Sub Rosa, a system of quasiperiodic rhombic substitution tilings with n-fold rotational symmetry

TL;DR

This paper proves the existence of quasiperiodic rhombic substitution tilings with n-fold rotational symmetry for any n, providing explicit substitution rules and demonstrating consistent interior tiling.

Contribution

It introduces a new class of quasiperiodic tilings with n-fold symmetry and explicitly constructs substitution rules for them, expanding the understanding of symmetric tilings.

Findings

Existence of quasiperiodic rhombic tilings with n-fold symmetry for all n

Explicit substitution rules for the edges of the rhombuses

Proof of consistent interior tiling with given edge substitutions

Abstract

In this paper we prove the existence of quasiperiodic rhombic substitution tilings with 2n-fold rotational symmetry, for any n. The tilings are edge-to-edge and use [n/2] rhombic prototiles with unit length sides. We explicitly describe the substitution rule for the edges of the rhombuses, and prove the existence of the corresponding tile substitutions by proving that the interior can be tiled consistently with the given edge substitutions.