Complexity of two-dimensional patterns

To describe quantitatively the complexity of two-dimensional patterns we introduce a complexity measure based on a mean information gain. Two types of patterns are studied: geometric ornaments and patterns arising in random sequential adsorption of discs on a plane (RSA). For the geometric ornaments analytical expressions for entropy and complexity measures are presented, while for the RSA patterns these are calculated numerically. We compare the information-gain complexity measure with some alternative measures and show advantages of the former one, as applied to two-dimensional structures. Namely, this does not require knowledge of the maximal entropy of the pattern, and at the same time sensitively accounts for the inherent correlations in the system.