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Undecidability of Translational Tiling of the 4-dimensional Space with a Set of 4 Polyhypercubes

Chao Yang, Zhujun Zhang

TL;DR

This work addresses the decidability of translational tiling of $\mathbb{Z}^n$ by a fixed tile set, building on Greenfeld–Tao results that periodic tilings are not guaranteed and that undecidability can arise when the dimension is part of the input. It introduces a novel encoder–selector–linker framework and a lifting technique to simulate Wang tilings within higher-dimensional tilings. The authors prove undecidability for tiling $3$-D space with five polycubes and lift this construction to $4$-D to obtain undecidability for tiling $4$-D space with four polyhypercubes, thereby advancing toward fixed-dimensional undecidability with a small tile set. These results push the boundary between decidable and undecidable translational tilings and outline a concrete pathway via reductions from Wang tilings.

Abstract

Recently, Greenfeld and Tao disprove the conjecture that translational tilings of a single tile can always be periodic [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, 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 shows that translational tiling of the $3$-dimensional space with a set of $5$ polycubes is undecidable. By introducing a technique that lifts a set of polycubes and its tiling from $3$-dimensional space to $4$-dimensional space, we manage to show that translational tiling of the $4$-dimensional space with a set of $4$ tiles is undecidable. This is a step towards the attempt to settle the conjecture of the undecidability of translational tiling of the $n$-dimensional space with a monotile, for some fixed $n$.

Undecidability of Translational Tiling of the 4-dimensional Space with a Set of 4 Polyhypercubes

TL;DR

This work addresses the decidability of translational tiling of by a fixed tile set, building on Greenfeld–Tao results that periodic tilings are not guaranteed and that undecidability can arise when the dimension is part of the input. It introduces a novel encoder–selector–linker framework and a lifting technique to simulate Wang tilings within higher-dimensional tilings. The authors prove undecidability for tiling -D space with five polycubes and lift this construction to -D to obtain undecidability for tiling -D space with four polyhypercubes, thereby advancing toward fixed-dimensional undecidability with a small tile set. These results push the boundary between decidable and undecidable translational tilings and outline a concrete pathway via reductions from Wang tilings.

Abstract

Recently, Greenfeld and Tao disprove the conjecture that translational tilings of a single tile can always be periodic [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, 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 shows that translational tiling of the -dimensional space with a set of polycubes is undecidable. By introducing a technique that lifts a set of polycubes and its tiling from -dimensional space to -dimensional space, we manage to show that translational tiling of the -dimensional space with a set of tiles is undecidable. This is a step towards the attempt to settle the conjecture of the undecidability of translational tiling of the -dimensional space with a monotile, for some fixed .
Paper Structure (11 sections, 4 theorems, 20 figures)

This paper contains 11 sections, 4 theorems, 20 figures.

Table of Contents

  1. Introduction
  2. Undecidability of Tiling $3$-dimensional Space
  3. Building Blocks
  4. The Set of $5$ Polycubes
  5. Proof of Theorem \ref{['thm_3d_new']}
  6. Undecidability of Tiling $4$-dimensional Space
  7. Building Blocks in $4$-dimensional Space
  8. Lifting the Tiles and Tilings
  9. The Set of $4$ Polyhypercubes
  10. Proof of Theorem \ref{['thm_main']}
  11. Conclusion

Key Result

Theorem 1

Wang's domino problem is undecidable.

Figures (20)

  • Figure 1: A set of $3$ Wang tiles
  • Figure 2: Translational tiling problem of $\mathbb{Z}^n$ with a set of $k$ tiles.
  • Figure 3: Level-1 layer diagram of $c$, a functional cube with a dent.
  • Figure 4: Level-1 layer diagram of $c^-$, a functional cube with a dent.
  • Figure 5: Level-1 layer diagram of $C$, a functional cube with a bump.
  • ...and 15 more figures

Theorems & Definitions (7)

  • Conjecture 1: Periodic Tiling Conjecture, gs16lw96s74
  • Theorem 1: b66
  • Theorem 2: yz24b
  • Theorem 3: Undecidability with Four Tiles
  • Theorem 4: Undecidability with Five Tiles
  • proof : Proof of Theorem \ref{['thm_3d_new']}
  • proof : Proof of Theorem \ref{['thm_main']}