Cyclotomic Aperiodic Substitution Tilings

Abstract

The class of Cyclotomic Aperiodic Substitution Tilings (CASTs) is introduced. Its vertices are supported on the 2 n -th cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Substitution matrices and minimal inflation multipliers of CASTs are discussed as well as practical use cases to identify specimen with individual dihedral symmetry D n or D 2 n , i.e., the tiling contains an infinite number of patches of any size with dihedral symmetry D n or D 2 n only by iteration of substitution rules on a single tile.

Keywords

• cyclotomic aperiodic substitution tiling
• discrete metric geometry
• quasiperiodic
• nonperiodic
• minimal inflation multiplier
• minimal scaling factor
• substitution matrix
• Girih tiling