to increase the separation between heptagons, and keep pentagons close to equator at the same time. The resulting structure should look like the ball with a long and thin hole (球中柱). The following figure generated by Chuang should be close to 球中柱. I wish I can find some people to make a beaded model for this compound. More than 1000 beads are needed! Probably I need to think about how to create an automatic weaving (beading) machine to do the job. Then there is no need to ask our sophomores make beaded nanotori in the service course II (服務課程 II) as our dept. head suggested.



Prof. Mou also mentioned that the long and thin hole puncturing through the north and south poles is a mathematically singularity. Naive thinking might suggest that we should put heptagons close to the region of singularity in order to simulate the negative Gaussian curvature; however, in the case of carbon hole-in-a-sphere, we have to have an equal number of pentagons far away from the sigularity, but lying around equator to satisfy the Euler-Poincare theorem V-E+F=2-2g=0, where g=1 is the genus of this system.