Abstract: The W-M fractal function and the structure-function method for determining fractal dimensions are introduced. Some factors of influencing the surface quality in the nanoscale cutting are ana-lyzed by the fractal theory. A series of testing and statistical analysis for the nanoscale cutting surface are done, and the major influence factors on the surface quality are especially analyzed in quantity. The determining method of the fractal dimensions for a set and its subsets and the relation between them are obtained and checked. Some issues on the application of the theory above in the real design are discussed.

Key words: Fractal theory, Nanoscale ultra-precision cutting, Set and subset, Structure function, Surface quality