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This is a Quantitative Study study.

September 24, 2025Open Access

Undecidability of Translational Tiling with 2 Polycubes

Key Points

It is proven that determining if two polycubes can tile ℤ³ by translation is undecidable.
Using a new technique, the study simulates two disconnected polycubes with two connected ones, expanding its results to higher dimensions.
For dimensions n ≥ 3, disconnected tiles can be represented by the same number of connected tiles, highlighting significant mathematical implications.
The implications of this result may extend to new understandings in tiling theory and its applications in various mathematical fields.

Abstract

In this paper, we prove that it is undecidable whether a set of two polycubes can tile Z³ by translation. The proof involves a new technique that allows us to simulate two disconnected polycubes with two connected polycubes. By expanding this technique to higher dimensions, we also prove that a set of disconnected tiles in Zⁿ can be simulated by the same number of connected tiles in Zⁿ for n 3.

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Cite This Study

Y. Kim (Fri,) studied this question.

synapsesocial.com/papers/68d6e16f8b2b6861e4c40116 https://doi.org/https://doi.org/10.48550/arxiv.2508.11725

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