On substitution tilings of the plane with n-fold rotational symmetry

A method is described for constructing, with computer assistance, planar substitution tilings that have n-fold rotational symmetry. This method uses as prototiles the set of rhombs with angles that are integer multiples of pi/n, and includes various special cases that have already been constructed by hand for low values of n. An example constructed by this method for n = 11 is exhibited; this is the first substitution tiling with 11-fold symmetry appearing in the literature.Comment: 17 pages, 10 figure

Cyclotomic Aperiodic Substitution Tilings

The class of Cyclotomic Aperiodic Substitution Tilings (CAST) is introduced. Its vertices are supported on the 2n-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 Dn or D2n, i.e. the tiling contains an infinite number of patches of any size with dihedral symmetry Dn or D2n only by iteration of substitution rules on a single tile.Comment: 60 pages, 31 figures. Parts of Theorem 2.1 (primitive substitution matrices) and Theorem 2.2 (proof of aperiodicity) were revised. A reference to [Hib15] was added, due to a prior claim regarding the generalized Lancon-Billard tilin