The periodic domino problem is undecidable in the hyperbolic plane

TL;DR

This paper proves that the problem of determining periodic tilings is undecidable in the hyperbolic plane, extending the known Euclidean results to hyperbolic geometry.

Contribution

It establishes the undecidability of the periodic tiling problem specifically within the hyperbolic plane, a new extension of prior Euclidean results.

Findings

Periodic tiling problem is undecidable in the hyperbolic plane

Extends Euclidean undecidability results to hyperbolic geometry

Confirms the complexity of tiling problems in non-Euclidean spaces

Abstract

In this paper, we consider the periodic tiling problem which was proved undecidable in the Euclidean plane by Yu. Gurevich and I. Koriakov in 1972. Here, we prove that the same problem for the hyperbolic plane is also undecidable.