On triangular paperfolding patterns

TL;DR

This paper studies triangular grid patterns created by paperfolding, demonstrating their generation via substitution rules, analyzing their mathematical properties, and proving uniform densities of basic triangles in these patterns.

Contribution

It introduces a method to generate triangular paperfolding patterns using substitution rules and analyzes their mathematical properties, including eigenvalues and densities.

Findings

Patterns can be generated by substitution rules.

Eigenvalues and eigenvectors of the associated matrices are computed.

Densities of all basic triangles are equal.

Abstract

We introduce patterns on a triangular grid generated by paperfolding operations. We show that in case these patterns are defined using a periodic sequence of foldings, they can also be generated using substitution rules and compute eigenvalues and eigenvectors of the corresponding matrices. We also prove that densities of all basic triangles are equal in these patterns.