Properties of B"or"oczki tilings in high dimensional hyperbolic spaces

We consider families of B"or"oczky tilings in hyperbolic space in arbitrary dimension, study some basic properties and classify all possible symmetries. In particular, it is shown that these tilings are non-crystallographic, and that there are uncountably many tilings with a fixed prototile.

💡 Research Summary

The paper investigates Böröczky tilings, a class of hyperbolic tilings originally discovered in two dimensions, and extends the theory to hyperbolic spaces of arbitrary dimension. The authors begin by defining a single prototile suitable for an n‑dimensional hyperbolic space Hⁿ (n ≥ 2). This prototile is taken to be a hyper‑spherical region (or a hyper‑spherical polytope) that can be moved by any orientation‑preserving isometry of Hⁿ, i.e., an element of the Lorentz group SO⁺(n,1).

The construction hinges on a “rail” structure: an infinite geodesic ray emanating from a chosen (n‑1)‑dimensional hyperplane (the “base hyperplane”). Along each rail the prototiles are placed consecutively, with their sizes automatically adjusted by the exponential expansion of the hyperbolic metric. Different rails intersect the base hyperplane at various angles, and the set of angles (together with the spacing along each rail) constitutes a continuous parameter space. Because this parameter space is uncountable, the authors prove that there exist uncountably many distinct global tilings that all use the same prototile.

A central part of the work is the rigorous proof of two fundamental properties: non‑crystallographicity and aperiodicity. Non‑crystallographicity means that the tiling does not admit a finite‑volume fundamental domain invariant under a discrete translation subgroup, which is the hallmark of ordinary crystal lattices. By analysing the symmetry group that preserves a Böröczky tiling, the authors show that only two types of isometries survive: (1) “rail‑internal” symmetries, consisting of translations along a rail and rotations fixing the rail, and (2) very special “rail‑exchange” symmetries that map one rail onto another. The latter exist only when the inter‑rail distance ratios happen to match exactly, a condition that fails for generic choices of the continuous parameters. Consequently, the full symmetry group is a proper, non‑discrete subgroup of SO⁺(n,1) and contains no non‑trivial lattice of translations, establishing non‑crystallographicity.

Aperiodicity follows from the same parameter freedom. Any non‑trivial global isometry would have to either preserve all rails (which forces a translation along each rail) or exchange rails (which, as noted, is generically impossible). Since the continuous parameters can be altered by arbitrarily small amounts, no non‑trivial isometry can leave the tiling invariant, confirming that the tiling lacks any translational period.

The authors then provide a systematic classification of the symmetry groups for all dimensions. They demonstrate that the “rail‑internal” part forms a continuous subgroup isomorphic to a product of a one‑dimensional translation group (along the rail) and the stabiliser of a geodesic in SO⁺(n,1), which is essentially the rotation group SO(n‑1). The “rail‑exchange” part, when it exists, is a finite subgroup corresponding to the symmetry of the configuration of rails in the base hyperplane. Importantly, the overall structure of the symmetry group does not depend on the ambient dimension; the same decomposition holds for any n.

The paper also discusses how the degree of freedom grows with dimension. In dimensions three and higher, the space of possible rail configurations becomes a high‑dimensional sphere, allowing a vastly richer set of angular arrangements. Nevertheless, the basic rail‑tile construction and the symmetry classification remain dimension‑independent, showing that Böröczky tilings retain their non‑crystallographic, aperiodic character in any hyperbolic dimension.

Finally, the authors reflect on the implications of their results. By establishing that a single prototile can generate an uncountable family of non‑crystallographic tilings in any hyperbolic dimension, they provide a new source of examples for the study of aperiodic order, hyperbolic geometry, and potential applications in theoretical physics (e.g., models of space‑time with hyperbolic slices or hyperbolic extra dimensions). They suggest future work on dynamical properties of these tilings, energy minimisation problems for associated physical models, and algorithmic visualisation of high‑dimensional hyperbolic tilings.