Substitution Discrete Plane Tilings with 2n-Fold Rotational Symmetry for Odd n

Abstract

We study substitution tilings that are also discrete plane tilings, that is, satisfy a relaxed version of cut-and-projection. We prove that the Sub Rosa substitution tilings with a 2n-fold rotational symmetry for odd \(n>5\) defined by Kari and Rissanen are not discrete planes—and therefore not cut-and-project tilings either. We then define new Planar Rosa substitution tilings with a 2n-fold rotational symmetry for any odd n, and show that these satisfy the discrete plane condition. The tilings we consider are edge-to-edge rhombus tilings. We give an explicit construction for the 10-fold case, and provide a construction method for the general case of any odd n. Our methods are to lift the tilings and substitutions to \({\mathbb {R}}^{{n}}\) using the lift operator first defined by Levitov, and to study the planarity of substitution tilings in \({\mathbb {R}}^{{n}}\) using mainly linear algebra, properties of circulant matrices, and trigonometric sums. For the construction of the Planar Rosa substitutions we additionally use the Kenyon criterion and a result on De Bruijn multigrid dual tilings.