The finite tiling problem is undecidable in the hyperbolic plane

TL;DR

This paper proves that the finite tiling problem remains undecidable when extended from the Euclidean plane to the hyperbolic plane, highlighting the problem's inherent computational complexity in different geometries.

Contribution

It extends the undecidability result of the finite tiling problem from Euclidean to hyperbolic geometry, a previously unresolved case.

Findings

Finite tiling problem is undecidable in the hyperbolic plane

Undecidability extends from Euclidean to hyperbolic geometry

Highlights computational complexity in different geometric contexts

Abstract

In this paper, we consider the finite tiling problem which was proved undecidable in the Euclidean plane by Jarkko Kari in 1994. Here, we prove that the same problem for the hyperbolic plane is also undecidable.