A proof of Ollinger's conjecture: undecidability of tiling the plane with a set of $8$ polyominoes

TL;DR

The paper settles Ollinger's conjecture by proving the undecidability of tiling the plane with a fixed set of $8$ polyominoes, via a reduction from Wang's domino problem. It introduces a novel orientation for mapping Wang tiles to polyominoes and a dent-based color-encoding mechanism, enabling an $8$-tile construction built from a meat/jaw/filler/teeth/links architecture. A base-$9$ scaling and a boundary-word scheme encode the colors and adjacency constraints, ensuring that a tiling exists exactly when the corresponding Wang tile set tiles the plane. This result establishes the smallest known tile-set size for undecidability in this translational tiling setting and opens questions about undecidability for smaller $k$ and in related spaces.

Abstract

We give a proof of Ollinger's conjecture that the problem of tiling the plane with translated copies of a set of $8$ polyominoes is undecidable. The techniques employed in our proof include a different orientation for simulating the Wang tiles in polyomino and a new method for encoding the colors of Wang tiles.

Figures (9)

• Figure 1: A set of $3$ Wang tiles.
• Figure 2: Wang tiles simulated by polyominoes.
• Figure 3: The meat.
• Figure 4: The jaw.
• Figure 5: Part of the jaw (zoom in).

Theorems & Definitions (5)

• Theorem 1: b66
• Theorem 2: o09
• Conjecture 1: Ollinger's conjectureo09
• Theorem 3
• proof : Proof of Theorem \ref{['thm_main']}