not strictly peri- odic， whose approximate periodicity corresponds to best approximations to a

real number. The quasi-periodicity of a

Moire fringe can be rigorously defined by expressing the ratio of the respective periods of

the two arrays in the form of a continued fraction expansion， and its quasi-periods can be

derived by approximating the ratio to convergents of different orders. Meanwhile， the observed

period is the lowest quasi - period with a degree of aperiodicity smaller

than an empirical constant. Based on this direct correspondence between a Moire fringe and

continued fraction of a real number， a set of rigorously-periodic Moire fringes and another set of “worst periodic”

golden-ratio fringes can be identified. The present work connects Moire fringes with the basic

properties of real numbers， and therefore pro- vides new understandings for the Moire fringe

phenomenon， and is of general significance to all sorts of periods su-perposition problems.