Translational Aperiodic Sets of 7 Polyominoes

TL;DR

The paper addresses the decidability of tiling the plane by translations of a fixed set of polyominoes. It achieves this by a reduction from Wang's domino problem, encoding Wang tiles into a seven-polyomino system using an encoder, linkers, a connector, and fillers to enforce color matching and connectivity, with the encoder size expressed as $2n(2t+1)\times 3$ where $t=\lceil \log_2 m\rceil$. The main contributions are the undecidability of planar tiling with $7$ polyominoes and the construction of a corresponding aperiodic seven-polyomino set, improving Ammann's historical $8$-tile bound. The work introduces an interlacing encoding technique that yields a rigid lattice of connectors and demonstrates that tilings exist iff the original Wang tiling exists, highlighting a significant advance in fixed-tile tiling theory and its implications for aperiodicity.

Abstract

Recently, two extraordinary results on aperiodic monotiles have been obtained in two different settings. One is a family of aperiodic monotiles in the plane discovered by Smith, Myers, Kaplan and Goodman-Strauss in 2023, where rotation is allowed, breaking the 50-year-old record (aperiodic sets of two tiles found by Roger Penrose in the 1970s) on the minimum size of aperiodic sets in the plane. The other is the existence of an aperiodic monotile in the translational tiling of $\mathbb{Z}^n$ for some huge dimension $n$ proved by Greenfeld and Tao. This disproves the long-standing periodic tiling conjecture. However, it is known that there is no aperiodic monotile for translational tiling of the plane. The smallest size of known aperiodic sets for translational tilings of the plane is $8$, which was discovered more than $30$ years ago by Ammann. In this paper, we prove that translational tiling of the plane with a set of $7$ polyominoes is undecidable. As a consequence of the undecidability, we have constructed a family of aperiodic sets of size $7$ for the translational tiling of the plane. This breaks the 30-year-old record of Ammann.

Figures (9)

• Figure 1: A set of $3$ Wang tiles
• Figure 2: Building blocks $l$, $r$, $L$ and $R$.
• Figure 3: Building blocks $Y^+$, $y^+$, $Y^-$ and $y^-$.
• Figure 4: Building blocks $X$ and $x$.
• Figure 5: Building blocks $A$ and $a$.

Theorems & Definitions (4)

• Theorem 1: b66
• Theorem 2: Undecidability with Seven Polyominoes
• Corollary 1: Aperiodicity with Seven Polyominoes
• proof : Proof of Theorem \ref{['thm_main']}