Self-Similar One-Dimensional Quasilattices

TL;DR

This paper investigates self-similar one-dimensional quasilattices, providing explicit formulas, geometric constructions, and classifications, which are essential for generating higher-dimensional quasicrystalline tilings like Penrose patterns.

Contribution

It introduces a closed-form floor expression for 1D quasilattices, classifies their equivalence types, and catalogs key self-similar quasilattices relevant for quasicrystal tilings.

Findings

Explicit floor form expression for 1D quasilattices

Classification into lattice equivalent, self-similar, and self-same classes

Tabulation of ten key self-similar quasilattices for tiling applications

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}: $x_{n}=S(n-\alpha)+(L-S)\lfloor \kappa(n-\beta)\rfloor$, where $L >S>0$ and $0<\kappa<1$ 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.