This work introduces the idea of symmetry order, which describes the rotational symmetry types of tilings in the hull of a given substitution. Definitions are given of the substitutions σ6 and σ7 which give rise to aperiodic primitive substitution tilings with dense tile orientations and which are invariant under six- and sevenfold rotations, respectively; the derivation of the symmetry orders of their hulls is also presented.
Say-awen, A. L. D., De Las Peñas, M. L. A. N., & Frettlöh, D. (2018). Primitive substitution tilings with rotational symmetries. Acta Crystallographica Section A: Foundations and Advances, 74(4), 388-398.