On Quasiperiodic Space Tilings, Inflation and Dehn Invariants

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 $τ= \frac{1+\sqrt{5}}{2}$ and $-\frac{1}τ$, and allow to reconstruct the inflation matrix uniquely.