Beware of 46-Fold Symmetry: The Classification of Two-Dimensional Quasicrystallographic Lattices

Abstract

The problem of classifying two-dimensional lattices with $N$-fold rotational symmetry for arbitrary (noncrystallographic) even $N$ is shown to be equivalent to a much-studied problem in algebraic number theory. When translated into crystallographic language, the number-theoretic results establish that except for 29 even numbers $N$ there are two or more distinct lattices. The smallest $N$ for which there is more than a single lattice, however, is $N=46$. We list every $N$ for which there is a unique lattice, and give the numbers of distinct lattices for all $N<100$.