Hyperbolic tiling neighborhoods in O(1) time

TL;DR

This paper presents a new combinatorial method to generate hyperbolic tilings and their adjacency graphs efficiently, enabling large-scale simulations without coordinate dependence.

Contribution

A novel approach that constructs hyperbolic tilings and their neighborhood graphs simultaneously using combinatoric rules, independent of lattice size.

Findings

Algorithmic complexity is independent of lattice size.

Implementation outperforms existing methods significantly.

Enables ultra large-scale numerical simulations.

Abstract

Tilings of the hyperbolic plane are of significant interest among many branches of mathematics, physics and computer science. Yet, their construction remains a non-trivial task. Current approaches primarily use tree-based recursive algorithms, which are fundamentally limited: they do not readily yield the neighborhood graph representing cell adjacencies, which is however required for many applications. We introduce a novel approach that allows to build hyperbolic tilings and their associated graph structure simultaneously, using only combinatoric rules without requiring an explicit coordinate representation. This allows to generate arbitrarily large, exact hyperbolic graphs, with an algorithmic complexity that does not depend on the lattice size. We provide an easy-to-use implementation which substantially outperforms existing methods, hence rendering ultra large-scale numerical simulations on these geometric structures accessible for the scientific community.