On Quasiperiodic Space Tilings, Inflation and Dehn Invariants

TL;DR

This paper explores the use of Dehn invariants to analyze inflation properties of quasiperiodic space tilings, focusing on golden tetrahedra and Mosseri-Sadoc tiles, revealing eigenvectors and eigenvalues related to inflation.

Contribution

It introduces Dehn invariants as a novel tool for studying inflation in quasiperiodic tilings and provides explicit inflation rules and eigenstructure analysis.

Findings

Dehn invariants help identify eigenvectors of the inflation matrix.

Inflation rules for decorated Mosseri-Sadoc tiles are derived.

Eigenvalues related to the golden ratio are obtained.

Abstract

We introduce Dehn invariants as a useful tool in the study of the inflation of quasiperiodic space tilings. The tilings by ``golden tetrahedra'' are considered. We discuss how the Dehn invariants can be applied to the study of inflation properties of the six golden tetrahedra. We also use geometry of the faces of the golden tetrahedra to analyze their inflation properties. We give the inflation rules for decorated Mosseri-Sadoc tiles in the projection class of tilings ${\cal T}^{(MS)}$. The Dehn invariants of the Mosseri-Sadoc tiles provide two eigenvectors of the inflation matrix with eigenvalues equal to $\tau = \frac{1+\sqrt{5}}{2}$ and $-\frac{1}{\tau}$, and allow to reconstruct the inflation matrix uniquely.