Prof. Mou asked me about the possiblity of making Mobius-like toroidal structures. The answer is yes, it is possible. The strain energy might be large though. Here is some discussions I wrote in my group blog on April 7, 2007.
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A few months agos, I have a discussion with Chuang on the toroidal carbon nanotubes consisted only hexagons. I mentioned that there are several possible ways to patch the two ends of a nanotube into a circular torus. Most people have discussed the simplest situation that does not allow the twisting of the nanotube. If the nanotube is either armchair or zigzag, we can view the resulting torus as consisting of many parallel armchaired or zigzagged strips.
But if the nanotube is either chiral or twisted zigzag (armchair), the situation is quite interesting from topological point of view. Firstly, if we loose the patching constraint (boundary condition), there could be many different tori that we can obtain. Secondly, the more striking thing is that the whole torus may only contain one strip, which is similar to the Mobius strip. This also remind me of the algorithm Fan used to generate the carbon nanotube by imagining that SWNT is made of polyacetylene. （I should ask Fan to comment on this point.）
Yesterday I read a section of Pickover's "This Mobius Strip" on the Mobius prismatic doughnut, in which Pickover has also noticed the same problem in different context.
Now there is no doubt that we should go ahead and look at the electronic structure, physical and optical properties for this kind of Mobius carbon tori with different boundary conditions.
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