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Undecidability of Translational Tiling with Three Tiles

Chan Yang, Zhujun Zhang

TL;DR

This work establishes undecidability for translational tiling in $4$-dimensional space using a fixed set of three connected tiles, extending prior results that linked higher dimensions to undecidability with one tile. The authors develop a lifting approach that preserves a $3$-D tiling framework while encoding colors and Wang-tile constraints into a time-augmented $4$-D construction, via an encoder, a linker, and a filler. A precise reduction from Wang’s domino problem shows that a tiling exists in spacetime precisely when the corresponding Wang tiles admit a plane tiling, thereby proving undecidability for the $4$-D case with three tiles. This result adds to the growing evidence that translational tiling with a fixed monotile may be undecidable in some fixed dimension, aligning with Greenfeld and Tao’s breakthroughs, and motivates further exploration of one- or two-tile cases and higher-dimensional analogs.

Abstract

Is there a fixed dimension $n$ such that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable? Several recent results support a positive answer to this question. Greenfeld and Tao disprove the periodic tiling conjecture by showing that an aperiodic monotile exists in sufficiently high dimension $n$ [Ann. Math. 200(2024), 301-363]. In another paper [to appear in J. Eur. Math. Soc.], they also show that if the dimension $n$ is part of the input, then the translational tiling for subsets of $\mathbb{Z}^n$ with one tile is undecidable. These two results are very strong pieces of evidence for the conjecture that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable, for some fixed $n$. This paper gives another supportive result for this conjecture by showing that translational tiling of the $4$-dimensional space with a set of three connected tiles is undecidable.

Undecidability of Translational Tiling with Three Tiles

TL;DR

This work establishes undecidability for translational tiling in -dimensional space using a fixed set of three connected tiles, extending prior results that linked higher dimensions to undecidability with one tile. The authors develop a lifting approach that preserves a -D tiling framework while encoding colors and Wang-tile constraints into a time-augmented -D construction, via an encoder, a linker, and a filler. A precise reduction from Wang’s domino problem shows that a tiling exists in spacetime precisely when the corresponding Wang tiles admit a plane tiling, thereby proving undecidability for the -D case with three tiles. This result adds to the growing evidence that translational tiling with a fixed monotile may be undecidable in some fixed dimension, aligning with Greenfeld and Tao’s breakthroughs, and motivates further exploration of one- or two-tile cases and higher-dimensional analogs.

Abstract

Is there a fixed dimension such that translational tiling of with a monotile is undecidable? Several recent results support a positive answer to this question. Greenfeld and Tao disprove the periodic tiling conjecture by showing that an aperiodic monotile exists in sufficiently high dimension [Ann. Math. 200(2024), 301-363]. In another paper [to appear in J. Eur. Math. Soc.], they also show that if the dimension is part of the input, then the translational tiling for subsets of with one tile is undecidable. These two results are very strong pieces of evidence for the conjecture that translational tiling of with a monotile is undecidable, for some fixed . This paper gives another supportive result for this conjecture by showing that translational tiling of the -dimensional space with a set of three connected tiles is undecidable.

Paper Structure

This paper contains 10 sections, 3 theorems, 1 equation, 22 figures.

Table of Contents

  1. Introduction
  2. Undecidability of Tiling $3$-dimensional Space
  3. Building Blocks
  4. The Set of Four Tiles
  5. Tiling Pattern and Proof of Theorem \ref{['thm_3d_new']}
  6. Undecidability of Tiling $4$-dimensional Space
  7. Building Blocks
  8. The Set of Three Tiles
  9. Proof of Theorem \ref{['thm_main']}
  10. Conclusion

Key Result

Theorem 1

Translational tiling of $4$-dimensional space with a set of $3$ polyhypercubes is undecidable.

Figures (22)

  • Figure 1: Translational tiling problem of $\mathbb{Z}^n$ with a set of $k$ tiles.
  • Figure 2: A set of $3$ Wang tiles
  • Figure 3: Level-1 layer diagram of $u$.
  • Figure 4: Level-1 layer diagram of $U$.
  • Figure 5: Level-1 layer diagram of $d$.
  • ...and 17 more figures

Theorems & Definitions (5)

  • Theorem 1: Undecidability with Three Tiles
  • Theorem 2: Undecidability with Four Tiles
  • Theorem 3: b66
  • proof : Proof of Theorem \ref{['thm_3d_new']}
  • proof : Proof of Theorem \ref{['thm_main']}