Reflection Quasilattices and the Maximal Quasilattice

TL;DR

This paper introduces reflection quasilattices as a generalization of Bravais lattices with reflection symmetry, classifies their existence in dimensions 2-4, and identifies a unique maximal reflection quasilattice in four dimensions.

Contribution

It defines reflection quasilattices, proves their existence only in dimensions 2-4, and characterizes the unique maximal reflection quasilattice in four dimensions.

Findings

Reflection quasilattices exist only in dimensions 2, 3, and 4.

There is a unique maximal reflection quasilattice in four dimensions.

Complete set of scale factors for reflection quasilattices in these dimensions.

Abstract

We introduce the concept of a {\it reflection quasilattice}, the quasiperiodic generalization of a Bravais lattice with irreducible reflection symmetry. Among their applications, reflection quasilattices are the reciprocal (i.e. Bragg diffraction) lattices for quasicrystals and quasicrystal tilings, such as Penrose tilings, with irreducible reflection symmetry and discrete scale invariance. In a follow-up paper, we will show that reflection quasilattices can be used to generate tilings in real space with properties analogous to those in Penrose tilings, but with different symmetries and in various dimensions. Here we explain that reflection quasilattices only exist in dimensions two, three and four, and we prove that there is a unique reflection quasilattice in dimension four: the "maximal reflection quasilattice" in terms of dimensionality and symmetry. Unlike crystallographic Bravais lattices, all reflection quasilattices are invariant under rescaling by certain discrete scale factors. We tabulate the complete set of scale factors for all reflection quasilattices in dimension $d>2$, and for all those with quadratic irrational scale factors in $d=2$.