Symmetries and fuzzy symmetries of Carbon nanotubes

Carbon nanotubes (CNTs) possess the fuzzy cylinder group characteristic. Comparing with the linear and planer molecules, there are included the fuzzy symmetry of the cylinder screw rotation (CSR) in relation to some higher ( $>$ 2) fold rotation axis. The CSR may be noted as the product of translation (T) and rotation (C). The CSR symmetry will be imperfect owing to the introduction of T. As the extent of whole translation is more than 10-fold than every time, the membership function of CNT in relation to CSR will be more than about 0.9, and such CNT may be seems as provided with the perfect CSR symmetry. For analyse the CNT we may using the cylindrical orthogonal curvilinear coordinate system. The MO ought to be provided with a pure irreducible representation, but the component of symmetry adapted atomic orbital (SA-AO) set may be not sole, and it is difficult to get and analyse the ‘pure’ $uppi $ -MO. There are some various AO (1S-, 2S-, 2Pz-, 2Pr-, 2Pt of carbon and 1S- for hydrogen)-set components in a certain MO. For the CNT with the same diameter and different length, the MO energy and the SA-AO component versus the relative serial number will be with the similar distribution. The MOs of CNT with higher fold C symmetry may be provided with two-dimensional irreducible representation. For the molecular skeleton and the MO which belong to one-dimensional irreduable representation, their membership functions in relation to the CSR with the product of the same T and different C would be equality. However, for the single MO which belong to two-dimensional irreducible representation that may be somewhat difference. The torus carbon nanotube (TCNT) may be provided the symmetry with the torus group and torus screw rotation (TSR), such symmetry would be or near be not rare in nature. Similar as the planer rectangle (called as the MH rectangle) may composed the Hückel- or Möbius-strip band, the more MH rectangles in the cylinder CNT may be composed the more Hückel- or Möbius-strip bands, such strip bands set may be called strip tube, meanwhile the fuzzy CSR symmetry will be transform to the perfect TSR symmetry. The intersecting line (Z-axis) of the MH rectangles will be transform to the common basic circle of these strip bands. When the CNT to form a TCNT, as one of the MH rectangle form a Hückel-strip band or an $n(t)$ -twisted Möbius-strip band itself, the other MH rectangle will be form the strip band with the same topological structure synchronously, and the set of these strip bands may be called the strip tube. The boundary closed curves of the strip band may reflect the torus group symmetrical characteristic of the relative strip bands. The closed curve may correspond to a cyclical group or subgroup. The number of carbon atomic pairs on the closed curve denoted the order of such group or subgroup. As the CNT to form the TCNT, it is different as the single MH rectangle, they may be to form the fractal-twisted Möbius-strip tube synchronously, in which the single Möbius-strip band may be formed from more one MH rectangle, however, single MH rectangle may enter into only one Möbius-strip band. As the hetero-CNT with the helical-structure distribution, such hetero-CNT may form the relative torus hetero-CNT, but according to the continuity of CNT tube side, a certain twisted to form Möbius-strip tube may often be required. There is some interaction between the distributional helical-structure and twisting way, such interaction may touch to the degree of tightness of the helical-structure distribution in torus hetero-CNT.