Self-Similar One-Dimensional Quasilattices

Abstract

We study 1D quasilattices, especially self-similar ones that can be used to generate two-, three- and higher-dimensional quasicrystalline tessellations that have matching rules and invertible self-similar substitution rules (also known as inflation rules) analogous to the rules for generating Penrose tilings. The lattice positions can be expressed in a closed-form expression we call {\it floor form}: $x_{n}=S(n-α)+(L-S)\lfloor κ(n-β)\rfloor$, where $L >S>0$ and $0<κ<1$ is an irrational number. We describe two equivalent geometric constructions of these quasilattices and show how they can be subdivided into various types of equivalence classes: (i) {\it lattice equivalent}, where any two quasilattices in the same lattice equivalence class may be derived from one another by a local decoration/gluing rule; (ii) {\it self-similar}, a proper subset of lattice equivalent where, in addition, the two quasilattices are locally isomorphic; and (iii) {\it self-same}, a proper subset of self-similar where, in addition, the two quasilattices are globally isomorphic (i.e.) identical up to rescaling). For all three types of equivalence class, we obtain the explicit transformation law between the floor form expression for two quasilattices in the same class. We tabulate (in Table I and Figure 5) the ten special self-similar 1D quasilattices relevant for constructing Ammann patterns and Penrose-like tilings in two dimensions and higher, and we explicitly construct and catalog the corresponding self-same quasilattices.