Reflection quasilattices and the maximal quasilattice
Abstract
We introduce the concept of a 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 followup 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 .
 Publication:

Physical Review B
 Pub Date:
 August 2016
 DOI:
 10.1103/PhysRevB.94.064107
 arXiv:
 arXiv:1604.06426
 Bibcode:
 2016PhRvB..94f4107B
 Keywords:

 Mathematical Physics;
 Condensed Matter  Materials Science;
 High Energy Physics  Theory
 EPrint:
 6 pages, no figures, important references added, matches Phys Rev B version