Primitive substitution tilings with rotational symmetries

Abstract

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 σ and σ 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.

Cyclotomic Aperiodic Substitution Tilings

“…Well known examples are the Pinwheel tiling as described by Radin and Conway in [Rad94,Rad99] and its relatives in [Sad98,Fre08a,Fre08b]. Recently Frettlöh, Sayawen and de las Peñas described "Substitution Tilings with Dense Tile Orientations and n-fold Rotational Symmetry" [FSdlP17] and continued the discussion in [SadlPF18]. For this reason we extend the results in [Pau17, Chapter 2] regarding Cyclotomic Aperiodic Substitution Tilings (CAST) to cases with DTO in Section 2.…”

“…of CASTs with DTO are derived in Section 4. In Section 5 examples with minimal inflation multiplier and individual dihedral symmetry D 2n are introduced for the cases n ∈ {2, 3, 4, 5, 6, 7} and compared to the results in [FSdlP17] and [SadlPF18].…”

“…For terms and definitions we stay close to [BG13], [FGH], [FSdlP17], [SadlPF18], [Pau17] and references therein:…”

On the frequency module of the hull of a primitive substitution tiling

“…An example of a substitution is the substitution ! (Frettlo ¨h et al., 2017;Say-awen et al, 2018;Say-awen, 2016) shown in Fig. 1.…”

“…Other tilings in X ! , particularly those tilings with rotational symmetry, are derived in Say-awen et al (2018).…”