The Domino Problem of the Hyperbolic Plane Is Undecidable

TL;DR

This paper proves that determining whether a set of regular polygon tiles can tile the hyperbolic plane is undecidable, extending the classical tiling problem's undecidability from Euclidean to hyperbolic geometry.

Contribution

It establishes the undecidability of the hyperbolic plane tiling problem using only regular polygons, building on prior Euclidean results and addressing a longstanding open question.

Findings

Hyperbolic tiling problem is undecidable.

Undecidability holds with only regular polygon tiles.

Extends classical tiling undecidability to hyperbolic geometry.

Abstract

In this paper, we prove that the general tiling problem of the hyperbolic plane is undecidable by proving a slightly stronger version using only a regular polygon as the basic shape of the tiles. The problem was raised by a paper of Raphael Robinson in 1971, in his famous simplified proof that the general tiling problem is undecidable for the Euclidean plane, initially proved by Robert Berger in 1966.