Aperiodicity, rotational tiling spaces and topological space groups

Abstract

We study the rotational structures of aperiodic tilings in Euclidean space of arbitrary dimension using topological methods. Classical topological approaches to the study of aperiodic patterns have largely concentrated just on translational structures, studying an associated space, the continuous hull, here denoted $\Omega_t$. In this article we consider two further spaces $\Omega_r$ and $\Omega_G$ (the rotational hulls) which capture the full rigid motion properties of the underlying patterns. The rotational hull $\Omega_r$ is shown to be a matchbox manifold which contains $\Omega_t$ as a sub-matchbox manifold. We develop new S-MLD invariants derived from the homotopical and cohomological properties of these spaces demonstrating their computational as well as theoretical utility. We compute these invariants for a variety of examples, including a class of 3-dimensional aperiodic patterns, as well as for the space of periodic tessellations of $\mathbb{R}^3$ by unit cubes. We show that the classical space group of symmetries of a periodic pattern may be recovered as the fundamental group of our space $\Omega_G$. Similarly, for those patterns associated to quasicrystals, the crystallographers' aperiodic space group may be recovered as a quotient of our fundamental invariant.