Substitution discrete plane tilings with $2n$-fold rotational symmetry for odd n

TL;DR

This paper investigates substitution tilings with 2n-fold rotational symmetry, proving certain existing tilings are not discrete planes and introducing new tilings that do satisfy the discrete plane condition for odd n.

Contribution

It demonstrates the non-discreteness of Sub Rosa tilings for odd n > 5 and constructs new Planar Rosa tilings with 2n-fold symmetry that are discrete planes.

Findings

Sub Rosa tilings with odd n > 5 are not discrete planes.

New Planar Rosa tilings with 2n-fold symmetry are discrete planes.

Explicit construction provided for the 10-fold case.

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 greater than 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.