Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal -module, where is the set of integers) containing the absolute frequency of each of its patches. The method involves deriving the tiling's edge types and vertex stars; in the process, a new substitution is introduced on a reconstructed set of prototiles.
Say-awen, A. L. D., Frettlöh, D., & De Las Peñas, M. L. A. N. (2022). On the frequency module of the hull of a primitive substitution tiling. Acta Crystallographica Section A: Foundations and Advances, A78(1), 36–55. https://doi.org/10.1107/S2053273321012572