About the domino problem in the hyperbolic plane from an algorithmic point of view

TL;DR

This paper proves that determining whether a set of Wang tiles can tile the hyperbolic plane is an undecidable problem, highlighting fundamental limits in hyperbolic tiling algorithms.

Contribution

It establishes the undecidability of the hyperbolic plane tiling problem using Wang tiles, extending classical results to hyperbolic geometry.

Findings

Hyperbolic plane tiling with Wang tiles is undecidable.

The result generalizes classical tiling undecidability to non-Euclidean geometry.

Implications for algorithmic approaches to hyperbolic tilings.

Abstract

In this paper, we prove that the general problem of tiling the hyperbolic plane with \`a la Wang tiles is undecidable.