The domino problem of the hyperbolic plane is undecidable, new proof

TL;DR

This paper presents a new, simplified proof that the tiling problem of the hyperbolic plane is undecidable, using fewer prototiles and a regular polygon, building on previous foundational work.

Contribution

It offers an improved, more efficient proof of undecidability for hyperbolic plane tilings, reducing complexity and prototile count compared to prior proofs.

Findings

Undecidability of hyperbolic plane tiling problem established

Proof uses only a regular polygon as the basic tile shape

Number of prototiles significantly reduced

Abstract

The present paper is a new version of the arXiv paper revisiting the proof given in a previous paper of the author published in 2008 proving 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. The present construction improves that of the recent arXiv paper. It also strongly reduces the number of prototiles.