# The Domino problem is undecidable on every rhombus subshift

## TL;DR

This paper proves that the Domino problem remains undecidable for all rhombus-shaped tilings, including Penrose tilings, by demonstrating its $\, ext{Pi}^0_1$-hardness and completeness in certain cases.

## Contribution

It extends the classical Domino problem to rhombus tilings and establishes its undecidability for all such subshifts, including well-known examples like Penrose tilings.

## Key findings

- The Domino problem is $\, ext{Pi}^0_1$-hard for all rhombus subshifts.
- The problem is $\, ext{Pi}^0_1$-complete when subshifts are given by computable forbidden patterns.
- Undecidability holds for a broad class of rhombus tilings, including Penrose tilings.

## Abstract

We extend the classical Domino problem to any tiling of rhombus-shaped tiles. For any subshift X of edge-to-edge rhombus tilings, such as the Penrose subshift, we prove that the associated X-Domino problem is $\Pi^0_1$ -hard and therefore undecidable. It is $\Pi^0_1$ -complete when the subshift X is given by a computable sequence of forbidden patterns.