Planar Rosa : a family of quasiperiodic substitution discrete plane tilings with $2n$-fold rotational symmetry

TL;DR

This paper introduces Planar Rosa, a family of quasiperiodic tilings with 2n-fold symmetry generated by primitive substitutions, and compares them to existing tilings, establishing new properties of their geometric structure.

Contribution

It presents the Planar Rosa family of tilings with discrete plane properties for all even n ≥ 4, expanding previous work on odd n and analyzing their geometric and substitution characteristics.

Findings

Planar Rosa tilings have 2n-fold rotational symmetry and are generated by primitive substitution.

These tilings are discrete plane tilings obtained via projection from higher dimensions.

Sub Rosa tilings by Kari and Rissanen do not satisfy the discrete plane condition for even n ≥ 4.

Abstract

We present Planar Rosa, a family of rhombus tilings with a $2n$-fold rotational symmetry that are generated by a primitive substitution and that are also discrete plane tilings, meaning that they are obtained as a projection of a higher dimensional discrete plane. The discrete plane condition is a relaxed version of the cut-and-project condition. We also prove that the Sub Rosa substitution tilings with $2n$-fold rotational symmetry defined by Kari and Rissanen do not satisfy even the weaker discrete plane condition. We prove these results for all even $n\geq 4$. This completes our previously published results for odd values of $n$.