Substitution tilings with dense tile orientations and n-fold rotational symmetry

TL;DR

This paper demonstrates the existence of primitive substitution tilings with dense tile orientations that are invariant under specific n-fold rotations, expanding understanding of symmetrical tiling patterns.

Contribution

It introduces a new method to prove dense tile orientations in substitution tilings with n-fold rotational symmetry for n=2,3,4,5,6,8.

Findings

Existence of primitive substitution tilings with dense tile orientations for specified n

Invariant under n-fold rotation for these tilings

Use of irrationality of angles in parallelograms to prove density

Abstract

It is shown that there are primitive substitution tilings with dense tile orientations invariant under n-fold rotation for n=2,3,4,5,6,8. The proof for dense tile orientations uses a general result about irrationality of angles in certain parallelograms.