Hierarchical fractal structure of perfect single-layer graphene

Abstract

The atomic lattice structure of perfect single-layer graphene that can actually be regarded as a kind of hierarchical fractal structure from the perspective of fractal geometry was studied for the first time. Three novel and special discoveries on hierarchical fractal structure and sets were unveiled upon examination of the regular crystal lattices of the single-layer graphene. The interior fractal-type structure was discovered to be the fifth space-filling curve from physical realm. Two efficient methods for calculating the fractal dimension of this fresh member was also provided. The outer boundary curve had a fractal dimension equal to one, and a multi-fractal structure from a naturally existing material was found for the first time. A series of strict self-similar hexagons comprised a rotating fractal set. These hexagons slewed at a constant counterclockwise angle α of 19.1° when observed from one level to the next higher level. From the perspective of fractal geometry, these pioneering discoveries added three new members to the existing regular fractal structures and sets. A fundamental example of a multi-fractal structure was also presented.

Keywords

hierarchical fractal structure / fractal dimension / the fifth space-filling curve / multi-fractal structure

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Acknowledgements

The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (Grant No. 10602028).

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg