Aperiodicity, rotational tiling spaces and topological space groups

TL;DR

This paper introduces new topological methods to analyze the rotational symmetries of aperiodic tilings, extending classical approaches to include full rigid motion properties and computing invariants for complex patterns.

Contribution

It develops novel topological invariants for rotational hulls of aperiodic tilings, enabling the recovery of symmetry groups and advancing the understanding of aperiodic and quasicrystal structures.

Findings

Rotational hulls are matchbox manifolds containing translational hulls.

New S-MLD invariants are derived from homotopical and cohomological properties.

Classical space groups are recovered as fundamental groups of the constructed spaces.

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.