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

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 R-n using the lift operator first defined by Levitov, and to study the planarity of substitution tilings in 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.

Automated summary

This study focuses on substitution tilings that are also discrete plane tilings. We prove that the Sub Rosa substitution tilings with a 2n-fold rotational symmetry are not discrete planes, and then define new Planar Rosa substitution tilings that satisfy the discrete plane condition. Our methods involve using the lift operator to adjust the tilings and substitutions, and applying linear algebra, properties of circulant matrices, and trigonometric sums to study the planarity of substitution tilings in R-n. The construction of the Planar Rosa substitutions also utilizes the Kenyon criterion and a result on De Bruijn multigrid dual tilings.