The Domino Problem of the Hyperbolic Plane for Regular Polygons

TL;DR

This paper classifies all finite sets of regular polygons that can tile the hyperbolic plane, proving the Domino Problem is decidable for these sets and discovering new weakly aperiodic protosets.

Contribution

It provides a complete classification of regular polygon sets for hyperbolic tilings and establishes the decidability of the Domino Problem for these sets.

Findings

Classification of all finite regular polygon sets tiling the hyperbolic plane

Decidability of the Domino Problem for these prototiles

Discovery of the first weakly aperiodic regular polygon protosets in hyperbolic space

Abstract

We provide a definitive classification of all finite sets of regular polygons that admit a tiling of the hyperbolic plane, thereby establishing the decidability of the Domino Problem for this class of prototiles. We show that admissibility is determined by a finite set of local and inductive combinatorial constraints. This classification further leads to the discovery of the first known examples of weakly aperiodic protosets consisting of regular polygons in $\mathbb{H}^2$.