Arithmetic-analytical expression of the Koch-type curves and their generalizations (I)

In this paper, by constructing the one-to-one correspondence between its IFS and the quaternary fractional expansion, the n -th iteration analytical expression and the limit representation of the family of the Koch-type curves with arbitrary angles are obtained. The distinction between our method and that of H. Sagan is that we provide the generation process analytically and represent it as a graph of a series function which looks like the Weierstrass function. With these arithmetic expressions, we further analyze and prove some of the fractal properties of the Koch-type curves such as the self-similarity, the Hölder exponent and with the property of continuous everywhere but differentiable nowhere. Then, we will show that the Koch-type curves can be approximated by different constructed generators. Based on the analytic transformation of the Koch-type curves, we also constructed more continuous but nowhere-differentiable curves represented by arithmetic expressions. This result implies that the analytical expression of a fractal has theoretical and practical significance.